Generation, estimation, and protection of novelquantum states of spin systems

Thesis

For the award of the degree of

DOCTOR OF PHILOSOPHY

Supervised by: Submitted by:

Prof. Arvind & Harpreet SinghDr. Kavita Dorai

Indian Institute of Science Education & Research MohaliMohali - 140 306

India(September 2017)

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DeclarationThe work presented in this thesis has been carried out by me under the guidance

of Prof. Arvind and Dr. Kavita Dorai at the Indian Institute of Science Education andResearch Mohali.

This work has not been submitted in part or in full for a degree, diploma or a fel-lowship to any other University or Institute. Whenever contributions of others are in-volved, every effort has been made to indicate this clearly, with due acknowledgementof collaborative research and discussions. This thesis is a bonafide record of originalwork done by me and all sources listed within have been detailed in the bibliography.

Harpreet Singh

Place :Date :

In our capacity as supervisors of the candidate’s PhD thesis work, we certify that theabove statements by the candidate are true to the best of our knowledge.

Dr. Kavita Dorai Dr. ArvindAssociate Professor ProfessorDepartment of Physical Sciences Department of Physical SciencesIISER Mohali IISER Mohali

Place : Place :Date : Date :

AcknowledgementsI owe a huge gratitude to my PhD advisors Dr. Kavita Dorai and Prof. Arvind for

teaching the basics of NMR technique and Quantum Information Processing. I mustadmit that without their help and support this thesis would not have been possible.

I would like to thank my doctoral committee member Dr. Abhishek Chaudhurifor his useful advices. I am grateful to Prof. N Sathyamurthy, the Director of IISERMohali for providing all kind of help needed for my research work. I am thankful tothe faculty of the Department of Physical Sciences of IISER Mohali for providing theirexcellent guidance during my course work.

I am grateful to the NMR and QCQI group IISER Mohali for discussions and pre-sentations. I would like to acknowledge the support from my group members: SatnamSingh, Navdeep Gogna, Amandeep Singh, Rakesh Sharma, Jyotsana Ojha, JaskaranSingh, Chandan Sharma, Akash Sherawat, Akshay Gaikwad, Rajendera Singh Bhati,Akansha Gautam, Amit Devra, Gaurav Saxena, Dileep Singh and Aditya Mishra, Am-rita Kumari and Matsyendra Nath Shukla. I am specially grateful to Ritabrata Sen-gupta, Debmayla Das and Shruti Dogra for the useful discussions and providing allkind of support in my research work.

I owe my special thanks to Dr. Paramdip Singh Chandi for helping us troubleshoot-ing the errors in the lab machines. At the same time, I am thankful to Dr B. S. Joshi(Application, Bruker India) for his technical support and guidance.

I have benefited a lot from discussions with Gopal Verma, Anshul Choudhary,Vivek Kohar, Kanika Pasrija and Satyam Ravi during course work. I am grateful tomy hostel friends Preetinder Singh, Bhupesh Garg, Nayyar Aslam, George Thomas,Navnoor Kaur Saran and Junaid Khan because of whom I enjoyed my hostel life.

I would like to thank my sisters Hardeep Kaur, Harveen Kaur and cousin JasjeetSingh for their immense support and encouragement. I would also like to thank mybrother-in-law Amandeep Singh for his continuous encouragement and guidance. Ibow my head to my parents and grandparents for their generous invaluable blessings.

I am obliged to NMR research facility of IISER Mohali for providing me the plat-form for carrying out my research. During this thesis, various experiments were per-formed on 600 MHz Bruker NMR spectrometers using QXI, TXI and BBI probeheads.

I would like to acknowledge the financial support from Council of Scientific &Industrial Research (CSIR) during my PhD. I would like to thank IISER Mohali forproviding fund for participation in EUROMAR 2015, held in Prague Czech Republic.

Finally, thanks to my wonderful niece Divjot Kaur for keeping the enthusiasm andpositivity alive, which was required for my PhD completion.

Harpreet Singh

Abstract

This thesis deals with the generation, estimation and preservation of novel quantumstates of two and three qubits on an NMR quantum information processor. Using themaximum likelihood ansatz, a method has been developed for state estimation such thatthe reconstructed density matrix does not have negative eigenvalues and the errors arewithin the space of valid density operators. Due to interactions with the environment,unwanted changes occur in the system, leading to decoherence. Controlling decoher-ence is one of the biggest challenges to be overcome to build quantum computers. Todecouple the quantum system from its environment, several experimental strategieshave been used. These strategies are based on our knowledge of system-environmentinteraction and states that need to be preserved. Considering the first case, where thesystem state is known but there is no knowledge about its interaction with the envi-ronment. To tackle decoherence in this case, the super-Zeno scheme is used and itsefficacy to preserve quantum states is demonstrated. The next situation considered isthat where only the subspace to which the system state belongs is known. To addresssuch a situation, the nested Uhrig dynamical decoupling scheme has been used. Thelater part of the thesis deals with situations where the state of the system as well as itsinteraction with the environment is known. In such situations, since the noise modelis known, decoupling strategies can be explicitly designed to cancel this noise. Usingthese decoupling strategies, the lifetime of time-invariant discord of two-qubit Bell-diagonal states has been experimentally extended. The decay of three-qubit entangledstates namely the GHZ state, the W state and the WW state are studied, and the noisemodel is constructed for the spin system. The experimentally observed and theoreticalexpected entanglement decay rates of these states are compared. Then, the dynamicaldecoupling scheme is applied to these states and remarkable protection is observed inthe case of the GHZ state and the WW state.

The contents of the thesis have been divided into seven chapters whose brief ac-count is sketched below:

v

0. Abstract

Chapter 1

This chapter provides an introduction to the field of NMR quantum computation andquantum information as well as the motivation for the present thesis. In addition to thebasics of NMR and quantum computation, recent developments in the field of quantumcomputation and quantum information are discussed.

Chapter 2

This chapter describes the utility of the maximum likelihood (ML) estimation schemeto estimate quantum states on an NMR quantum information processor. Various sep-arable and entangled states of two and three qubits are experimentally prepared, andthe density matrices are reconstructed using both the ML estimation scheme as wellas standard quantum state tomography (QST). Further, an entanglement parameter isdefined to quantify multiqubit entanglement and entanglement is estimated using boththe QST and the ML estimation schemes.

Chapter 3

This chapter experimentally demonstrates the freezing of evolution of quantum statesin one- and two-dimensional subspaces of two qubits on an NMR quantum informa-tion processor. The state evolution is frozen and leakage of the state from its subspaceto an orthogonal subspace is successfully prevented using super-Zeno sequences. Thesuper-Zeno scheme comprises a set of radio frequency (rf) pulses, punctuated by pre-selected time intervals. The efficacy of the scheme is demonstrated by preservingdifferent types of states, including separable and maximally entangled states in one-and two-dimensional subspaces of a two-qubit system. The changes in the experi-mental density matrices are tracked by carrying out full state tomography at severaltime points. For the one-dimensional case, the fidelity measure is used and for thetwo-dimensional case, the leakage (fraction) into the orthogonal subspace is used as aqualitative indicator to estimate the resemblance of the density matrix at a later timeto the initially prepared density matrix. For the case of entangled states, an entangle-ment parameter is computed additionally to indicate the presence of entanglement inthe state at different times. The experiments demonstrate that the super-Zeno schemeis able to successfully confine state evolution to the one- or two-dimensional subspacebeing protected.

vi

Chapter 4

In this chapter, the efficacy of a three-layer nested Uhrig dynamical decoupling (NUDD)sequence to preserve arbitrary quantum states in a two-dimensional subspace of thefour-dimensional two-qubit Hilbert space is experimentally demonstrated on an NMRquantum information processor. The effect of the state preservation is studied firston four known states, including two product states and two maximally entangled Bellstates. Next, to evaluate the preservation capacity of the NUDD scheme, it is ap-plied to eight randomly generated states in the subspace. Although, the preservationof different states varies, the scheme on the average performs very well. The completetomographs of the states at different time points are used to compute fidelity. The statefidelities using NUDD protection are compared with those obtained without using anyprotection.

Chapter 5

The discovery of the intriguing phenomenon that certain kinds of quantum correlationsremain impervious to noise up to a specific point in time and then suddenly decay, hasgenerated immense recent interest. In this chapter, dynamical decoupling sequencesare exploited to prolong the persistence of time-invariant quantum discord in a systemof two NMR qubits decohering in independent dephasing environments. Noise chan-nels affecting the considered spin system of the molecule are characterized and eachspin of the spin system is mainly affected by the independent phase damping channel.Bell-diagonal quantum states are experimentally prepared on a two-qubit NMR pro-cessor, and robust dynamical decoupling schemes are applied for state preservation.It is demonstrated that these schemes are able to successfully extend the lifetime oftime-invariant quantum discord.

Chapter 6

This chapter demonstrates the experimental protection of different classes of tripartiteentangled states, namely the maximally entangled GHZ and W states and the WWstate, using dynamical decoupling. The states are created on a three-qubit NMR quan-tum information processor and allowed to evolve in the naturally noisy NMR envi-ronment. The tripartite entanglement is monitored at each time instant during stateevolution, using negativity as an entanglement measure. It is observed that the W stateis the most robust while the GHZ-type states are the most fragile against the natural de-coherence present in the NMR system. The WW state which is in the GHZ-class, yetstores entanglement in a manner akin to the W state, surprisingly turns out to be morerobust than the GHZ state. The experimental data are best modeled by considering the

vii

0. Abstract

main noise channel to be an uncorrelated phase damping channel acting independentlyon each qubit, along with a generalized amplitude damping channel. Using dynam-ical decoupling, a significant protection of entanglement for GHZ state is achieved.There is a marginal improvement in the state fidelity for the W state (which is alreadyrobust against natural system decoherence), while the WW state shows a significantimprovement in fidelity and protection against decoherence.

Chapter 7

This chapter provides some general remarks on the problems covered in the thesis.Possible future applications of the state protection techniques used in this thesis andthe new avenues of research they open up are described. The overall contribution ofthis thesis in the context of the study of decoherence and state preservation techniquesin quantum information processing, is summarized.

viii

List of Publications

1. Harpreet Singh, Arvind and Kavita Dorai. Experimental protection againstevolution of states in a subspace via a super-Zeno scheme on an NMR quantuminformation processor. Phys. Rev. A, 90, 052329 (2014).

2. Harpreet Singh, Arvind and Kavita Dorai. Constructing valid density matriceson an NMR quantum information processor via maximum likelihood estimation.Phys. Lett. A, 380 3051 (2016).

3. Harpreet Singh, Arvind and Kavita Dorai. Experimental protection of arbitrarystates in a two-qubit subspace by nested Uhrig dynamical decoupling. Phys.Rev. A, 95, 052337 (2017).

4. Harpreet Singh, Arvind and Kavita Dorai. Experimentally freezing quantumdiscord in a dephasing environment using dynamical decoupling. EPL, 118,50001 (2017).

5. Rakesh Sharma, Navdeep Gogna,Harpreet Singh, Kavita Dorai. Fast profil-ing of metabolite mixtures using chemometric analysis of a speeded-up 2D het-eronuclear correlation NMR experiment. RSC Adv., 7, 29860 (2017).

6. Harpreet Singh, Arvind and Kavita Dorai. Evolution of tripartite entangledstates in a decohering environment and their experimental protection using dy-namical decoupling. PhysRevA, 97, 022302 (2018).

7. Amit Devra, Prithviraj Prabhu,Harpreet Singh and Kavita Dorai. Efficient ex-perimental design of high-fidelity three-qubit quantum gates via genetic pro-gramming.Quantum Inf Process, 17, 67 (2018).

8. Harpreet Singh, Arvind and Kavita Dorai. Multiple-quantum relaxation inthree-spin homonuclear and heteronuclear systems: An integrated Redfield andLindblad master-equation approach. (Manuscript in preparation 2017).

Contents

Abstract v

List of Figures xv

List of Tables xxv

1 Introduction 11.1 Quantum computing and quantum information processing . . . . . . . 4

1.1.1 Quantum bit . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 N -qubit quantum register . . . . . . . . . . . . . . . . . . . 61.1.3 Density matrix representation . . . . . . . . . . . . . . . . . 61.1.4 Quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.5 Quantum measurement . . . . . . . . . . . . . . . . . . . . . 9

1.2 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . 101.3 NMR quantum computing . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 NMR qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.3 Quantum gate implementation in NMR . . . . . . . . . . . . 181.3.4 Numerical techniques for quantum gate optimization . . . . . 191.3.5 Measurement in NMR . . . . . . . . . . . . . . . . . . . . . 22

1.4 Evolution of quantum systems . . . . . . . . . . . . . . . . . . . . . 231.4.1 Closed quantum systems . . . . . . . . . . . . . . . . . . . . 241.4.2 Open systems . . . . . . . . . . . . . . . . . . . . . . . . . . 251.4.3 Quantum noise channels . . . . . . . . . . . . . . . . . . . . 27

1.4.3.1 Generalized amplitude damping channel . . . . . . 271.4.3.2 Phase damping channel . . . . . . . . . . . . . . . 281.4.3.3 Depolarizing channel . . . . . . . . . . . . . . . . 29

1.4.4 Nuclear spin relaxation . . . . . . . . . . . . . . . . . . . . . 301.4.5 Longitudinal relaxation . . . . . . . . . . . . . . . . . . . . . 30

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CONTENTS

1.4.6 Transverse relaxation . . . . . . . . . . . . . . . . . . . . . . 311.4.7 Bloch-Wangness-Redfield relaxation theory . . . . . . . . . . 31

1.5 Decoherence suppression . . . . . . . . . . . . . . . . . . . . . . . . 351.5.1 Hahn echo . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.5.2 CPMG DD sequence . . . . . . . . . . . . . . . . . . . . . . 371.5.3 Uhrig DD sequence . . . . . . . . . . . . . . . . . . . . . . . 381.5.4 Super-Zeno Scheme . . . . . . . . . . . . . . . . . . . . . . 391.5.5 Nested Uhrig dynamical decoupling . . . . . . . . . . . . . . 39

1.6 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 39

2 State tomography on an NMR quantum information processor via maxi-mum likelihood estimation 412.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.1 NMR quantum state tomography . . . . . . . . . . . . . . . . 432.1.2 Maximum likelihood estimation . . . . . . . . . . . . . . . . 47

2.2 Comparison of quantum state estimation via ML estimation and stan-dard QST schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.0.1 Fidelity measure and state estimation . . . . . . . . 512.2.1 Comparison of separable states estimation . . . . . . . . . . . 522.2.2 Comparison of entangled states estimation . . . . . . . . . . . 53

2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Experimental protection of quantum states via a super-Zeno scheme 593.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 The super-Zeno scheme . . . . . . . . . . . . . . . . . . . . . . . . . 613.3 Experimental implementations of super-Zeno scheme . . . . . . . . . 64

3.3.1 NMR system details . . . . . . . . . . . . . . . . . . . . . . 643.3.2 Super-Zeno scheme for state preservation . . . . . . . . . . . 66

3.3.2.1 Preservation of product states: . . . . . . . . . . . . 663.3.2.2 Preservation of entangled states . . . . . . . . . . . 693.3.2.3 Estimation of state fidelity . . . . . . . . . . . . . . 72

3.3.3 Super-Zeno for subspace preservation . . . . . . . . . . . . . 733.3.3.1 Preservation of product states in the subspace . . . 753.3.3.2 Preservation of an entangled state in the subspace . 793.3.3.3 Estimating leakage outside subspace . . . . . . . . 80

3.3.4 Preservation of entanglement . . . . . . . . . . . . . . . . . . 803.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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CONTENTS

4 Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences 834.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 The NUDD scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3 Experimental protection of two qubits using NUDD . . . . . . . . . . 91

4.3.1 Experimental implementation of the NUDD scheme . . . . . 914.3.2 NUDD protection of known states in the subspace . . . . . . 934.3.3 NUDD protection of unknown states in the subspace . . . . . 99

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 Experimentally preserving time-invariant discord using dynamical decou-pling 1055.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Measure for correlations of two qubits . . . . . . . . . . . . . . . . . 1065.3 Characterization of noise channels . . . . . . . . . . . . . . . . . . . 108

5.3.1 Bell-diagonal states under a dephasing channel . . . . . . . . 1105.4 Time invariant quantum correlations . . . . . . . . . . . . . . . . . . 1105.5 Experimental realization of time-invariant discord . . . . . . . . . . . 111

5.5.1 NMR System . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.5.2 Observing time-invariant discord . . . . . . . . . . . . . . . . 111

5.6 Protection of time-invariant discord using dynamical decoupling schemes1145.6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Dynamics of tripartite entanglement under decoherence and protectionusing dynamical decoupling 1236.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.2 Dynamics of tripartite entanglement . . . . . . . . . . . . . . . . . . 124

6.2.1 Tripartite entanglement under different noise channels . . . . 1246.2.2 NMR system . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2.3 Construction of tripartite entangled states . . . . . . . . . . . 1296.2.4 Decay of tripartite entanglement . . . . . . . . . . . . . . . . 132

6.3 Protecting three-qubit entanglement via dynamical decoupling . . . . 1406.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7 Summary and future outlook 147

Bibliography 149

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CONTENTS

xiv

List of Figures

1.1 Bloch sphere representation of a qubit. . . . . . . . . . . . . . . . . . 51.2 (a) NMR tube with sample oriented in a strong static magnetic fieldB0

along the z-axis and time-dependent magnetic field B1(t) along the x-axis. (b) The number of spins precessing around the direction parallelto the field are more than the number antiparallel to the field direction,which creates a bulk magnetization M0. . . . . . . . . . . . . . . . . 10

1.3 Populations of energy levels of a two-qubit system of a (a) thermalequilibrium state and (b) a pseudopure state. . . . . . . . . . . . . . . 15

1.4 Plot of rf pulse amplitude and phase with time, optimized with GRAPEfor CSWAP gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5 Rotation of Bulk magnetization using resonance. B1(t) applied in -x-direction for duration such that total rotation is 90 . . . . . . . . . . 22

1.6 (a) Representation of a closed system; the circle around the systemdepicts no interaction between the system and environment. (b) Rep-resentation of an open system; the dashed circle around the systemshows that the system and environment are interacting. . . . . . . . . 24

1.7 Model of closed quantum systems. . . . . . . . . . . . . . . . . . . . 251.8 Model of open quantum system consisting of two parts: the principal

system and its environment. . . . . . . . . . . . . . . . . . . . . . . . 261.9 Evolution of the bulk magnetization under Hahn echo sequence. (a)

Initially thermal equilibrium bulk magnetization in z direction is rep-resented by an arrow. (b) Bulk magnetization rotated to −y axis using90 pulse, (c) A delay of time t is given and arrows represent differentspins, (d) 180 is applied to spins due to which a spin precessing fastwill fall behind a spin precessing slowly, and (e) after time t, all thespins are in the same direction. . . . . . . . . . . . . . . . . . . . . . 36

1.10 Pulse sequence of CPMG DD sequence for a cycle of duration T andin one cycle eight pulses are applied; filled bars represent π rotationpulses and τ is the duration between two consecutive π pulses. . . . . 37

xv

LIST OF FIGURES

1.11 Pulse sequence for UDD sequence for a cycle time of T and the num-ber of π pulses in one cycle of sequence is eight. Filled bars representπ rotation pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) |00〉 state, with a computed fidelity of 0.9937 using stan-dard QST and a computed fidelity of 0.9992 using ML method forstate estimation. (b) 1√

2(|00〉 + |01〉) state, with a computed fidelity

of 0.9928 using standard QST and a computed fidelity of 0.9991 usingML method for state estimation. The rows and columns are labeled inthe computational basis ordered from |00〉 to |11〉. . . . . . . . . . . . 51

2.2 Real (left) and imaginary (right) parts of the experimental tomographsof the entangled state 1√

2(|01〉+ |10〉) reconstructed (a) using standard

QST and (b) using ML estimation. The fidelities computed using stan-dard QST and using ML method for state estimation are 0.9933 and0.9999 respectively. The rows and columns are labeled in the compu-tational basis ordered from |00〉 to |11〉. . . . . . . . . . . . . . . . . 52

2.3 Real (left) and imaginary (right) parts of the experimental tomographsof the three-qubit maximally entangled state |W 〉 = 1√

3(i|001〉+|010〉+

|100〉), reconstructed (a) using standard QST with a computed fidelityof 0.9833 and (b) using ML estimation with a computed fidelity of0.9968. The rows and columns are labeled in the computational basisordered from |000〉 to |111〉. . . . . . . . . . . . . . . . . . . . . . . 54

2.4 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) 1√

2(|00〉 + |11〉) state. Tomographs (b)-(e) depict the state

at T = 0.04, 0.08, 0.12, 0.16s, with the tomographs on the left and theright representing the state estimated using standard QST and usingML estimation, respectively. The rows and columns are labeled in thecomputational basis ordered from |00〉 to |11〉. . . . . . . . . . . . . 55

2.5 Plot of the entanglement parameter η with time, using standard QSTand ML protocols for state reconstruction, computed for the 1√

2(|00〉+

|11〉) state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.1 (a) Molecular structure of cytosine with the two qubits labeled as H1

and H2 and tabulated system parameters with chemical shifts νi andscalar coupling J12 (in Hz) and relaxation times T1 and T2 (in seconds)(b) NMR spectrum obtained after a π/2 readout pulse on the thermalequilibrium state. The resonance lines of each qubit are labeled by thecorresponding logical states of the other qubit and (c) NMR spectrumof the pseudopure |00〉 state. . . . . . . . . . . . . . . . . . . . . . . 65

xvi

LIST OF FIGURES

3.2 (a) Quantum circuit for preservation of the state |11〉 using the super-Zeno scheme. ∆i = xit, (i = 1...5) denote time intervals punctuatingthe unitary operation blocks. Each unitary operation block containsa controlled-phase gate (Z), with the first (top) qubit as the controland the second (bottom) qubit as the target. The entire scheme is re-peated N times before measurement (for our experiments N = 30).(b) Block-wise depiction of the corresponding NMR pulse sequence.A z-gradient is applied just before the super-Zeno pulses, to clean upundesired residual magnetization. The unfilled and black rectanglesrepresent hard 1800 and 900 pulses respectively, while the unfilled andgray-shaded conical shapes represent 1800 and 900 pulses (numericallyoptimized using GRAPE) respectively; τ12 is the evolution period un-der the J12 coupling. Pulses are labeled with their respective phasesand unless explicitly labeled, the phase of the pulses on the second(bottom) qubit are the same as those on the first (top) qubit. . . . . . 67

3.3 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) |11〉 state, with a computed fidelity of 0.99. (b)-(e) depictthe state at T = 0.61, 3.03, 5.46, 7.28 s, with the tomographs on theleft and the right representing the state without and after applying thesuper-Zeno preserving scheme, respectively. The rows and columnsare labeled in the computational basis ordered from |00〉 to |11〉. . . . 68

3.4 (a) Quantum circuit for preservation of the singlet state using the super-Zeno scheme. ∆i, (i = 1...5) denote time intervals punctuating theunitary operation blocks. The entire scheme is repeatedN times beforemeasurement (for our experiments N = 10). (b) NMR pulse sequencecorresponding to one unitary block of the circuit in (a). A z-gradientis applied just before the super-Zeno pulses, to clean up undesiredresidual magnetization. The unfilled rectangles represent hard 1800

pulses, the black filled rectangles representing hard 900 pulses, whilethe shaded shapes represent numerically optimized (using GRAPE)pulses and the gray-shaded shapes representing 900 pulses respectively;τ12 is the evolution period under the J12 coupling. Pulses are labeledwith their respective phases and unless explicitly labeled, the phase ofthe pulses on the second (bottom) qubit are the same as those on thefirst (top) qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xvii

LIST OF FIGURES

3.5 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) 1√

2(|01〉 − |10〉) (singlet) state, with a computed fidelity of

0.99. (b)-(e) depict the state at T = 0.85, 2.54, 4.24, 5.93 s, with thetomographs on the left and the right representing the state without andafter applying the super-Zeno preserving scheme, respectively. Therows and columns are labeled in the computational basis ordered from|00〉 to |11〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.6 Plot of fidelity versus time of (a) the |11〉 state and (b) the 1√2(|01〉 −

|10〉) (singlet) state, without any preserving scheme and after the super-Zeno preserving sequence. The fidelity of the state with the super-Zenopreservation remains close to 1. . . . . . . . . . . . . . . . . . . . . 72

3.7 Plot of signal intensity versus time of the |11〉 state and the 1√2(|01〉 −

|10〉) (singlet) state, after the super-Zeno preserving sequence. . . . . 73

3.8 (a) Quantum circuit for preservation of the 01, 10 subspace using thesuper-Zeno scheme. ∆i, (i = 1...5) denote time intervals punctuatingthe unitary operation blocks. The entire scheme is repeated N timesbefore measurement (for our experiments N = 30). (b) NMR pulsesequence corresponding to the circuit in (a). A z-gradient is appliedjust before the super-Zeno pulses, to clean up undesired residual mag-netization. The unfilled rectangles represent hard 1800 pulses; τ12 isthe evolution period under the J12 coupling. Pulses are labeled withtheir respective phases. . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.9 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) |10〉 state in the two-dimensional subspace 01, 10, with acomputed fidelity of 0.98. (b)-(e) depict the state at T = 1.15, 3.45, 5.75, 7.48s, with the tomographs on the left and the right representing the statewithout and after applying the super-Zeno preserving scheme, respec-tively. The rows and columns are labeled in the computational basisordered from |00〉 to |11〉. . . . . . . . . . . . . . . . . . . . . . . . 76

3.10 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) |01〉 state in the two-dimensional subspace 01, 10, with acomputed fidelity of 0.99. (b)-(e) depict the state at T = 1.15, 3.45, 5.75, 7.48s, with the tomographs on the left and the right representing the statewithout and after applying the super-Zeno preserving scheme, respec-tively. The rows and columns are labeled in the computational basisordered from |00〉 to |11〉. . . . . . . . . . . . . . . . . . . . . . . . 77

xviii

LIST OF FIGURES

3.11 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) 1√

2(|01〉 − |10〉) (singlet) state in the two-dimensional sub-

space 01, 10, with a computed fidelity of 0.98. (b)-(e) depict the stateat T = 1.15, 3.46, 5.77, 7.50 s, with the tomographs on the left and theright representing the state without and after applying the super-Zenopreserving scheme, respectively. The rows and columns are labeled inthe computational basis ordered from |00〉 to |11〉. . . . . . . . . . . 78

3.12 Plot of leakage fraction from the |01〉, |10〉 subspace to its orthog-onal subspace |00〉, |11〉 of (a) the |10〉 state and (b) the 1√

2(|01〉 −

|10〉) (singlet) state, without any preservation and after applying thesuper-Zeno sequence. The leakage to the orthogonal subspace is min-imal (remains close to zero) after applying the super-Zeno scheme. . 79

3.13 Plot of entanglement parameter η with time, with and without applyingthe super-Zeno sequence, computed for (a) the 1√

2(|01〉−|10〉) (singlet)

state, and (b) the same singlet state when embedded in the subspace|01〉, |10〉 being preserved. . . . . . . . . . . . . . . . . . . . . . . 81

4.1 (a) Circuit diagram for the three-layer NUDD sequence. The inner-most UDD layer consists of X0 control pulses, the middle layer com-prises X1 control pulses and the outermost layer consists of Xφ pulses.The entire NUDD sequence is repeatedM times; ∆i are time intervals.(b) NMR pulse sequence to implement the control pulses for X0 andX1 UDD sequences. The values of the rf pulse phases φ1 and φ2 areset to x and y for the X0 and to −x and −y for the X1 UDD sequence,respectively. (c) NMR pulse sequence to implement the control pulsesfor the Xφ UDD sequence. The filled rectangles denote π/2 pulseswhile the unfilled rectangles denote π pulses, respectively. The timeperiod τ12 is set to the value (2J12)−1, where J12 denotes the strengthof the scalar coupling between the two qubits. . . . . . . . . . . . . . 90

4.2 (a) Structure of isotopically enriched chloroform-13C molecule, withthe 1H spin labeling the first qubit and the 13C spin labeling the secondqubit. The system parameters are tabulated alongside with chemicalshifts νi and scalar coupling J12 (in Hz) and NMR spin-lattice andspin-spin relaxation times T1 and T2 (in seconds). (b) NMR spectrumobtained after a π/2 readout pulse on the thermal equilibrium state and(c) NMR spectrum of the pseudopure |00〉 state. The resonance linesof each qubit in the spectra are labeled by the corresponding logicalstates of the other qubit. . . . . . . . . . . . . . . . . . . . . . . . . 91

xix

LIST OF FIGURES

4.3 Plot of fidelity versus time for (a) the |01〉 state and (b) the |10〉) state,without any protection and after applying NUDD protection. The fi-delity of both the states remains close to 1 for upto long times, afterNUDD protection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) |01〉 state, with a computed fidelity of 0.98. (b)-(e) depict thestate at T = 1.02, 2.04, 3.06, 4.08 s, with the tomographs on the leftand the right representing the state without any protection and afterapplying NUDD protection, respectively. The rows and columns arelabeled in the computational basis ordered from |00〉 to |11〉. . . . . . 94

4.5 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) |10〉 state, with a computed fidelity of 0.97. (b)-(e) depict thestate at T = 1.02, 2.04, 3.06, 4.08 s, with the tomographs on the leftand the right representing the state without any protection and afterapplying NUDD protection, respectively. The rows and columns arelabeled in the computational basis ordered from |00〉 to |11〉. . . . . . 95

4.6 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) 1√

2(|01〉−|10〉) state, with a computed fidelity of 0.99. (b)-(e)

depict the state at T = 0.28, 0.55, 0.83, 1.10 s, with the tomographs onthe left and the right representing the state without any protection andafter applying NUDD protection, respectively. The rows and columnsare labeled in the computational basis ordered from |00〉 to |11〉. . . . 96

4.7 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) 1√

2(|01〉+|10〉) state, with a computed fidelity of 0.99. (b)-(e)

depict the state at T = 0.28, 0.55, 0.83, 1.10 s, with the tomographs onthe left and the right representing the state without any protection andafter applying NUDD protection, respectively. The rows and columnsare labeled in the computational basis ordered from |00〉 to |11〉. . . . 97

4.8 Plot of fidelity versus time for (a) the Bell singlet state and (b) the Belltriplet state, without any protection and after applying NUDD protec-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.9 NMR pulse sequence for the preparation of arbitrary states. The se-quence of pulses before the vertical dashed line achieve state initial-ization into the |00〉 state. The values of flip angles θ and φ of the rfpulses are the same as the θ and φ values describing a general statein the two-dimensional subspace P = |01〉, |10〉. Filled and un-filled rectangles represent π

2and π pulses respectively, while all other

rf pulses are labeled with their respective flip angles and phases; theinterval τ12 is set to (2J12)−1 where J12 is the scalar coupling. . . . . 99

xx

LIST OF FIGURES

4.10 Geometrical representation of eight randomly generated states on aBloch sphere belonging to the two-qubit subspace P = |01〉, |10〉.Each vector makes angles θ, φ with the z and x axes, respectively. Thestate labels RS-i (i = 1..8) are explained in the text. . . . . . . . . . 100

4.11 Bar plots of fidelity versus time of eight randomly generated states(labeled RS-i, i = 1..8), without any protection (cross-hatched bars)and after applying NUDD protection (red solid bars): (a) RS-1, (b)RS-2, (c) RS-3, (d) RS-4, (e) RS-5, (f) RS-6, (g) RS-7 and (h) RS-8.(i) Bar plot showing average fidelity of all eight randomly generatedstates, at each time point. . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1 (a) Quantum circuit for the initial pseudopure state preparation, fol-lowed by the block for BD state preparation. The next block depictsthe DD scheme used to preserve quantum discord. The entire DD se-quence is repeated N times before measurement. (b) NMR pulse se-quence corresponding to the quantum circuit. The rf pulse flip anglesare set to α = 46 and β = 59.81, while all other pulses are labeledwith their respective angles and phases. . . . . . . . . . . . . . . . . 112

5.2 Time evolution of total correlations (triangles), classical correlations(squares) and quantum discord (circles) of the BD state: (a) Simula-tion, (b) Experimental plot without applying any preservation. . . . . 114

5.3 Real (left) and imaginary (right) parts of the experimental tomographsof the (a) Bell Diagonal (BD) state, with a computed fidelity of 0.99.(b)-(e) depict the state at T = 0.06, 0.12, 0.17, 0.23s, with the tomo-graphs on the left and the right representing the simulated and exper-imental state, respectively. The rows and columns are labeled in thecomputational basis ordered from |00〉 to |11〉. . . . . . . . . . . . . 115

5.4 Time evolution of total correlations (triangles), classical correlations(squares) and quantum discord (circles) of the BD state: Experimentalplots using CPMG preserving sequences (a) CPMG with τ = 0.57 msand (b) CPMG with τ = 0.38 ms. . . . . . . . . . . . . . . . . . . . 116

xxi

LIST OF FIGURES

5.5 NMR pulse sequence corresponding to DD schemes (a) XY4(s), (b)XY8(s), (c) XY16(s), and (d) KDDxy , with time delays between pulsesdenoted by τ4 , τ8, τ , τk , respectively. All the pulses are of flip angleπ and are labeled with their respective phases. The pulses are appliedsimultaneously on both qubits. The superscript ‘2’ in the KDD xysequence denotes that one unit cycle of this sequence contains twoblocks of the ten-pulse block represented schematically, i.e., a totalof twenty pulses. The shorter duration proton pulses and the longerduration carbon pulses are centered on each other and the various timedelays (τ4, τ8 , τ , τk) in all the DD schemes are tailored to the gapbetween two consecutive carbon pulses. . . . . . . . . . . . . . . . . 117

5.6 Time evolution of total correlations (triangles), classical correlations(squares) and quantum discord (circles) of the BD state: Experimentalplots using (a) XY4(s) with τ4 = 0.58 ms and (b) XY8(s) with τ8 =0.29 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.7 Real (left) and imaginary (right) parts of the experimental tomographsof the (a)-(e) depict the BD state at N = 20, 40, 60, 80, 100, with thetomographs on the left and the right representing the BD state after ap-plying the XY4 and XY8 scheme, respectively. The rows and columnsare labeled in the computational basis ordered from |00〉 to |11〉. . . . 119

5.8 Time evolution of total correlations (triangles), classical correlations(squares) and quantum discord (circles) of the BD state: Experimentalplots using (a) XY16(s) with τ = 0.145 ms and (b) KDDxy with τk =0.116 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.9 Real (left) and imaginary (right) parts of the experimental tomographsin (a)-(e) depict the Bell Diagonal (BD) state atN = 20, 40, 60, 80, 100,with the tomographs on the left and the right representing the BD stateafter applying the XY16 and KDDxy preserving DD schemes, respec-tively. The rows and columns are labeled in the computational basisordered from |00〉 to |11〉. . . . . . . . . . . . . . . . . . . . . . . . 121

5.10 Plot of time evolution of entanglement without applying any preserva-tion (filled circle), CPMG with τ = 0.57 ms (empty circle), XY4(s)(filled rectangle), XY8(s) (empty rectangle), XY16(s) (filled triangle)and KDDxy (empty triangle),respectively. . . . . . . . . . . . . . . . 122

xxii

LIST OF FIGURES

6.1 Simulation of decay of tripartite entanglement parameter negativityN(3) of the GHZ state (blue squares), the W state (red circles) and theWW state (green triangles) under the action of (a) amplitude damp-ing (Pauli σx) channel, (b) bit-phase flip (Pauli σy) channel (c) phasedamping (Pauli σz) channel and (d) isotropic noise (depolarizing) chan-nel. The κ parameter denotes inverse of the decoherence time. . . . . 125

6.2 (a) Molecular structure of trifluoroiodoethylene molecule and tabu-lated system parameters with chemical shifts νi and scalar couplingsJij (in Hz), and spin-lattice relaxation times T1 and spin-spin relax-ation times T2 (in seconds). (b) NMR spectrum obtained after a π/2readout pulse on the thermal equilibrium state. and (c) NMR spectrumof the pseudopure |000〉 state. The resonance lines of each qubit arelabeled by the corresponding logical states of the other qubit. . . . . 127

6.3 NMR pulse sequence used to prepare pseudopure state ρ000 startingfrom thermal equilibrium.The pulses represented by black filled rect-angles are of angle π. The other rf flip angles are set to θ1 = 5π

12,

θ2 = π6

and δ = π4. The phase of each rf pulse is written below each

pulse bar. The evolution interval τij is set to a multiple of the scalarcoupling strength (Jij). . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4 (Quantum circuit showing the sequence of implementation of the single-qubit local rotation gates (labeled by R), two-qubit controlled-rotationgates (labeled by CR) and controlled-NOT gates required to constructthe (a) GHZ state (b) W state and (c) WW state. . . . . . . . . . . . . 129

6.5 The real (left) and imaginary (right) parts of the experimentally tomo-graphed (a) GHZ-type state, with a fidelity of 0.97. (b) W state, with afidelity of 0.96 and (c) WW state with a fidelity of 0.94. The rows andcolumns encode the computational basis in binary order from |000〉 to|111〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.6 Time dependence of the tripartite negativity N(3) for the three-qubitsystem initially experimentally prepared in the (a) GHZ state (squares)(b) W state (circles) and (c) WW state (triangles) (the superscript expdenotes “experimental data”). The fits are the calculated decay of neg-ativity N(3) of the GHZ state (solid line), the WW state (dashed line)and the W state (dotted-dashed line), under the action of the modeledNMR noise channel (the superscript cal denotes “calculated fit”). TheW state is most robust against the NMR noise channel, whereas theGHZ state is most fragile. . . . . . . . . . . . . . . . . . . . . . . . . 133

xxiii

LIST OF FIGURES

6.7 The real (left) and imaginary (right) parts of the experimentally to-mographed density matrix of the state at the time instances when thetripartite negativity N

(3)123 approaches zero for the (a) GHZ state at t =

0.55 s (b) W state at t = 0.90 s and (c) WW state at t = 0.67 s.The rows and columns encode the computational basis in binary order,from |000〉 to |111〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.8 NMR pulse sequence corresponding to (a) XY-16(s) and (b) KDDxy

DD schemes (the superscript 2 implies that the set of pulses inside thebracket is applied twice, to form one cycle of the DD scheme). Thepulses represented by black filled rectangles (in both schemes) are ofangle π, and are applied simultaneously on all three qubits (denotedby F i, i = 1, 2, 3). The angle below each pulse denotes the phase withwhich it is applied. Each DD cycle is repeated N times, with N largeto achieve good system-bath decoupling. . . . . . . . . . . . . . . . . 141

6.9 Plot of the tripartite negativity (N(3)123) with time, computed for the (a)

GHZ-type state, (b) W state and (c) WW state. The negativity wascomputed for each state without applying any protection and after ap-plying the XY-16(s) and KDDxy dynamical decoupling sequences.Notethat the time scale for part (a) is different from (b) and (c) . . . . . . . 143

xxiv

List of Tables

4.1 Results of applying NUDD protection on eight randomly generatedstates in the two-dimensional subspace. Each random state (RS) istagged with a number for convenience, and its corresponding (θ,φ)angles are given in the column alongside. The fourth column displaysthe time at which the state fidelity approaches ≈ 0.8 without NUDDprotection and the last column displays the time for which state fidelityapproaches ≈ 0.8 after applying NUDD protection. . . . . . . . . . . 102

LIST OF TABLES

xxvi

Chapter 1

Introduction

Quantum computing and quantum information is an area which has grown tremen-dously over the past two decades; it comprises the study and implementation of the in-formation processing tasks that can be efficiently performed using a quantum mechan-ical system. Quantum computers are able to accomplish computational tasks whichare not possible to carry out on classical computers. The encoding of n bits of classicalinformation requires at least n bits of classical resources. However, because of thequantum superposition principle, quantum mechanical systems can in principle havea better encoding efficiency than classical systems [1, 2]. In 1981, R. Feynman pro-posed the idea of a ‘quantum computer’ and showed that a classical computer wouldexperience an exponential slowdown while simulating a quantum phenomenon, whilea quantum computer would not [3]. In 1985, D. Deutsch, took Feynman’s ideas furtherand defined two models of quantum computation; he also devised the first quantumalgorithm. One of Deutsch’s ideas is that quantum computers could take advantageof the computational power present in many “parallel universes” and thus outperformconventional classical algorithms [4]. In 1994, P. Shor demonstrated two importantproblems; the problem of finding the prime factors of an integer, and the so-called‘discrete logarithm’ problem, both of which could be solved efficiently on a quantumcomputer [1, 5]. Shor’s results clearly indicate the power of quantum computers. Fur-ther in 1996, L. Grover showed that a search algorithm for an unsorted database on aquantum computer is quadratically faster then its classical counterpart [6]. The mostpopular model of a quantum computer is based on qubits which are two-level quan-tum systems, with a qubit being a basic unit of quantum information. In 2000, D. P.DiVincenzo proposed a list of requirements for the realization of an actual quantumcomputer [7]: a scalable physical system, ability to initialize the system to any quan-tum state, a universal set of quantum gates that can be implemented, qubit-specificmeasurement and sufficiently long coherence times (relative to the gate implementa-

1

1. Introduction

tion times).

Till date, no quantum hardware completely fulfills these criteria. Several quantumcomputing experiments have been performed using optical photons [8], optical cav-ity [9], ion traps[10], superconducting qubits [11] nitrogen-vacancy centers [12] andnuclear magnetic resonance (NMR) techniques [13]. In an optical photon quantumcomputer, the qubits are represented by the polarization of a photon. The initial stateis prepared by creating single photon states by attenuating light. Quantum gates areapplied using beam-splitters, phase shifters and nonlinear Kerr media. Measurementis done by detecting single photons using a photomultiplier tube [1]. In an opticalcavity, qubits are represented by the polarizations of a photon and initial state prepa-ration is similar to that of an optical photon quantum computer. Quantum gates areapplied using beam-splitters, phase shifters and a cavity QED system, comprised of aFabry-Perot cavity containing a few atoms, to which the field is coupled. In trappedion quantum computers, ions are allowed to be cooled down to the extent that theirvibrational state is sufficiently close to having zero photons and a qubit is realizedby the hyperfine state of an atom and lowest level vibrational modes of the trappedatoms [14]. Quantum gates here are constructed by applying laser pulses. Measure-ment is done by measuring populations of hyperfine states [15]. In superconductingquantum computers, qubits are represented by the phase, charge and flux qubits. In thecharge qubit, different energy levels correspond to an integer number of Cooper pairson a superconducting island [16]. In the flux qubit, the energy levels correspond todifferent integer numbers of magnetic flux quanta trapped in a superconducting ring.In the case of a phase qubit, the energy levels correspond to different quantum chargeoscillation amplitudes across a Josephson junction, where the charge and the phase areanalogous to momentum and position correspondingly of a quantum harmonic oscil-lator [17]. Quantum gates are implemented using microwave pulses. The nitrogen-vacancy center (NV center) is a point defect in a diamond which offers access to anisolated quantum system that can be controlled at room temperature. The 14N NV−

center has an electronic spin component of S = 1 and a nuclear spin component ofI = 1, which gives spin eigenvalue level count of nine. Using these energy levelswe can realize qubits. Resonant microwave pulses allow full quantum control of thestate of the center. Measurement can be done using optical and electrical detectionmethods. The NV center is of course affected by the absolute temperature and tem-perature changes, it has useful properties at room temperature, which make it suitablefor a range of applications such as quantum sensors at room temperature, includingquantum computing [18, 19, 20].

In May 2016, IBM Corporation has placed a five-qubit quantum computer on theIBM Cloud to run algorithms and experiments, and explore tutorials and simulationsaround what might be possible with quantum computing. It is a universal five-qubit

2

quantum computer based on superconducting transmon qubits. After 18 months, IBMhas also brought online a five- and sixteen-qubit system for public access through theIBM Q experience and is currently working further for upgradation of qubits. Recently,Intel has also announced a 49-qubit quantum chip. So quantum computer is no morea theoretical concept but now a physical computational machine and in the near futurewill be ready to solve real problems [21].

This thesis uses NMR as a tool for performing quantum information processingtasks. NMR quantum computing has provided a good testbed for implementing var-ious quantum information processing protocols. In NMR, the chemical shifts of dif-ferent spins are used to address the spins individually in frequency space and externalradio frequency pulses are used for quantum control [22, 23]. For quantum informa-tion processing we require pure quantum states. However, an NMR spin system atroom temperature is far from pure, since the separation between the spin energy levelsis ~ω which much less than kBT . Therefore the initial state of an ensemble of nuclearspins is nearly random. However, for performing computational tasks we can initial-ize the system into a pseudopure state [24] which mimics a pure state. Using radiofrequency pulses and the couplings between the spins any unitary operator can be im-plemented. Further, the compensations of errors due to pulse imperfections and offseterror can be performed via numerically optimized pulses using GRAPE and geneticalgorithms [25, 26, 27, 28] which make the NMR technique an excellent test bed forthe implementation of quantum algorithms [29, 30, 31, 32, 33, 34], quantum simula-tions [35, 36, 37, 38, 39, 40], the study of decoherence [41, 42, 43] and many otherquantum information processing applications [44, 45, 46, 47, 48, 49, 50, 51, 52, 53].

In this thesis, we first tackle the problem of negative eigenvalues occurring duringthe reconstruction of density matrix from the experimental data. We experimentallyprepared quantum states relevant for quantum information processing and reconstructvalid state density matrices on an NMR quantum information processor of two andthree qubits. In NMR quantum information processing [1, 22], information is encodedin the quantum state of an ensemble of nuclei. Theoretically, reconstruction of thestate density matrix is possible if we have infinite copies of the spin system. However,only a finite but large number of copies of the spin system are available. Furthermore,due to experimental errors such as detection pulse errors and temperature fluctuations,copies of the spin system are slightly different [54]. If not properly handled, it can leadto a situation where the standard state tomography may give rise to an unphysical state.To tackle this problem, we use the maximum likelihood method [55, 56, 57, 58] whichalways gives a valid state density matrix close to the experimental data and resolvesthis issue of unphysical states [59]. In the rest of the thesis, we focus on the differentstrategies to cancel out system-environment interactions. First we deal with a situationwhere we are aware of the system state but have no knowledge about its interaction

3

1. Introduction

with the environment. It is then required to consider all the possible interactions bywhich system in a given state can interact with the environment. We use the super-Zeno scheme for state protection [60, 61, 62, 63], In this scheme, we construct aninverting pulse which has information about the state and use a train of these invertingpulses punctuated by unequal intervals of time to protect the system state. Then weconsider a situation where only the subspace is known to which system state belongsinstead of the exact state and its interaction with the environment, and to resolve thisproblem we use nested Uhrig dynamical decoupling (NUDD) schemes [64, 65, 66].The NUDD scheme consists of nesting of protection layers to cancel all the possibleinteractions that the state can have with the environment. We next move on to situa-tions where we have knowledge of the state of the system as well as its interaction withthe environment. We study the evolution of the state of the system in the presence ofintrinsic NMR noise and then fit its decay to a noise model to characterize the noise.In the two- and three-qubit systems studied, each qubit of the system is modeled as be-ing affected by an independent phase and amplitude damping noise channel [67], withthe noise being dominated by the phase damping channel. We use the Knill dynami-cal decoupling (KDDxy ) scheme and XY16 [68, 69] dynamical decoupling scheme totackle the dephasing noise. These pulse sequences are robust against pulse angle errorsand offsets errors. We apply KDDxy and XY16 sequences on experimentally preparedtwo-qubit Bell-diagonal states and see the effect on the lifetime of time-invariant dis-cord [70, 71]. We also apply these sequences on experimentally prepared three-qubitGHZ, W and WW states and observe the decay of entanglement and its subsequentsuppressing using dynamical decoupling.

1.1 Quantum computing and quantum information pro-cessing

Although computational algorithms are conceived mathematically, a computer whichexecutes these algorithms has to be a physical device. The most common model ofquantum computation is a generalization of the classical circuit model known as quan-tum circuit model. A quantum circuit is an instruction for carrying out the preparationof an input state, applying a set of quantum gates which cause a unitary evolution andmeasuring the output state. The input state is prepared on a quantum register, whichis the quantum analog of the classical processor register. A classical register of sizeof n comprises of n flip flops which can have 2n possible classical states. A quantumregister of size n comprises of n two-level quantum systems which are interacting witheach other and due to superposition can have infinite possible states.

4

1.1 Quantum computing and quantum information processing

1.1.1 Quantum bitThe basic unit of classical information is a bit. Classical digital computers processinformation in a discrete form. It operates on data that are expressed in binary code i.e.0 and 1. A bit can have two states either 0 or 1 and therefore it can be easily physicallyrealized on a two-state device. Quantum computing and quantum information are builtupon an analogous concept, the qubit i.e. quantum bit [1]. The two possible logicalstates for a qubit can be |0〉 and |1〉 states. However, the most general qubit state isgiven by:

|ψ〉 = α|0〉+ β|1〉 (1.1)

The state of a qubit is a vector in a two-dimensional complex vector space; |0〉 and|1〉 form the orthogonal basis for this vector space. The complex numbers α and β aresuch that |α|2 + |β|2 = 1. We cannot determine the values of α and β by measurementson a single qubit.

We can rewrite Eq 1.1 as

|ψ〉 = eiγ(cosθ

2|0〉+ eiφsin

θ

2|1〉) (1.2)

|1〉

|0〉b

b

b|ψ〉

x

y

z

θ

φ

Figure 1.1: Bloch sphere representation of a qubit.

with new real variables θ, φ, and γ. Global phase eiγ can be ignored because it has noobservable effects and we can write

|ψ〉 = cosθ

2|0〉+ eiφsin

θ

2|1〉 (1.3)

5

1. Introduction

where θ and φ define a point on the unit three-dimensional sphere, known as the Blochsphere, as shown in Fig.[1.1]. Each of the infinite points on the surface of the spherecorresponds to a state of the qubit.

1.1.2 N -qubit quantum register

A quantum register of size n comprises of n qubits which are interacting with eachother. The most general state of such a register is a superposition of 2n basis elementswhich is given by,

|φ〉 =∑j

αj|φ1j〉 ⊗ |φ2

j〉 ⊗ · · · ⊗ |φnj 〉 (1.4)

where |φij〉 refers to ith qubit in jth term of the superposition, |φij〉 ∈ |0〉, |1〉 and αjare the complex coefficients such that

∑j |αj|2 = 1. If a state |ψ〉 can be expressed as

|ψ1〉 ⊗ |ψ2〉 ⊗ · · · ⊗ |ψn〉 where |ψi〉 = αi|0〉+ βi|1〉, then the state is called separableand if not then it is entangled. Entanglement is an intrinsically quantum mechanicalphenomenon and it plays a crucial role in various QIP protocols. Experimental re-alization of quantum registers is one of the biggest challenges in building a quantumcomputer. Up to now only a few qubit quantum registers have been physically realized.For instance in linear optics ten qubits [72], in trapped ion fourteen qubits [15] and inNMR twelve qubits have been realized [73].

1.1.3 Density matrix representation

The state of the system can not be reconstructed if we have a single copy of a qubit. Ona measurement of a qubit with state |ψ〉 = α|0〉+β|1〉 possible outcomes will be 0 or 1with probability |α|2 and |β|2, respectively. To compute probabilities |α|2 and |β|2, weneed either a large number of measurements on a qubit with repeated state preparationas is done in single-photon quantum computing or a simultaneous measurement ofa large number of copies of the qubit as done in NMR quantum computing. In anensemble it may be possible that all the spins are in same state |ψ〉 and this type ofensemble is called pure ensemble. It may be possible that with p1 probability spins arein |ψ1〉, with p2 probability spins are in |ψ2〉 and so on, this type of ensemble is called amixed ensemble. The density matrix formulation is very useful in describing the stateof an ensemble quantum system such as an ensemble of spins in NMR [74].

For an ensemble with the pi probability to be in |ψi〉 state, the density operator isgiven as

ρ =∑i

pi|ψi〉〈ψi| (1.5)

6

1.1 Quantum computing and quantum information processing

where∑

i pi = 1. If all the members of an ensemble are in the same state |ψ〉 or for apure ensemble, the density operator is given as

ρpure = |ψ〉〈ψ| (1.6)

A density operator ρ has to satisfy three important properties: ρ is Hermitian, i.e.,ρ = ρ†, all the eigenvalues of the ρ is positive, and Tr[ρ] = 1. For a pure stateensemble Tr[ρ2] = 1 and for a mixed state ensemble Tr[ρ2] < 1. The most generalstate of a single qubit can be written as,

ρ =I + ~r.~σ

2(1.7)

where I is a identity matrix, ~r is a three-dimensional Bloch vector with |~r| ≤ 1, ~σ =σxx+ σyy + σz z and σis are the Pauli matrices. All the pure states can be representedas points on the surface of the Bloch sphere and mixed states are represented by pointsinside the Bloch sphere.

1.1.4 Quantum gatesThe building block of a digital circuit of a classical computer are the logic gates fore.g. NOT, OR, NOR and NAND. The analogous building blocks of a quantum circuitare quantum gates. Quantum gates being unitary, are reversible, as opposed to theclassical logic gates which may be irreversible and hence dissipative. The action ofquantum gates can be realized by a unitary operator U (UU† = I). It has been shownthat a set of gates that consists of all one-qubit quantum gates [U(2)] and the two-qubitexclusive-OR gate is universal in the sense that all unitary operations can be expressedas compositions of such gates [75]. One such set of universal quantum gates is theHadamard gate (H), a phase rotation gate R(cos−1(3

5)) and a two-qubit controlled-

NOT gate. Once a basis is chosen, quantum gates are represented as matrices. Thefollowing are some of the important quantum gates:

Hadamard gateThe Hadamard gate is a single-qubit gate and it maps the basis state |0〉 to |+〉 = |0〉+|1〉√

2

and |1〉 to |−〉 = |0〉−|1〉√2

. It creates a superposition, which means that a measurementwill have equal probability to become either 1 or 0. The matrix representation ofHadamard gate is:

H =1√2

(1 11 −1

)(1.8)

Pauli-X gate (NOT gate)The Pauli-X gate maps the basis state |0〉 to |1〉 and |1〉 to |0〉. The matrix representation

7

1. Introduction

of Pauli-X gate is:

X =

(0 11 0

)(1.9)

Pauli-Y gateThe Pauli-Y gate maps the basis state |0〉 to i|1〉 and |1〉 to −i|0〉. The matrix repre-sentation of Pauli-Y gate is:

Y =

(0 −ii 0

)(1.10)

Pauli-Z gateThe Pauli-Z gate leaves the basis state |0〉 unchanged and maps |1〉 to−|1〉. The matrixrepresentation of Pauli-Z gate is:

Z =

(1 00 −1

)(1.11)

Square root of NOT gate (√

NOT)The√

NOT gate maps the basis state |0〉 to 12

((1 + i)|0〉+ (1− i)|1〉) and |1〉 to12

((1− i)|0〉+ (1 + i)|1〉). The matrix representation of square root of NOT gate is:

√NOT =

1

2

(1 + i 1− i1− i 1 + i

)(1.12)

Phase shift gateThe phase shift gate leaves the basis state |0〉 unchanged and maps |1〉 to eiφ|1〉. Thematrix representation of the phase shift gate is:

Rφ =

(1 00 eiφ

)(1.13)

SWAP gateThe SWAP gate is a two-qubit gate which leaves the basis states |00〉 and |11〉 un-changed. It maps |01〉 to |10〉 and |10〉 to |01〉. The matrix representation of the SWAPgate is:

SWAP =

1 0 0 00 0 1 00 1 0 00 0 0 1

(1.14)

Controlled NOT gateThe controlled NOT (CNOT) gate is two-qubit gate which leave the basis state |00〉

8

1.1 Quantum computing and quantum information processing

and |01〉 unchanged. It maps |10〉 to |11〉 and |11〉 to |10〉. The matrix representationof CNOT gate is:

CNOT =

1 0 0 00 1 0 00 0 0 10 0 1 0

(1.15)

1.1.5 Quantum measurementThe standard measurement schemes in quantum information and quantum computationuse projective measurements which is described below. Later we will take up the issueof ensemble measurements on an NMR quantum information processor, which arenon-projective in nature.

Consider a quantum system in a pure state specified by a vector |ψ〉 in an n-dimensional Hilbert space. Let us suppose that one performs a projective measure-ment of an observable M on it. In the formalism of quantum mechanics, associatedwith the observable M is a Hermitian operator M , where |m1〉, |m2〉, . . . , |mn〉 denotethe eigenvectors of the operator M with m1, m2,. . . , mn as the respective eigenvalues.If the eigenvalue spectrum of the observable M is nondegenerate then

|ψ〉 = c1|m1〉+ c2|m2〉+ · · ·+ cn|mn〉 with∑i

|ci|2 = 1 (1.16)

where c1, c2, . . . , cn are complex numbers. Upon a projective measurement of theobservable M on such a system, an outcome mi is obtained with a probability |ci|2 andthe state of the system collapses to the corresponding eigenvector |mi〉.

A projective measurement is described by a complete set of projectors Πn whereΠn = |mn〉〈mn| with

∑n Π†nΠn = 1. If the state of the quantum system is |ψ〉

immediately before the measurement then the probability that m occurs is given by

p(n) = 〈ψ|Π†nΠn|ψ〉, (1.17)

and the state of the system after measurement is

Πn|ψ〉√〈ψ|Π†nΠn|ψ〉

. (1.18)

For instance, consider the measurement of a qubit with state |φ〉 = α|0〉 + β|1〉in the computational basis. The measurement is defined by the two measurement op-erators Π0 = |0〉〈0| and Π1 = |1〉〈1|. The measurement operators are Hermitian i.e.Π†1 = Π1 and Π†2 = Π2. The probability of obtaining the measurement outcome 0 is

p(0) = 〈φ|Π†0Π0|φ〉 = 〈φ|0〉〈0|0〉〈0|φ〉 = |α|2 (1.19)

9

1. Introduction

and the qubit state will collapse to |0〉. Similarly, the probability of obtaining themeasurement outcome 1 is p(1)=|β|2 and the qubit state will collapse to |1〉.

1.2 Nuclear Magnetic ResonanceNuclear magnetic resonance (NMR) describes a phenomenon wherein, an ensembleof nuclear spins precessing in a static magnetic field, absorb and emit radiation in theradiofrequency range in resonance with their Larmor frequencies [22]. In the quantummechanical formalism, the spin magnetization is a vector operator represented by ~Iwhere I is a dimensionless operator representing the total angular momentum of thenuclear spin. Atomic nuclei with non-zero spin also possess a magnetic dipole momentµ which is given as

µ = γn~I, (1.20)

where γn is called the gyromagnetic ratio of the nucleus, which is a fundamental prop-erty of the nucleus.

①

②

③

①

③

②

(a) (b)

Figure 1.2: (a) NMR tube with sample oriented in a strong static magnetic field B0 alongthe z-axis and time-dependent magnetic field B1(t) along the x-axis. (b) The numberof spins precessing around the direction parallel to the field are more than the numberantiparallel to the field direction, which creates a bulk magnetization M0.

10

1.2 Nuclear Magnetic Resonance

A nuclear spin with I 6= 0 when placed in a magnetic field of strength B0 appliedalong the z-axis precesses as shown in Fig. 1.2(a). The Hamiltonian of interactionbetween the spin and the magnetic field is given by,

H = −µ.B0z = −γn~B0Iz = −~ωnIz (1.21)

The spins precess about the z-axis with a characteristic frequency called Larmor fre-quency ωn = −γnB0 (in rad s−1) as shown in Fig. 1.2(b). The magnetic field B0 isapplied along the z-direction and all the quantum operators act in the subspace spannedby the magnetic quantum number |m〉 where m = −I,−I + 1, . . . , I − 1, I . Underthe action of the Hamiltonian H , the expectation values of the angular momentum op-erators in the plane perpendicular to the z-direction i.e. 〈Ix〉 and 〈Iy〉 show oscillatorybehavior with time, with a frequency ωn, whereas 〈Iz〉 is stationary. The eigenvaluesof the Hamiltonian H are given by:

Em = −m~ωn (1.22)

For a nucleus with spin I , there are (2I+1) energy levels equally spaced by the amount~ωn.

For an ensemble of identical nuclei which are not perfectly isolated from the envi-ronment and surrounded by the lattice is at a temperature T . The interactions of nucleiwith the lattice lead to a thermal equilibrium state and the population of each energylevel in this state is given by the Boltzmann distribution. For a two-level system I = 1

2,

with the population n− and n+ of the m = −12

and m = 12

levels, respectively

n−n+

= e−~ωn/kBT (1.23)

where kB is the Boltzmann constant and T is the absolute temperature of the ensemble.The Boltzmann factor e−~ωn/kBT for protons (1H) in a magnetic field of 14.1 Tesla atroom temperature is very close to unity. The fractional difference of populations isabout 1 part in 105. This slight difference in the populations of m = −1

2and m = 1

2

levels cause the net magnetization along the z-direction. For n spin-1/2 nuclei thethermal equilibrium magnetization is given by:

M0 =µ0γ

2n~2B0

4kBT(1.24)

Since the Larmor frequency depends on the gyromagnetic ratio γn, each nucleus hasits own characteristic Larmor frequency. Nuclear spins in a molecule are surroundedby the electronic environment, which leads to shielding of the magnetic field, the socalled “chemical shift”, with the effective magnetic field being given by

Beff = B0(1− σ0) (1.25)

11

1. Introduction

where σ0 is the isotropic chemical shift tensor.There are several terms in the nuclear spin Hamiltonian which encompass different

spin-spin interactions such as the scalar coupling term HJ , the dipolar coupling termHDD, and the quadrupolar coupling termHQ. The scalar coupling interactionHJ arisefrom the hyperfine interactions between the nuclei and local electrons. A pair of nucleiexhibit dipole-dipole interaction HDD by inducing local magnetic fields at the site ofeach other through space. In an isotropic liquid at room temperature, molecules tumblevery fast, thus averaging the intramolecular dipolar coupling to zero. The quadrupolarcoupling HQ is exhibited by nuclei with spin > 1/2 which possess an asymmetriccharge distribution [76].Radio frequency field interaction and the resonance phenomenon:- The Larmorfrequencies of the nuclear spins in a static magnetic field of a few Tesla are of theorder of MHz. The transition between the different spin states can be induced by aradio frequency (rf) oscillating magnetic field [2].

~Brf = 2B1cos(ωrf t+ φ)x, (1.26)

where ωrf is the frequency of the magnetic field and φ is the phase.

Hrf = −µ. ~Brf = −γn~Ix (2B1cos(ωrf t+ φ)) (1.27)

We can rewrite ~Brf as a superposition of two fields rotating in opposite directions.

~Brf = B1(cos(ωrf t+φ)x+ sin(ωrf t+φ)y) +B1(cos(ωrf t+φ)x− sin(ωrf t+φ)y),(1.28)

For the simplicity, we assume φ = 0 and analyze Eq. 1.28 in a coordinate system thatrotates around the static magnetic field at the frequency ωrf . In this rotating frame

~Brotrf = B1x+B1(cos(2ωrf t)x− sin(2ωrf t)y) (1.29)

We can observe that one of the two components is now static and the other is rotatingat twice the rf field frequency (which can be neglected) [77]. We can transform Hrf

into rotating frame using the unitary operator

U(t) = eiωntIz/~, (1.30)

Hrotrf = U−1HrfU + i~U−1U = −~(ωn − ωrf )Iz − ~ω1Ix

where ω1 = γnB1. If the phase φ 6= 0 then

Hrotrf = −~(ωn − ωrf )Iz − ~ω1Ixcosφ+ Iysinφ. (1.31)

The evolution of the quantum ensemble under the effective field in the rotating frameis described by

ρrot(t) = e−iHrotrf tρrot(0)eiH

rotrf t, (1.32)

where ρrot(0) is density matrix of state at time t.

12

1.3 NMR quantum computing

1.3 NMR quantum computingIn 1997, D. G. Cory and I. L. Chuang independently proposed a NMR quantum com-puter that can be programmed much like a quantum computer [78, 79]. Their compu-tational model uses an ensemble quantum computer wherein the results of a measure-ment are the expectation values of the observables. This computational model can berealized by NMR spectroscopy on macroscopic ensembles of nuclear spins. Severalquantum algorithms have been implemented on an NMR quantum computer such asthe Grover search algorithm [30], realization of Shor algorithm [80], implementationof the Deutsch-Jozsa algorithm using noncommuting selective pulses [31] and manymore till date. A qubit in an NMR quantum computer is realized by a spin-1/2 nucleus.The NMR spectrometer consists of a superconducting magnet which applies a highmagnetic field in the z-direction and rf coils for exciting the spins and receiving theNMR signal from the relaxing spin ensemble. When the sample is placed in the mag-netic field, the spins interact with the magnetic field, and energy levels split dependingupon the size of the spin system. At room temperature, these energy levels are popu-lated according to the Boltzmann distribution and thus the system is in a mixed stateat thermal equilibrium. This poses a difficult challenge for quantum computing, whichrequires pure states as initial quantum states. This difficulty is circumvented in NMRquantum computing by creating a “pseudopure” state as an initial state, which mimicsa pure state. Using the rf pulses and interaction between the spins, quantum gates areimplemented and as a result of the computation the NMR signal was recorded which isan average magnetization the in x and y directions. This signal is directly proportionalto the expectation values of some elements of the basis set of the qubits. With theapplication of rf pulses rotating individual spins, the expectation of all the elements inthe basis set can be calculated. From these expectation values, we can reconstruct thedensity matrix. Further, recent developments in NMR in the area of control of spin dy-namics via rf pulses makes it possible to implement quantum gates for NMR quantumcomputing with high fidelities. A nuclear spin is well separated from its environmentdue to which it exhibits long coherence times. Even with all these merits, one majorlimitation of liquid state NMR quantum computers is scalability. In the following sec-tions, state initialization, implementation of quantum gates and measurement in NMRquantum computing are discussed.

1.3.1 NMR qubitsConsider an ensemble of N spin-1/2 nuclei tumbling in a liquid and placed in a mag-netic field B0. The Hamiltonian H of this system is given as

H = −ω0Iz (1.33)

13

1. Introduction

where Iz = σz/2. The eigenstate and eigenvalues ofH are |0〉, |1〉 and ω0/2,−ω0/2respectively. The energy difference between the two levels is given by ∆E = ~ω0

Hence such a two-level system acts as a single NMR qubit. For a system of n interact-ing spins-1/2 in a magnetic field the Hamiltonian is given by:

H0 =n∑i=1

ωiIiz + 2π

n∑i<j

JijIi.Ij (1.34)

where Jij is the scalar coupling between the spins and ωi is the Larmor frequency. If|ωi − ωj| >> 2π|jij| then the NMR qubits are weakly coupled and the Hamiltonianfor such a system is

H0 =n∑i=1

ωiIiz + 2π

n∑i<j

JijIiz.I

jz (1.35)

1.3.2 InitializationAny QIP task begins by initializing the system into a pure state. In NMR QIP, an N -qubit ensemble of spins at room temperature has a population distribution of energylevels given by the Boltzmann distribution [1]. All the energy levels are almost equallypopulated and the initial state is mixed. Under the high temperature approximation theinitial state of the system is given by:

ρeq ≈1

2N(I + ε∆ρeq) (1.36)

where I is an identity matrix of 2N × 2N, ε(≈ 10−5) is a purity factor and ∆ρeq is adeviation density matrix. The problem of pure states in NMR can be overcome bypreparing a pseudopure state which is isomorphic to a pure state [78]. An ensembleof a pure state is given by ρpure = |ψ〉〈ψ| and the corresponding pseudopure state isgiven by

ρpps =1− ε2N

I + ε|ψ〉〈ψ| (1.37)

A pseudopure state in NMR can be prepared by several methods such as spatialaveraging, temporal averaging and logical labelling; all based on the idea of preparing

2N − 1 energy levels with equal population and with one energy level being morepopulated than the other energy levels as shown for two qubits in Fig.1.3.

To implement most quntum algorithms we need to create an entangled state. Thepower of quantum computers largely depends on entanglement. There has been alongstanding debate about the existence of entanglement in spin ensembles at hightemperature as encountered in NMR experiments. There are two ways to look at the

14

1.3 NMR quantum computing

|11〉

|10〉|01〉

|00〉

|10〉|01〉

|11〉

|00〉

(a) (b)

Figure 1.3: Populations of energy levels of a two-qubit system of a (a) thermal equilibriumstate and (b) a pseudopure state.

situation. Entangled states in such ensembles are obtained via unitary transformationson pseudopure states. If we consider the entire spin ensemble, given that the numberof spins that are involved in the pseudopure state is very small compared to the totalnumber of spins, it has been shown that the overall ensemble is not entangled [81, 82].However, one can take a different point of view and only consider the subensembleof spins that have been prepared in the pseudopure state, and as far as these spins areconcerned entanglement genuinely exists [13, 83]. The states that we have createdin this thesis for our experiments are entangled in this sense, and hence may not beconsidered as entangled if one works with the entire ensemble. Therefore, one has tobe aware and cautious about this aspect while dealing with these states. These statesare sometimes referred to as being pseudoentangled.Temporal averaging technique is based on the fact that quantum operations are linearand the observables measured in NMR are traceless. Experimentally, the temporal av-eraging scheme relies on adding the computational results of multiple experiments,where each experiment starts off with a different state preparation pulse sequencewhich permutes the populations [22]. For a two-spin system this technique beginswith the density matrix

ρ1 =

p1 0 0 00 p2 0 00 0 p3 00 0 0 p4

where p1, p2, p3 and p4 are populations of the normalized density operator ρ1, with∑4

i=1 pi = 1. U1 and U2 are operators constructed from controlled-NOT gates toobtain a state with the permuted populations:

15

1. Introduction

ρ2 = U1.ρ1.U†1 =

p1 0 0 00 p3 0 00 0 p4 00 0 0 p2

and

ρ3 = U2.ρ1.U†2 =

p1 0 0 00 p4 0 00 0 p2 00 0 0 p3

Since the readout is linear with respect to the initial state, all three permuted densitymatrices are added to realize the pseudopure state ρ = ρ1 + ρ2 + ρ2.

ρ =

3p1 0 0 00 p2 + p3 + p4 0 00 0 p2 + p3 + p4 00 0 0 p2 + p3 + p4

Rewriting ρ

ρ = p2+p3+p43

I + 13

4p1 − 1 0 0 0

0 0 0 00 0 0 00 0 0 0

ρ =

1

3(1− p1)I + (4p1 − 1)|00〉〈00| (1.38)

where ρ is the effective pure state corresponding to |00〉.Spatial averaging technique uses rf pulses and pulsed field gradients (PFG) to pre-pare pseudopure states. The PFG kills the magnetization in the plane perpendicularto its applied direction by randomizing the spin magnetization in that plane and spinmagnetization is retained only in the direction along which the PFGs are applied. Fora two-qubit homonuclear system (homonuclear meaning spins belonging to the samespecies) the pseudopure state ρ00 can be prepared from an initial thermal state usingthe following steps:

I1z + I2

z

(π3

)2x−−→ I1z + 1

2I2z +

√3

2I2y

Gz−−→ I1z + 1

2I2z

(π4

)1x−−→ 1√2I1z – 1√

2I1y + 1

2I2z

16

1.3 NMR quantum computing

12J12−−→ 1√

2I1z + 1√

22I1

xI2z + 1

2I2z

(π4

)1−y−−−→ 1√2I1z – 1√

2I1x + 1√

22I1

xI2z + 1

22I2

zI2z + 1

2I2z

Gz−−→ 12(I1

z + I2z + 2I1

zI2z)

where I1i = 1

2σi ⊗ I , I2

i = 12I ⊗ σi, σi with i = x, y, z are Pauli matrices, J12 is the

scalar coupling constant between two spins and Gz is a PFG along the z-axis whichkills all the magnetization in the xy-plane.Logical labeling technique uses one qubit of n-qubits to label the state while theother n − 1 qubits are placed in a pseudopure configuration [79]. To illustrate thelogical labeling technique, let us consider a homonuclear three-qubit system at thermalequilibrium with its deviation density matrix

∆ρeq =

3 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 −1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 −1 0 00 0 0 0 0 0 −1 00 0 0 0 0 0 0 −3

The relative population of the eigenstates are:

State |000〉 |001〉 |010〉 |011〉 |100〉 |101〉 |110〉 |111〉Relative population 6 4 4 2 4 2 2 0

Assuming the first qubit as a label, the first four eigenstates can be perceived as a two-qubit system with the label qubit in the state |0〉 and the other four eigenstates can beconsidered as a two-qubit system with the label qubit in the state |1〉. First a CNOT21

gate is applied (the second qubit being a control qubit and the first qubit being thetarget qubit) and then a CNOT31 gate is applied (the third qubit being the control qubitand the first qubit as the target qubit). The action of these two gates results in a finalrelative state population:

State |000〉 |001〉 |010〉 |011〉 |100〉 |101〉 |110〉 |111〉Relative population 6 2 2 2 4 4 4 0

The deviation part of the pseudopure state density matrix of the two qubits corre-sponding to label 0 is ∆ρ0 = 4|00〉〈00| − I and corresponding to label 1 is ∆ρ1 =I − 4|11〉〈11|. In this thesis we will use the spatial averaging technique throughout forpseudopure state preparation.

17

1. Introduction

1.3.3 Quantum gate implementation in NMR

Section 1.1.4 dealt with the mathematical description of quantum gates. This sectionwill explain the physical implementation of unitary gates on an NMR quantum com-puter. In traditional NMR techniques, spin states are manipulated by using rf pulsesor by free evolution under the internal nuclear spin interactions. It was shown in Sec-tion 1.2 that a spin which satisfies the resonance condition can be rotated about theφ axis by applying an rf pulse along the φ axis with high precession. Due to this,any quantum gate can be implemented in NMR with high fidelity using rf excitationpulses and interaction between the spins. The action of an on-resonance rf pulse witharbitrary phase φ and duration tp is given by

(θ)Iφ = exp(−iω1tpIφ) = exp(−iθIφ) (1.39)

where Iφ = Ixcos(φ) + Iysin(φ) and θ = ω1tp. The rf excitation pulse rotates aspin on-resonance with an angle θ along the φ axis. A single-qubit gate can hence beimplemented using this set of rotations. Some examples of NMR implementations ofsingle-qubit gates are:

• Hadamard gate (H) can be implemented by a spin-selective pulse π pulse alongthe x-axis and a π

2pulse along the y-axis.

H

π2ππ2π

xy

• Pauli-X gate (NOT gate) can be implemented by a single spin-selective π pulsealong the x-axis.

X

ππ

18

1.3 NMR quantum computing

Rφ

φπ2

π2

yx −x

• Phase shift gate (Rφ) can be implemented by a single spin-selective φ pulse alongthe z-axis. However in NMR we can only apply rf pulses along an axis in the xyplane, so a φ rotation along the z-axis is typically decomposed as a pulse cascade(π

2)x(φ)y(

π2)−x.

For the implementation of multi-qubit gates, we use the spin-spin interaction term inthe Hamiltonian along with single-qubit gates. An NMR pulse sequence for the two-qubit CNOT gate is (π

2)2−y(

π2)1,2−z

14J12

(π)1,2y

14J12

(π)1y(π2)y; where 1

J12denotes an evolu-

tion period under the coupling Hamiltonian.

Control

Target

π ππ2

y y−z

−y−z y y

1/4j 1/4j

1.3.4 Numerical techniques for quantum gate optimizationNMR quantum gates can be realized by the application of rf pulses and interactionsbetween the spins. For heteronuclear coupled spins (where the spins under considera-tion belong to different nuclear species), due to the large Larmor frequency differencebetween the spins and the availability of multi-channel rf coils in the spectrometerhardware, it is easy to experimentally implement individual spin-selective rotations.However, for homonuclear spins (where the spins under consideration belong to the

19

1. Introduction

same nuclear species), it is difficult to selectively manipulate an individual spin, dueto much smaller differences in the chemical shifts of the spins. The traditional wayof exciting individual spins in NMR is by using a shaped pulse which is usually oflong duration and results in a low experimental gate fidelity. To tackle this problem,one possible solution is optimization of quantum gates using numerical techniques.The most commonly used numerical optimization techniques are strongly modulatedpulses, genetic algorithms and the gradient ascent pulse engineering (GRAPE) algo-rithm. This thesis mainly uses the GRAPE algorithm for gate optimization so it isdiscussed in detail, while the other techniques are discussed briefly.Strongly modulated pulses (SMPs) is a procedure for finding high-power pulses thatstrongly modulate the dynamics of the system to precisely craft a desired unitary op-eration [84]. It uses the knowledge of the internal Hamiltonian and the form of theexternal Hamiltonian to generate the parameter values to determine the desired gate.SMPs make use of the Nelder-Mead Simplex algorithm [85] to minimize the qualityfactor by searching through the mathematical parameter space. It generates a controlsequence as a cascade of rf pulses with fixed power, transmitter frequency, initial phaseand pulse duration.Genetic algorithms (GAs): These are stochastic search algorithms based on the con-cept of natural selection, a process which drives the biological evolution [86]. GAsmodify the population of the individual solution at each step using the biological in-spired operations such as selection, mutation, crossover etc. to evolve towards anoptimal solution. At each step, the algorithm calculates the fitness of every individualsolution and the algorithm runs until the desired fitness is achieved. In quantum infor-mation processing, GAs have been used to optimize quantum algorithms [87, 88, 89],for quantum entanglement [90] and for optical dynamical decoupling [91]. GAs havealso been used to optimize the pulse sequences for unitary transformations on an NMRquantum information processor [27, 28].Gradient ascent pulse engineering: To construct the desired unitary quantum gateUtgt using the GRAPE algorithm [92], we assume a closed system, with the propagatorU evolving under the Hamiltonian H according to

d

dtU = −iHU. (1.40)

Solving this equation leads to

Uopt =N∏j=1

Uj (1.41)

and

Uj = exp

−i∆t

(H0 +

m∑k=1

uk(j)Hk

)(1.42)

20

1.3 NMR quantum computing

where H0 is the system Hamiltonian, Hk is the rf control Hamiltonian and the controlamplitudes uk are constant, i.e., during the jth step the amplitude uk(j) of the kthcontrol Hamiltonian is given by uk(j). If T is the total pulse duration of the unitarygate then for simplicity the total time T is discretized in N equal steps and ∆t = T/N .So, the problem is to find the optimal amplitudes uk(j) of the rf fields. The actualpropagator Uopt is identical to the desired operator UD when ||Uopt − Utrg||2 = 0 andin an optimization we will search for its minimum. Expanding further

||U − UD||2 = Tr(Uopt − Utgt)†(Uopt − Utgt)†= 2TrI − 2ReTrU †tgtUopt, (1.43)

Hence our task is equivalent to maximization of

Φ = ReTrU †tgtUopt. (1.44)

Further it is not necessary to exactly reproduce Utgt. It serves equally well to reproducethe target operator up to a global phase factor eiφUtgt. Thus the task is equivalent tothe maximization of

Φ0 = |TrU †tgtUopt|2 (1.45)

This performance function increases if we choose

uk(j) −−→ uk(j) + ε δΦ0

δuk(j)

where δΦ0

δuk(j)= 2∆tImTrU †tgtUN . . . Uj+1HkUj . . . U1TrU †optUtgt and ε is a

small step size.The basic GRAPE algorithm consist of the following steps:

1. Guess initial controls uk(j).

2. Evaluate Φ0.

3. Evaluate δΦ0

δuk(j)and update the m×N control amplitudes uk(j).

4. With these as the new controls, iterate to step 2.

The algorithm is terminated if the change in the performance index Φ0 is smaller thana chosen threshold value.

In Fig. 1.4, plot of rf pulse amplitude and phase with time for GRAPE optimizedCSWAP gate is shown with fidelity 0.9995. We used the three 19F spins of the trifluo-roiodoethylene (C2F3I) molecule as NMR sample. With GRAPE we can tackle errorsdue to rf inhomogeneity, off-set and flip angle by optimization.

21

1. IntroductionAmplitu

de

Time(s)

0

2

4

6

8

10

0 0.01 0.02 0.03

Phase(d

eg)

Time(s)

0

100

200

300

0 0.01 0.02 0.03

Figure 1.4: Plot of rf pulse amplitude and phase with time, optimized with GRAPE forCSWAP gate.

1.3.5 Measurement in NMRA conventional detection of the NMR signal is a so-called ensemble weak measure-ment, as the weak interaction of spins with radio-frequency coil does not change sig-nificantly the quantum states of the spins in the process of measuring the total spinmagnetization. A direct projective measurement is not possible on NMR quantumcomputer. However, some experiments have been done to simulate projective mea-surements in NMR [93, 94].

z

y

x

z

y

x

M0

Mxy

t

90o-x

ωFT

FID Signal NMR Spectrum

Figure 1.5: Rotation of Bulk magnetization using resonance. B1(t) applied in -x-directionfor duration such that total rotation is 90

As described in Section 1.2, when the nuclear spins are placed in the magnetic field B0

along the z-axis, the average of the the magnetic moments µ of the nuclei at thermalequilibrium produces a bulk magnetization. With the application of rf pulses, this bulkmagnetization is rotated from the z-axis to the xy plane, where this rotated bulk mag-netization precesses about the z-axis with a Larmor frequency ω0. The precessing bulkmagnetization causes a change in magnetic flux in the rf coil which in turn produces

22

1.4 Evolution of quantum systems

a signal voltage as shown in Fig. 1.5. The recorded signal is proportional to the timerate of change of the magnetic flux linking an inductor that is a part of a tuned circuit.Due to relaxation processes with time the magnetization in the xy plane decays andthe resultant signal also decays with time (called free induction decay (FID)) as shownin Fig 1.5. If the quality factor of the coil is not too high, the recorded signal may beregarded as a time record of the instantaneous bulk magnetization that is transverse (inthe xy-plane ) to the applied static field (which is in the z-axis). This rf signal is mixeddown with a phase-sensitive detector, and the signal has both real (x) and imaginary(y) components. The time-domain signal of the transverse magnetization is given as

S(t) ∝ Tr

ρ(t)

∑k

(σkx + iσky)

(1.46)

where∑

k(σkx − iσky is the detection operator, σkx and σky are Pauli spin operatorsproportional to the x and y components of the magnetization due to kth spin and ρ(t) isa reduced density operator which represents the average state of a single molecule [95].The Fourier transform of Eq.1.46 gives the signal in the frequency domain which rep-resent spectral lines at well-defined frequencies. These spectral lines are characteristicof the spin system used.

The state density matrix ρ, at any instant t, can be reconstructed by systematicallymeasuring the NMR signal. This process of state reconstruction is called quantum statetomography (QST) [95, 96]. Any general normalized state state density matrix can bewritten as

ρ =

(a1 a2 + ia3

a2 − ia3 1− a1

)The NMR signal is proportional to ρ12 = a2 + ia3 and can be measured by measuringthe intensity of the peaks from the real and imaginary parts of the spectra. The realintensity is proportional to a2 and the imaginary intensity is proportional to a3. Formeasuring a1, a π/2 pulse along the y axis is applied and the real intensity of the peakof spectra is proportional to a1 [96]. The reconstruction of density matrix is discussedin detail in Chapter 2.

1.4 Evolution of quantum systemsQuantum systems which do not interact with the outside world are called closed sys-tems. In reality however, no physical system is an entirely closed system, except per-haps the universe as a whole. Real systems suffer from unwanted interactions withtheir environment. These adverse interactions show up as noise in the quantum sys-tem. So, it is important to understand and control such a noise process in order to build

23

1. Introduction

realistic quantum information processors. The tools traditionally used by physicists forthe description of open quantum systems are master equations, Langevin equations andstochastic differential equations. Another potent tool, which simultaneously addressesa broad range of physical scenarios is the mathematical formalism of quantum opera-tions. With this formalism not only nearly closed systems which are weakly coupledto their environments but also the systems which are strongly coupled to their envi-ronments can be modeled. Quantum operations formalism is well adapted to describediscrete state change, that is, transformations between an initial state ρ to final state ρ′,without explicit reference to the passage of time [1].

Figure 1.6: (a) Representation of a closed system; the circle around the system depicts nointeraction between the system and environment. (b) Representation of an open system;the dashed circle around the system shows that the system and environment are interacting.

1.4.1 Closed quantum systems

The dynamics of a closed quantum system in a pure state is governed by the Schrodingerequation

i~∂

∂t|ψ(x, t)〉 = Hsys|ψ(x, t)〉 (1.47)

where ψ(x, t) is the wave function, Hsys the Hamiltonian, and ~ is Planck’s constant.In NMR closed systems, unitary evolution is governed by the Liouville-von Neumannequation

ρ(t) = − i~

[ Hsys, ρ(t)] (1.48)

The solution to Eq. (1.47) and Eq. (1.48) is given by

|ψ(t)〉 = U(t)|ψ(0)〉 (1.49)ρ(t) = U(t)ρ(0)U(t)† (1.50)

24

1.4 Evolution of quantum systems

where |ψ(0)〉 and ρ(0) is a state of the system at time t=0 and

U(t) = Texp

[− i~

∫ t

0

Hsysdt

](1.51)

is a unitary operator. The dynamics of a closed quantum system can be described by aunitary transformation.

Uρ ρU U †

Figure 1.7: Model of closed quantum systems.

A model of a closed system is presented in Fig. 1.7 where the unitary transforma-tion is represented as a box into which the input state ρ enters and from which theoutput state ρ′ = U.ρ.U † exits.

1.4.2 Open systemsThe standard approach for deriving the equations of motion for a system interactingwith its environment is to expand the scope of the system to include the environment.The combined quantum system is then closed, and its evolution is governed by the VonNeumann equation.

ρtot(t) = − i~

[ Htot, ρtot(t)] (1.52)

Here, we assume that the initial state of the total system can be written as a separablestate ρtot = ρsys⊗ ρenv and Htot = Hsys +Henv +Hint is the total Hamiltonian, whichincludes the original system Hsys, the environment Henv, and interaction between thesystem and its environment Hamiltonian Hint. The solution to Eq.(1.52) is given by

ρtot(t) = U(t)ρtot(0)U(t)† (1.53)

Since we are interested in the dynamics of the principal system ρsys, we can extract theinformation about the system by taking the partial trace over the environment

ρsys = Trenv[U(t)(ρsys ⊗ ρevn)U(t)†] (1.54)

The most general trace-preserving and completely positive form of this evolution Eq.(1.52)is the Lindblad master equation for the reduced density matrix ρsys = Trenv[ ρtot] . TheLindblad equation is the most general form for a Markovian master equation, and it is

25

1. Introduction

ρenv

ρsys

U

E(ρsys)

Figure 1.8: Model of open quantum system consisting of two parts: the principal systemand its environment.

very important for the treatment of irreversible and non-unitary processes, from dissi-pation and decoherence to the quantum measurement process.

˙ρsys(t) = − i~

[ Hsys, ρtot(t)] +∑i,α

(Li,αρL†i,α +

1

2L†i,αLi,α, ρ) (1.55)

where the Lindblad operator Li,α =√ki,ασ

(i)α acts on the ith qubit and describes

decoherence, and σ(i)α denotes the Pauli matrix of the ith qubit with α= x, y, z. The

constant ki,α is approximately equal to the inverse of decoherence time.If we consider only the first term on the right hand side of Eq.(1.55), we obtain

the Liouville-von Neumann equation. This term is the Liouvillian and describes theunitary evolution of the density operator. The second term on the right hand side ofthe Eq.(1.55) is the Lindbladian and it emerges when we take the partial trace (a non-unitary operation) over the degrees of freedom of the environment. The Lindbladiandescribes the non-unitary evolution of the density operator and the Lindblad opera-tors can be understood to represent the system contribution to the system-environmentinteraction.

In Fig. 1.8, we have a system in state ρsys and environment in state ρenv which togetherform a closed system and sent into a box which represents the unitary operator on totalsystem with the final state exiting the box being E(ρ). It is to be noted that E(ρ) maynot be related by unitary transformation to the initial system state ρsys. The reducedstate of the system alone can be obtained by taking a partial trace over the environment

E(ρsys) = Trevn[U(ρsys ⊗ ρevn)U †] (1.56)

Now, let us consider |ek〉 to be an orthonormal basis for the state space of the en-vironment and the initial state of the environment can be written as ρevn = |e0〉〈e0|.

26

1.4 Evolution of quantum systems

Using Eq.(1.56), we get

E(ρ) =∑k

〈ek|U [ρ⊗ |e0〉〈e0|]U †|ek〉

=∑k

EkρE†k, (1.57)

where Ek = 〈ek|U |e0〉 is the Kraus operator. Eq.(1.57) is known as the operator-sum representation of E. The operators Ek are known as operation elements for thequantum operation E, which satisfy∑

k

E†kEk = I (1.58)

1.4.3 Quantum noise channelsQuantum channels are convex-linear and completely positive trace preserving maps Ewhich transform the initial state ρ of a quantum system to another state E(ρ).

E : ρ → E(ρ)

E(ρ) =∑k

EkρE†k (1.59)

where Ek’s are the Kraus operators. Quantum channels can be used to describe thetransformation occurring in the state of a system due to the system-environment inter-action. The interaction of a single qubit with its environment can be described by threequantum noise channels which are: phase damping, generalized amplitude dampingand depolarizing channel.

1.4.3.1 Generalized amplitude damping channel

The generalized amplitude damping channel describes the dissipative interactions be-tween the system and its environment which cause interconversions of populationsfrom ground state to excited states and vice versa at finite temperature [1]. For a singlequbit, the Kraus operators for this channel are given by:

E1 =√p

(1 00√

1− a

)(1.60)

E2 =√p

(0√a

0 0

)(1.61)

E3 =√

1− p( √

1− a 00 1

)(1.62)

E4 =√

1− p(

0 0√a 0

)(1.63)

27

1. Introduction

where E1 and E2 operators are responsible for the process by which the populationfrom its excited state will decay to its ground state, E3 and E4 operators are responsi-ble for the reverse process in which populations convert from the ground state to theexcited state, p is the probability of finding population in the ground state at thermalequilibrium and a = 1 − e−γt where ‘γ’ is the decay constant. The action of theseKraus operators on the density matrix ρ is given by:

E(ρ) = E1ρE†1 + E2ρE

†2,

ρ → E(ρ),(ρ11 ρ12

ρ21 ρ22

)→

(ρ′11 ρ

′12

ρ′21 ρ

′22

),

(1.64)

where

ρ′11 = e−γt + p(1− e−γt)ρ11 + p(1− e−γt)ρ22,

ρ′22 = (1− p)(1− e−γt)ρ11 + pe−γtρ22,

ρ′12 = e−γt/2ρ12,

ρ′21 = e−γt/2ρ21.

(1.65)

The Lindblad operator corresponding to the generalized amplitude damping chan-nel is given as

LGD =

√γ

2

(0

√p√

(1− p) 0

)(1.66)

In NMR, γ=1/ T1 where T1 is longitudinal relaxation time (explained in Section 1.4.5).The calculation becomes simple by assuming a high temperature approximation wherep = 1/2.

1.4.3.2 Phase damping channel

Phase damping (PD) channel is a non-dissipative channel, which mainly describes theloss of coherence without loss of energy. In this channel, the relative phase between|0〉 and |1〉 remains unchanged with some probability p or is inverted (φ → φ + π)with probability 1 − p. If the system is in state |0〉 or |1〉, it will be unaffected by thischannel. However, if it is in |ψ〉 = α|0〉+ β|1〉 it gets entangled with the environmentwhich destroys all the coherences but the probability of finding the qubit in state |0〉 or

28

1.4 Evolution of quantum systems

|1〉 does not change. The Kraus operators are given by:

E1 =√p

(1 00 1

)(1.67)

E2 =√

1− p(

1 00 −1

)(1.68)

where p = 1 − exp(−λt) and λ is the decay rate. The action of these operatorstransform the initial state ρ to the final state E(ρ)

E(ρ) = E1ρE†1 + E2ρE

†2, (1.69)

ρ → E(ρ), (1.70)(ρ11 ρ12

ρ21 ρ22

)→

(ρ11 exp(−λt)ρ12

exp(−λt)ρ21 ρ22

), (1.71)

Under the action of the phase damping channel, the off-diagonal elements decay anddiagonal elements remain unaffected.

The Lindblad operator corresponding to the phase damping channel is given as

Lph =

√λ

2

(1 00 −1

)(1.72)

In NMR, λ=1/T2 where T2 is the transverse relaxation time (explained in Section 1.4.6).

1.4.3.3 Depolarizing channel

Under the action of the depolarizing channel, the qubit remains intact with probability1− p while with probability p an identity type of noise occurs. The Kraus operator forthe depolarizing channel is given by:

E1 =√

1− p(

1 00 1

),

E2 =

√p

3

(0 11 0

),

E3 =

√p

3

(0 −ii 0

),

E4 =

√p

3

(1 00 −1

),

29

1. Introduction

where p = 1− exp(−dt) and d is the decay rate. The action of these Kraus operatorschange the initial state ρ to the final state E(ρ),

ρ → E(ρ),

E(ρ) = E1ρE†1 + E2ρE

†2 + E3ρE

†3 + E4ρE

†4,(

ρ11 ρ12

ρ21 ρ22

)→

((2p

3+ (1− 4p

3))ρ11 (1− 4p

3)ρ12

(1− 4p3

)ρ21 (2p3

+ (1− 4p3

))ρ22

), (1.73)

(1.74)

We can further simplify Eq.(1.74) to

E(ρ) =λ

2I + (1− λ)ρ (1.75)

where λ = 4p3

and I is identity matrix.The Lindblad operator corresponding to the depolarizing channel is

LD =

√δ

3(σx + σy + σz) (1.76)

1.4.4 Nuclear spin relaxationThe bulk spin magnetization which is along the z-axis at thermal equilibrium, can berotated to some other direction by the application of rf pulses. Over time the magne-tization returns to the z-axis due to relaxation processes, which are explained by thefamous Bloch equations, describing T1 and T2 relaxation processes.

1.4.5 Longitudinal relaxationLongitudinal relaxation is the process by which the longitudinal component of spinmagnetization returns to its equilibrium value, after a perturbation. In this process,energy is exchanged between the system of nuclear spins and its environment, whichis called the lattice. This process is also known as spin-lattice relaxation. The phe-nomenological equation describing this process is of the form:

dMz

dt=

M0 −Mz

T1

(1.77)

where T1 is known as the longitudinal or the spin-lattice relaxation time and M0 is thethermal equilibrium magnetization. The solution of the above equation is

Mz = M0(1− e−t/T1),

30

1.4 Evolution of quantum systems

when the M0 is tilted to the xy plane, then Mz(0) = 0.For measuring T1, the inversion recovery experiment is commonly used, where the

spin magnetization is first inverted such that Mz(0)=-M0:

Mz = M0(1− 2e−t/T1). (1.78)

1.4.6 Transverse relaxationTransverse relaxation is the process that leads to the disappearance of the coherencesnamely the xy-magnetization. The phenomenological equation describing the decayof the transverse magnetization in the rotating frame can be written as:

dMx,y

dt= −Mx,y

T2

(1.79)

where T2 is called the transverse relaxation time. The solution of this equation is

Mx,y = M0e−t/T2 (1.80)

where M0 is the initial value of the transverse magnetization after the application of a90 rf pulse.

1.4.7 Bloch-Wangness-Redfield relaxation theoryThis relaxation model uses a quantum mechanical approach to describe the systemparameters while the surrounding environment is described classically. The main lim-itation of this approximation is that at equilibrium the energy levels are predicted tobe equally populated. The theory is formally valid only in the high-temperature limit.For finite temperatures, corrections are required to ensure that the correct equilibriumpopulations are reached. However these corrections are significant only in the case ofvery low temperatures [76, 97, 98, 99].

The von Neumann-Liouville equation, which describes the time evolution of themagnetic resonance phenomenon using spin density matrix ρ(t) is given by

dρ(t)

dt= −i[H0 +H1(t), ρ(t)] (1.81)

where H0 is the time-independent part of the Hamiltonian which contains the spinHamiltonian and H1(t) describes the time-dependent perturbations.

It is convenient to remove the explicit dependence on H0 by rewriting the densityoperator ρ(t) in a new reference frame, called the interaction frame:

ρ∗ = exp(iH0(t))ρ(t)exp(−iH0(t)) (1.82)

31

1. Introduction

It is possible to rewrite Eq.(1.81) in the interaction frame:

dρ∗(t)

dt= −i[H∗1 (t), ρ∗(t)] (1.83)

To solve Eq.(1.83) the following assumptions are required:

1. The ensemble average of H∗1 (t) is zero.

2. ρ∗(t) and H∗1 (t) are not correlated, with this assumption it is possible to takethe ensemble average of the fluctuations of the Hamiltonian and quantum statesindependently.

3. τc << t << 2/R, where τc is the correction time relevant for H∗1 (t) and R isthe relevant relaxation rate constant.

4. For the system to relax towards the thermal equilibrium, ρ∗(t) has to be replacedby ρ∗(t)− ρ0, where ρ0 is the density operator at equilibrium.

Using these assumption, the R.H.S in Eq.(1.83) can be replaced by an integral:

dρ∗(t)

dt= −

∫ ∞0

[H∗1 (t), [H∗1 (t− τ), ρ∗(t)− ρ0]]dτ (1.84)

where the overbar represents the ensemble average. The third assumption allows theintegral to run to infinity and with the assumption that the fluctuations of the Hamilto-nian are not correlated with the density matrix, we can calculate the ensemble averageover the stochastic Hamiltonian independently from ρ∗(t).

For transforming Eq.(1.84) back in the lab frame, the stochastic HamiltonianH∗1 (t)has to be decomposed as the sum of the random functions of the spatial variable F q

k (t)and tensor spin operators Aqk:

H1(t) =k∑

q=−k(−1)qF−qk (t)Aqk (1.85)

The tensor spin operators are chosen to be spherical tensor operators because of theirtransformations properties under rotations. For the Hamiltonians of interest in NMRspectroscopy, the rank of the tensor k is one or two. These operators can be furtherdecomposed as a sum of basis operators:

Aqk =∑p

Aqkp (1.86)

32

1.4 Evolution of quantum systems

where the components Aqkp satisfy [H0, Aqkp] = ωqpA

qkp. The transformation of Aqk in

the interaction frame:

Aq∗k = exp(iH0t)Aqkexp(−iH0t) =

∑p

Aqkpexp(iωqpt) (1.87)

Using Eq.(1.85) and Eq.(1.87) we can rewrite Eq.(1.84)

dρ∗(t)

dt= −

∑q,q′

∑p,p′

(−1)q+q′expi(ωqp + ωq

′

p′)t[Aq′

kp′ , [Aqkp, ρ

∗(t)− ρ0]]∫ ∞0

F−qk (t)F−qk (t− τ)dτ (1.88)

If q 6= −q, the two random processes F−q′

k (t) and F−q′

k (t) are assumed to be statisti-cally independent, due to which the ensemble average vanishes, unless q′ = −q.

dρ∗(t)

dt= −

k∑q=−k′

∑p,p′

expi(ωqp − ωq′

p′)t[Aqkp′ , [Aqkp, ρ∗(t)− ρ0]]∫ ∞0

F−qk (t)F−qk (t− τ)exp(iωqpτ)dτ (1.89)

Further it is to be noted that terms in which |ωqp − ωqp| >> 0 oscillate much faster thanthe typical time scales of the relaxation phenomena will not affect the evolution. In theabsence of degenerate eigenfrequencies, terms in Eq.(1.89) do not vanish when p = p′.Hence

dρ∗(t)

dt= −

k∑q=−k′

∑p

[Aqkp, [Aqkp, ρ

∗(t)− ρ0]]∫ ∞0

F−qk (t)F−qk (t− τ)exp(iωqpτ)dτ (1.90)

The terms F−qk (t)F−qk (t− τ) are correlation functions. The real part of the integral inEq.( 1.90) is the power spectral density function

jq(ω) = 2Re

∫ ∞0

F−qk (t)F−qk (t− τ)exp(iωτ)

The imaginary part of the integral in Eq.( 1.90) is the power spectral density function

kq(ω) = Im

∫ ∞0

F−qk (t)F−qk (t− τ)exp(iωτ)

33

1. Introduction

In the high-temperature limit, the equilibrium density matrix is proportional to H0.Thus, using Eq. (1.87), the double commutator [[A−qkp , A

qkp], ρ0]] = 0

dρ∗(t)

dt= −1

2

k∑q=−k

∑p

[A−qkp , [Aqkp, ρ

∗(t)−ρ0]]jq(ωqp)+ik∑q=0

∑p

[[A−qkp , Aqkp], ρ

∗(t)]kq(ωqp)

(1.91)By transforming the above equation in lab frame

dρ(t)

dt= −i[H0, ρ(t)]− i[∆, ρ(t)]− Γ(ρ(t)− ρ0) (1.92)

where the relaxation superoperator is

Γ = −1

2

k∑q=−k

∑p

[A−qkp , [Aqkp, ]]j

q(ωqp) (1.93)

∆ is the dynamic frequency shift operator that accounts for second-order frequencyshifts of the resonance lines

∆ = −k∑q=0

∑p

[A−qkp , Aqkp]k

q(ωqp) (1.94)

This term can be incorporated into the Hamiltonian to obtain the final result, known asmaster equation:

dρ(t)

dt= −i[H0, ρ(t)]− Γ(ρ(t)− ρ0) (1.95)

In the calculation of relaxation rates it is often convenient to expand Eq.(1.95) in termsof the basis operators used to expand the density operator

dbr(t)

dt=∑s

−iΩrsbs(t)− Γrs[bs(t)− bs0] (1.96)

where Ωrs are characteristic frequencies defined as

Ωrs =〈Br|[H0,Bs]〉〈Br|Bs〉

(1.97)

Γrs are the rate constant for relaxation between the operator Bs and Br

Γrs =〈Br|ΓBs〉〈Br|Bs〉

= −1

2

k∑q=−k

∑p

〈Br|[A−qkp , [Aqkp,Bs]]〉

〈Br|Bs〉

jq(ωqp) (1.98)

34

1.5 Decoherence suppression

and

br(t) =〈Br|ρ(t)〉〈Br|Bs〉

(1.99)

The diagonal elements Γrr are auto-relaxation, while off-diagonal elements Γrs, arecross-relaxation rates.Because it is assumed that only terms satisfying q = −q givenon-zero contributions to Eq.(1.88), cross-relaxation can occur only between operatorswith the same coherence order. In addition, because of the secular approximation inEq.(1.90), cross-relaxation between off-diagonal terms is forbidden in the absence ofdegenerate transitions. These two features give rise to a characteristic block shape inthe relaxation superoperator, known as Redfield kite.

1.5 Decoherence suppressionThe coherent superposition of states in combination with the quantization of observ-ables, represents the one of the most fundamental features that mark the departure ofquantum mechanics from the classical realm [100]. Quantum coherence in multi-qubitsystems embodies the essence of entanglement and is an essential ingredient quantuminformation processing.A coherent quantum state in contact with environment losescoherence i.e. decoherence and entangle states are much more fragile to it. In NMR,T2 gives an estimation of the decoherence time of the system and in liquid state NMRit is the of order of seconds. There is a longstanding debate on quantumness of NMRspin-systems states, the presence of nonzero discord in some of the states of NMRspin-systems indicates the intrinsic quantumness of nuclear spin systems even at hightemperatures [83, 101].

Preserving quantum coherence is an important task in quantum information anddifferent techniques have been developed to suppress decoherence. These techniquesare broadly categorized as quantum error correction [102], decoherence free subspaces[103, 104] and dynamical decoupling (DD) methods [105, 106]. In particular, the DDtechnique is an important technique which suppresses the decoherence by eliminat-ing the system-bath coupling. The idea comes from spin-echo pulses in NMR wherestatic but nonuniform couplings can be compensated for perfectly by a single π pulsein the middle of the time interval [107]. The idea of the spin echo was expanded tosuppress dynamic interactions with the environment by using periodic π pulses or byperiodic Carr-Purcell cycles. The Carr-Purcell sequence was further modified to com-pensate errors due to π pulses and Carr-Purcell-Meiboom-Gill sequence (CPMG) wasdevised [108]. A more sophisticated technique namely the Uhrig dynamical decou-pling sequence was devised and it was shown that instead of applying π pulses at equalintervals of time if π pulses are applied at unequal intervals of time then the sequenceshows better preservation [109]. One of the advantages of the DD technique is that no

35

1. Introduction

extra qubits are required unlike other techniques. Most DD preserving sequences areconstructed to take care of dephasing type noise. In NMR language, T2 type relaxationis considered and noise due to T1 relaxation is ignored.

z

x

z

x

y y

z

x

y

x

z

x

y

z

x

y

90x

180x

After time t

(a) (b) (c)

After time t

(d) (e)

Figure 1.9: Evolution of the bulk magnetization under Hahn echo sequence. (a) Initiallythermal equilibrium bulk magnetization in z direction is represented by an arrow. (b) Bulkmagnetization rotated to−y axis using 90 pulse, (c) A delay of time t is given and arrowsrepresent different spins, (d) 180 is applied to spins due to which a spin precessing fastwill fall behind a spin precessing slowly, and (e) after time t, all the spins are in the samedirection.

1.5.1 Hahn echo

This technique was constructed by E. L. Hahn for suppressing time-independent noisein a system of isolated spins [107]. In an NMR setup, the static magnetic fieldB0 along

36

1.5 Decoherence suppression

the z axis has a spatial inhomogeneity due to which different spins in the ensemble ex-perience different magnetic field and hence precess with different Larmor frequencies(which cause spin dephasing). To tackle this problem Hahn devised the spin-echo se-quence as shown in Fig. 1.9. Initially at thermal equilibrium the bulk magnetization isin the z direction. With the application of an rf pulse, a (π/2)x rotation is applied tothe bulk magnetization. After time t a spin experiencing a greater magnetic field willbe ahead of a spin which experiences a smaller magnetic field as shown in Fig. 1.9(c).Then, a π pulse is applied due to which the slow moving spins come close to and thefast moving spins move away from the y axis. After time t all the spins precess in thesame direction as shown in Fig. 1.9(e).

1.5.2 CPMG DD sequence

In the Carr-Purcell(CP) sequence a series of rf pulses are applied: the first pulse flipsthe magnetization through π

2angle with a π

2pulse along the x-axis, and the following

train of equidistant pulses flip the magnetization through π with the π pulse alongthe x-axis [110]. In the actual application of the Carr and Purcell method for themeasurement of long relaxation times, it was found that the amplitude adjustment ofthe π pulses was critical. This is because a small deviation from the exact π value givesa cumulative error in the results. In the Carr-Purcell-Meiboom-Gill(CPMG) sequence:the first pulse flips the magnetization through π

2angle with a π

2pulse along a y-axis, and

a series of π pulses along the x axis are applied at times t = (2n+1)τ2

, (n = 0, 1, 2 . . . ).The CPMG sequence was able to suppress the time-dependent noise.

0 τ2

3τ2

5τ2

7τ2

9τ2

11τ2

13τ2

15τ2

T

Figure 1.10: Pulse sequence of CPMG DD sequence for a cycle of duration T and in onecycle eight pulses are applied; filled bars represent π rotation pulses and τ is the durationbetween two consecutive π pulses.

In the CPMG sequence a train of equidistant π pulses are applied on a qubit. InFig. 1.10 a train of eight pulses are applied in one cycle of duration T . The morethe number of pulses in a cycle, the better is the decoherence suppression.

37

1. Introduction

1.5.3 Uhrig DD sequenceThe Uhrig DD (UDD) sequence is an optimal DD scheme and was first constructed byUhrig for a pure dephasing spin-boson model [109], which uses N π pulses applied attime intervals Tj

Tj = Tsin2 jπ

2(N + 1)for j = 1, 2, . . . N, (1.100)

to eliminate the dephasing up to order O(TN+1); hence the UDD technique suppresseslow-frequency noise. The CPMG sequence is a UDD sequence of order N = 2. If foran interval of duration T , two π pulses are applied the CPMG sequence, it will elim-inate dephasing up to order O(T 3). Further, the proof of the universality of the UDDin suppressing the pure dephasing or the longitudinal relaxation of a qubit coupled toa generic bath has been given [111].

0 T1 T2 T3 T4 T5 T6 T7 T8 T

Figure 1.11: Pulse sequence for UDD sequence for a cycle time of T and the number ofπ pulses in one cycle of sequence is eight. Filled bars represent π rotation pulses.

Yang and Liu considered ideal UDD pulse sequences for a Hamiltonian of the form

H = C + σz ⊗ Z, (1.101)

where σz is the qubit Pauli matrix along the z-direction, and C and Z are bath oper-ators. This Hamiltonian describes a pure dephasing model as it contains no qubit flipprocesses and therefore leads to no longitudinal relaxation but only transverse dephas-ing. Defining two unitary operator U (N)

± as follows:

U(N)± (T ) = e−i[C±(−1)NZ](T−TN )

× e−i[C±(−1)(N−1)Z](TN−TN−1) · · ·× e−i[C∓Z](T2−T1)e−i[C±Z]T1 (1.102)

Yang and Liu proved that for Tj satisfying Eq. (1.100), we must have(U

(N)−

)†U

(N)+ = 1 +O(TN+1), (1.103)

38

1.6 Organization of the thesis

i.e., the product of(U

(N)−

)†and U (N)

+ differs from unity only by the order of O(TN+1)

for sufficiently small T [111]. In Fig.1.11, eight π pulses are applied with UDD tim-ing preserving qubit coherence up to order O(T 9), whereas the CPMG sequence inFig.1.10 is able to preserve coherences up to order O(T 3).

1.5.4 Super-Zeno SchemeThe super-Zeno scheme is an algorithm for suppressing the transitions of a quantummechanical system, initially prepared in a subspace P of the full Hilbert space of thesystem, to outside this subspace by subjecting it to a sequence of unequally spacedshort-duration pulses [60]. These durations were calculated numerically the leakageprobability from subspace P to its orthogonal subspace was minimized, and surpris-ingly the durations matched with Eq.(1.100). This scheme is experimentally imple-mented in Chapter3. This scheme efficiently cancels all the noise affecting the state,and the preservation is up to the order O(TN+1), where N denotes the number of in-verting pulses J which are applied. The construction of this inverting pulse J dependsthe on subspace P.

1.5.5 Nested Uhrig dynamical decouplingThe nested Uhrig dynamical decoupling (NUDD) scheme is an extension of the super-Zeno scheme for the case where instead of a state, only the subspace to which thestate belongs is known [64]. For protecting an unknown state in a known subspace,nesting of UDD protecting sequences are done in a a smart way such that these nestingof layers cancels all the possible interactions which affect states belonging to the P

subspace. The inverting pulses of each layer are constructed on the basis of subspaceto be protected, and these inverting pulses are applied at the UDD time points given byEq.(1.100). The NUDD scheme is very sensitive to the nesting of layers, and for thetime interval T with N pulses in each layer, protection is achieved of O(TN+1). TheNUDD scheme is discussed in detail in Chapter 4.

1.6 Organization of the thesisThis thesis deals with the estimation of experimentally prepared quantum states andprotection of these states using different decoupling strategies. The thesis is organizedas follows: Chapter 2 demonstrates the utility of the Maximum Likelihood (ML) esti-mation scheme to estimate quantum states on an NMR quantum information processor.We experimentally prepare separable and entangled states of two and three qubits, and

39

1. Introduction

reconstruct the density matrices using both the ML estimation scheme as well as stan-dard quantum state tomography (QST). Further, we define an entanglement parameterto quantify multiqubit entanglement and estimate entanglement using both the QSTand the ML estimation schemes. Chapter 3 demonstrates the efficacy of the super-Zenoscheme for the preservation of a state by freezing state evolution (one-dimensional sub-space protection) and subspace preservation by preventing leakage of population to anorthogonal subspace (two-dimensional subspace protection). Both kinds of protectionschemes are experimentally demonstrated on separable as well as on maximally entan-gled two-qubit states. Chapter 4 demonstrates the efficacy of NUDD scheme for theprotection of arbitrary states of the known subspace. Chapter 5 focuses on extendingthe lifetime of time-invariant discord using dynamical decoupling schemes. Chapter6 is based on modeling the noise and experimental protection of a three-qubit systemusing dynamical decoupling. Chapter 7 summarizes the work done in this thesis anddiscusses the future prospects.

40

Chapter 2

State tomography on an NMRquantum information processor viamaximum likelihood estimation

2.1 Introduction

Classically, a state is assigned to a physical system by determining the phase spacepoint corresponding to the configuration of the system and measuring the relevant sys-tem parameters in a non-invasive manner. For quantum systems, non-invasive mea-surements are not possible and therefore an ensemble of identically prepared systemsare required for state estimation. The quantum state cannot be known from a sin-gle measurement and the ‘no-cloning’ theorem renders impossible the possibility ofmaking several copies of the state and using them to make different measurementson the same state. Quantum state estimation is hence intrinsically a statistical pro-cess [112, 113]. In quantum information and experimental quantum computing, thecomplete estimation of a quantum state from a set of measurements on a finite ensem-ble is a hot topic of research. Several schemes have been proposed and implemented forquantum state estimation [114, 115, 116]. Due to the finite size of ensembles, a phys-ical situation may even have two different candidate states and there is always someambiguity associated with the estimated state. Such uncertainties and ambiguities, ifnot treated properly can lead to a self-contradictory estimation, where the estimatedstate is not even a valid quantum state. The density operator provides a convenientway to describe a quantum system whose state is not a pure state. It is a positive, Her-mitian operator with Tr[ρ] = 1 and Tr[ρ2] ≤ 1. For a pure ensemble Tr[ρ2] = 1 andfor a mixed ensemble Tr[ρ2] < 1. In NMR, the quantum state tomography (QST) of n

41

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

coupled spins is carried out by measuring free induction decays (FIDs) after applyinga set of preparatory pulses and applying a Fourier transformation to obtain spectrallines [95, 96, 117, 118]. Fitting these spectral lines, yields complex amplitudes atwell defined frequencies which are characteristic of the spin system. These complexamplitudes are measured experimentally for each pulse in the preparatory pulse set.Using this experimental data a state density matrix is reconstructed. However, the re-constructed density matrix with the standard QST procedure leads to an unphysicaldensity matrix i.e., some eigenvalues could turn out to be negative. This unphysicaldensity matrix makes no sense, as even if there are experimental errors they shouldbe within the space of allowed density operators. An ad hoc way to circumvent thisproblem is to add a multiple of identity to this density matrix so that the eigenvaluesare positive. A scheme that redresses this issue of reconstructed density matrices thatare unphysical, is the maximum-likelihood (ML) estimation scheme, which obtains apositive definite estimate for the density matrix by optimizing a likelihood functionalthat links experimental data to the estimated density matrix along with the constraintthat the density matrix should be positive at every point of optimization [57, 58, 119].

The ML estimation scheme begins with a guess quantum state and improves theestimate based on the measurements made; the more the number of measurements, thebetter is the state estimate.

This issue of unphysical density matrices was first pointed out in quantum opticswhere state reconstruction based on the inversion of measured data could not guaran-tee the positive definiteness of the reconstructed density matrix. Hence an algorithmfor quantum-state estimation based on the maximum-likelihood (ML) estimation wasproposed [56, 120]. A general method proposed for quantum state estimation basedon the ML approach can be applied to multi-mode radiation fields as well as to spinsystems [55]. An ensemble of spin-1

2particles was observed repeatedly using Stern-

Gerlach devices with varying orientations and the state of an ensemble was recon-structed via ML estimation [121] . A simple iterative algorithm for ML estimation ofthe quantum state was derived [122]. A tomographic protocol for a two-qubit systemwas recently constructed based on the measurement of 16 generalized Pauli operatorswhich is maximally robust against errors [123]. A refined iterative ML algorithm wasalso proposed to reconstruct a quantum state and applied to the tomography of opticalstates and entangled spin states of trapped ions [124]. Other quantum state estimationalgorithms include Bayesian mean estimation [125], least-squares inversion [126], nu-merical strategies for state estimation [126] and linear regression estimation [127]. Ifthe size of the ensemble is infinite, the estimation procedure will yield the unique truequantum state of the system. However, such an ensemble is never achievable in anylaboratory setting, as one can only perform measurements on a finite ensemble. Asa result, the estimated state will be different from the true state and depends on the

42

2.1 Introduction

details of the estimation procedure. Quantum state reconstruction on a finite numberof copies of a quantum system with informationally incomplete measurements, as arule, does not yield a unique result. A reconstruction scheme was derived where boththe likelihood and the von Neumann entropy functionals were maximized (MLME) inorder to systematically select the most-likely estimator with the largest entropy, thatis, the least-bias estimator, was consistent with a given set of measurement data. ThisMLME estimation protocol was applied to time-multiplexed detection tomography andlight-beam tomography [128, 129].

In this chapter, the utility of the ML estimation scheme has been demonstrated toperform quantum state estimation on an NMR quantum information processor. Sepa-rable and entangled states of two and three qubits are experimentally prepared, and thedensity matrices are reconstructed using both the standard QST and the ML estimationschemes. For the quantification of entanglement in multiqubit systems an entangle-ment parameter is defined and it is shown that the standard QST method overestimatesthe residual state entanglement at a given time, while the ML estimation method givesa correct estimate of the amount of entanglement present in the state.

2.1.1 NMR quantum state tomographyThe basic aim of quantum state tomography (QST) is to completely reconstruct an un-known state via a set of measurements on an ensemble of identically prepared states.Any density matrix ρ of n qubits in a 2n-dimensional Hilbert space can be uniquelydetermined using 4n − 1 independent measurements and the state of the system as de-scribed by its density operator ρ can be reconstructed by performing a set of projectivemeasurements on multiple copies of the state [29, 130]. Determining all the elementsof ρ would involve making repeated measurements of the same state in different mea-surement bases, until all the elements of ρ are determined [119, 131, 132].

In NMR we cannot perform projective measurements and instead measure the ex-pectation values of certain fixed operators over the entire ensemble. Therefore, werotate the state via different unitary transformations before performing the measure-ment to collect information about different elements of the density matrix [29]. Thestandard tomographic protocol for NMR uses the Pauli basis to expand an n qubit ρ,

ρ =3∑i=0

3∑j=0

...3∑

k=0

cij...kσi ⊗ σj ⊗ ....σk (2.1)

where c00...0 = 1/2n and σ0 denotes the 2 × 2 identity matrix while σ1, σ2 and σ3

are standard Pauli matrices. The measurements of expectation values allowed in anNMR experiment combined with unitary rotations leads to the determination of thecoefficients cij..k.

43

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

In an NMR experiment, we measure the signal induced in the detection coils whilethe nuclear spins precess freely in a strong applied magnetic field. This signal is calledthe free induction decay (FID) and is proportional to the time rate of change of mag-netic flux. This time-domain signal can be expressed in terms of the expectation valuesof the transverse magnetization [95]:

S(t) ∝ Tr

ρ(t)

∑k

(σkx − iσky)

(2.2)

where σkx and σky are the Pauli spin operators proportional to the x and y componentsof the magnetization of the kth spin and ρ(t) is the instantaneous density operator attime t during the FID. The recorded signal represents an average over a large numberof identical molecules of the sample.

The transformation of an initial density operator ρ0 under applied pulses UP andunder free-evolution Hamiltonian H for time t is given by:

ρ(t) = e−iHtUPρ0U†P e

iHt. (2.3)

Then Eq.(2.2) reduces to

Sp(t) ∝ Tr

e−iHtUPρ0U

†P e

iHt∑k

(σkx − iσky), (2.4)

using the linearity of the trace and its invariance with respect to cyclic permutation ofthe operators, we can write

Sp(t) ∝ Tr

ρ0U

†P e

iHt∑k

(σkx − iσky)e−iHtUP

(2.5)

In NMR spectroscopy, the initial density operator ρ0 is known and it representsthe thermal equilibrium state then from the signal we can determine the Hamiltonian.However in state tomography the reverse is true i.e. the Hamiltonian is known and thestate ρ0 is unknown. The Hamiltonian of n weakly coupled spins-1

2is given by,

H = −n∑k=1

ωkIkz + 2πn∑k=2

k−1∑j=1

JjkIjzIkz (2.6)

where ωk/2π is the Larmor frequency of the kth spin, and Jjk is the spin-spin couplingconstant between jth and kth spins.For a single spin system, the Hamiltonian H1 consists of a Zeeman term only, whichcan be written as

H1 =1

2ωσz (2.7)

44

2.1 Introduction

where ω is the Larmor frequency of the nuclear spin in the external magnetic field.Then, the NMR signal can be written as

S1p(t) ∝ Tr

ρ0U

†P e

i 12ωσzt

∑k

(σkx − iσky)e−i12ωσztUP

(2.8)

Without applying any pulse i.e. Up = I , the NMR signal is

S1I (t) ∝ Tr[ρ0σx]− iTr[ρ0σy] eiωt. (2.9)

after applying 90o pulse along x i.e. Up = 90ox, the NMR signal is

S1X(t) ∝ Tr[ρ0σx]− iTr[ρ0σz] eiωt. (2.10)

after applying 90o pulse along y i.e. Up = 90oy, the NMR signal can be written as

S1Y (t) ∝ Tr[ρ0σz]− iTr[ρ0σy] eiωt. (2.11)

Applying a Fourier transformation on Eq.(2.9, 2.10 and 2.11) then the ensembleaverage of operators σx, σy and σz can be obtained

〈σx〉 = c avg(Re[SI(ω)],Re[SX(ω)]), (2.12)〈σy〉 = c avg(−Im[SI(ω)],−ImSY (ω)]), (2.13)〈σz〉 = c avg(−Im[SX(ω)],Re[SY (ω)]). (2.14)

Where avg(a, b) means an average of a and b. The factor c depends on the experi-mental details such as the receiver gain and the number of spins. After determining thefactor c, the density operator of a single spin can be estimated as

ρ =1

2I + 〈σx〉σx + 〈σy〉σy + 〈σz〉σz (2.15)

For a two-spin system, the Hamiltonian can be written as

H2 =1

2ω1σ1z +

1

2ω2σ2z +

π

2J12σ1zσ2z (2.16)

where J12 is the scalar coupling constant. After inserting H2 in Eq.(2.5) and solvingfew steps the NMR signal due to spin 1 can be written as

SP,1(t) ∝ 1

2(ei(ω1−πJ12)t, ei(ω1+πJ12)t)×

(1 11 −1

)(Trρ0σ1−

Trρ0σ1−σ2z

)(2.17)

45

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

Where σ1− = U †P (σ1x − iσ1y)UP . Fourier transformation of SP,1(t) leads to,(SP,1(ω1 − πJ12)SP,1(ω1 + πJ12)

)∝ 1

2

(1 11 −1

)(Trρ0σ1−

Trρ0σ1−σ2z

)(2.18)

Similarly for spin 2,(SP,2(ω2 − πJ12)SP,2(ω2 + πJ12)

)∝ 1

2

(1 11 −1

)(Trρ0ω2−

Trρ02ω1zω2−

)(2.19)

The density matrix of a two-spin system can be expanded in terms of Pauli basisoperators I1 ⊗ I2 . . . σ1z ⊗ σ2z, as follows:

ρ =1

4(I1 ⊗ I2 + 〈σ1x ⊗ I2〉σ1x ⊗ I2 + . . .

+〈σ1z ⊗ σ2y〉σ1z ⊗ σ2y + 〈σ1z ⊗ σ2z〉σ1z ⊗ σ2z) (2.20)

and for the estimation of ρ we need to calculate the expectation values 〈σ1x ⊗ I2〉,. . . ,〈σ1z⊗σ2z〉. From the NMR spectra we can calculate the peaks intensities i.e. SP,1 andSP,2 and rewriting the Eq.(2.18), and Eq.(2.19)

(Trρ0I1−

Trρ02I1−I2z

)∝(

1 11 −1

)(SP,1(ω1 − πJ12)SP,1(ω1 + πJ12)

)(2.21)

(Trρ0I2−

Trρ02I1z I2−

)∝(

1 11 −1

)(SP,2(ω2 − πJ12)SP,2(ω2 + πJ12)

)(2.22)

we can calculate the expectation values. Taking UP = II and inserting the experimen-tally measured peak intensities of spin 1 in Eq.(2.21), we get the expectation values〈σ1x ⊗ I2〉, 〈σ1y ⊗ I2〉, 〈σ1x ⊗ σ2z〉, and 〈σ1y ⊗ σ2z〉. On inserting the peak intensitiesof spin 2 in Eq.(2.22) we get the expectation values 〈I1⊗σ2x〉, 〈I1⊗σ2y〉, 〈σ1z⊗σ2x〉,and 〈σ1z ⊗ σ2y〉. Similarly we can measure other expectation values by changing thepreparatory pulse. With the set UP = II, IX, IY,XX, we can measure all the ex-pectation values required to reconstruct the density matrix, where II corresponds to“no operation” on both spins, IX(Y ) corresponds to a “no operation” on the first spinand a 90 rf pulse of phase X(Y ) on the second spin, and XX corresponds to a 90 rfpulse of phase X on both spins.

As an example for a two-qubit system, we created the quantum state 1√2(|00〉 +

|01〉) and reconstructed it using standard QST. We experimentally generated twenty-five density matrices for this state, and computed the mean and the variance. The

46

2.1 Introduction

reconstructed density matrix ρQST

turned out to be

ρQST

=

0.484 0.508 + i0.028 −0.025− i0.029 −0.025 + i0.019

0.508− i0.027 0.516 0.025 + i0.003 0.009 + i0.030−0.025 + i0.029 0.025− i0.003 −0.039 + i0.000 −0.025− i0.011−0.025− i0.019 0.009− i0.030 −0.025 + i0.011 0.039 + i0.000

(2.23)

The above density matrix ρQST

, reconstructed using the standard QST protocol, isnormalized and Hermitian and its eigenvalues are 1.011±0.008, 0.052±0.025, 0.016±0.008,−0.079± 0.018. The errors in the reconstructed density matrix using the QSTmethod show up in the third decimal place. As is evident from the eigenvalues, the re-constructed density matrix is not positive, and furthermore, Tr(ρ2

QST) = 1.031±0.009.

It is clear from the above data that the negativity of the eigen value is statistically sig-nificant and is due to the way we have carried out state estimation. Density matricesthat represent physical quantum states must have the property of positive definitenesswhich, in conjunction with the properties of normalization and Hermiticity, impliesthat all the eigenvalues must lie in the interval [0,1] and sum to 1 i.e. 0 ≤ Tr(ρ2) ≤ 1.Clearly, the above density matrix which is reconstructed by the standard QST protocolviolates these conditions. Due to its negative eigenvalues it has as a strange feature thatTr(ρ2) > Tr(ρ). The obvious reasons for this problem are experimental inaccuracies,which implies that the actual magnetization values recorded in an NMR experimentdiffer from those that can be obtained from the Eq. (2.20). However, in a correctestimation scheme the experimental inaccuracies should lead to an error in the stateestimation by giving a state which is close to the actual state with some confidencelevel and should not give a non-state! An ad hoc way to circumvent this problem isto add a multiple of identity to this density matrix so that the eigenvalues are positive.However, this kind of addition is completely ad hoc, and leads to non-optimal esti-mates and one should be able to do better. We turn to this issue in the next section viathe maximum likelihood estimation method.

2.1.2 Maximum likelihood estimation

The example in the previous section illustrates that density matrices which are to-mographed using standard QST may not correspond to a physical quantum state. Toaddress this problem, the maximum likelihood (ML) estimation method was developedto ensure that the reconstructed density matrix is always positive and normalized [119].The ML estimation method estimates the entire quantum state, by finding the param-eters that are most likely to match the experimentally generated data and maximizinga specific target function; a priori knowledge of the density matrix can also be incor-

47

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

porated into the method. The main advantage of this method is that at every stage ofthe estimation process the density matrix is positive and normalized and therefore rep-resents a valid physical situation. The construction of a valid density operator throughmaximum likelihood estimation consists of the following steps:

1. The density operator is first obtained from a lower triangular matrix T such thatρ = T †T , here T is a function of real variables t1, t2, t3, . . . , t4n and n is thenumber of qubits. With this ρ(t1, t2, t3, . . . , t4n) will be always Hermitian andpositive.

2. A “likelihood function” is then constructed which quantifies how close the den-sity operator ρ(t1, t2, t3, . . . , t4n) is with respect to the experimental data. Thislikelihood function is a function of ti and experimental data ni and can be writtenas L(t1, t2, . . . , t4n ;n1, n2, . . . , n4n).

3. Using standard numerical optimization techniques, the optimum set of variablest(opt)1 , t

(opt)2 , t

(opt)3 , . . . , t

(opt)4n is obtained, for which the function L(t1, t2, . . . , t4n ;n1, n2

, . . . , n16) has the maximum value. The best estimated density operator is ρ(t(opt)1 , t

(opt)2 ,

t(opt)3 , . . . , t

(opt)4n ).

For a system of two qubits, the density matrix can be written in a compact formfollowing Eq. (2.1):

ρ =3∑j=0

3∑k=0

njkσj ⊗ σk (2.24)

where njk are real coefficients determining the state.A physical density matrix ρ has to be Hermitian, positive and must have trace equal

to unity. Such a density matrix can be written in terms of a lower triangular matrixT [119]

ρ(t1, t2, . . . , t16) = T †T Tr(T †T ) = 1 (2.25)

For a two-qubit system the lower triangular matrix T from which we obtain ρ has15 independent real parameters (one parameter from the 16 is eliminated due to thetrace condition), and can be written as

T =

t1 0 0 0

t5 + it6 t2 0 0t11 + it12 t7 + it8 t3 0t15 + it16 t13 + it14 t9 + it10 t4

(2.26)

48

2.1 Introduction

Given a valid density matrix as described in [119], it is possible to invert Eq. (2.25)to obtain the matrix T

T =

√∆

M(1)11

0 0 0

M(1)12√

M(1)11 M

(2)11,22

√M

(1)11

M(2)11,22

0 0

M(2)12,23

√ρ44

√M

(2)11,23

M(2)11,22

√ρ44

√M

(2)11,22

√M

(2)11,22

ρ440

ρ41√ρ44

ρ42√ρ44

ρ43√ρ44

√ρ44

(2.27)

where ∆ = Det(ρ), M(1)ij is the first minor of ρ (the determinant of the 3 × 3 matrix

formed by deleting the ith and jth columns of the ρ matrix), M(2)ij,kl is the second minor

of ρ (the determinant of the 2 × 2 matrix formed by deleting the ith and kth rows andjth and lth columns of the ρ matrix with i 6= j and k 6= l). From the experimental datawe obtain a set of expectation values njk = 〈σj ⊗ σk〉 = Tr((σj ⊗ σk)ρ).

The noise in a complex NMR signal acquired from a single receiving coil usingquadrature detection is uncorrelated (white) and Gaussian [133]. So, it is assumed thatthe experimental noise has a Gaussian probability distribution and the probability ofobtaining a set of measurement results for the set of expectation values njk is

P (n11, · · ·n33) = A

3,3∏j=0,k=0

exp

[−(njk − njk)2

2σ2jk

](2.28)

where A is a normalization constant and σjk is the standard deviation of the mea-sured variable njk (approximately given by

√njk). The next step in the ML estimation

method is to maximize the likelihood that the physical density matrix ρ will give riseto the experimental data njk. Since ln(x) is an increasing function, the maxima ofthe likelihood and the log of the likelihood coincide. Rather than finding the maxi-mum value of the probability P , the optimization problem gets simplified by findingthe maximum of its logarithm. Here we neglect the dependence of the normalizationconstant on t1, t2, . . . , t16 , which only weakly affects the solution for the most likelystate. So, we need to maximize

−∑3,3j=0,k=0

(njk(t1,···t16)−njk(t1,···t16))2

2σ2jk

.

Mathematically, if x0 is a maxima of function f then max(f(x0)) = min(−f(x0)),thus the optimization problem is reduced to finding the minimum of a “likelihoodfunction”

L(t1, · · · t16) =

3,3∑j=0,k=0

(njk(t1, · · · t16)− njk(t1, · · · t16))2

2σ2jk

(2.29)

49

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

Strictly speaking, for a system of NMR coupled qubits, the functional defined inEq. (2.29) is not a “likelihood” since the NMR experiment measures expectation valuesbut can be considered to be a Gaussian approximation of likelihood [119].

For a system of two qubits, the optimum set of variables topt1 , topt2 , . . . , topt16 whichminimizes this likelihood function can be determined using numerical optimizationtechniques. We used the MATLAB routine “lsqnonlin” [134] to find the minimumof the likelihood function. To execute this routine, one requires the initial estimationof the value of t1, t2, . . . , t16. Since a sixteen parameter optimization can be tricky,it is important to use a good initial guess for parameters. A reasonable way is tofirst estimate the state using the standard method, and obtain the values of t′is usingthe Eq. (2.27). Since the state may not be a physically allowed state, the parametersobtained in this manner are not necessarily real. Thus for our initial guess we dropthe imaginary part and use the real parts of each of the t′is as the initial estimate to gointo the optimization routine. We used the same experimentally generated 1√

2(|00〉 +

|01〉) state (as a mean of twenty-five experimental density matrices as described inthe example given in the earlier subsection), and re-computed the density matrix nowusing the ML estimation method, and obtained:

ρML

=

0.488 0.487 + i0.001 0.002− i0.012 −0.002 + i0.012

0.487− i0.001 0.487 0.002− i0.012 −0.002 + i0.0120.002 + i0.012 0.002 + i0.012 0.013 −0.013 + i0.000−0.002− i0.012 −0.002− i0.012 −0.013− i0.000 0.013

(2.30)

The eigen values of this matrix are 0.975±0.001, 0.025±0.001, 0.001±0.000, 0.000±0.000 and are all positive and furthermore Tr(ρ2

ML) = 0.950± 0.004. The errors inthe reconstructed density matrix using the ML estimation method show up in the thirddecimal place. While the density matrix reconstructed using QST was unphysical, theML reconstruction led to a valid density matrix.

2.2 Comparison of quantum state estimation via MLestimation and standard QST schemes

We performed state estimation of several different quantum states of two and threequbits, constructed on an NMR quantum information processor, using the ML esti-mation method. The results were compared every time with the results obtained byreconstruction using the standard QST protocol.

50

2.2 Comparison of quantum state estimation via ML estimation and standardQST schemes

0001

1011 00

0110

11

000110

11 0001 10

11

000110

11 0001

1011

000110

11 0001

1011

0001

1011 00

0110

11

0001

1011 00

0110

11

0001

1011 00

0110

11

0001

10

11 0001

1011

(a)Standard QST

Using ML

(b)Standard QST

Using ML

Real Imaginary Real Imaginary

Figure 2.1: Real (left) and imaginary (right) parts of the experimental tomographs of the(a) |00〉 state, with a computed fidelity of 0.9937 using standard QST and a computedfidelity of 0.9992 using ML method for state estimation. (b) 1√

2(|00〉+ |01〉) state, with a

computed fidelity of 0.9928 using standard QST and a computed fidelity of 0.9991 usingML method for state estimation. The rows and columns are labeled in the computationalbasis ordered from |00〉 to |11〉.

2.2.0.1 Fidelity measure and state estimation

The fidelity measures commonly used in NMR quantum computing are:

1. The fidelity measure F1 is computed by measuring the overlap between theoret-ically expected and experimentally measured states [135]:

F1 =Tr(ρtheoryρexpt)√

(Tr(ρ2theory))

√(Tr(ρ2

expt))(2.31)

2. The fidelity measure F2 is computed by measuring the projection between thetheoretically expected and experimentally measured states using the Uhlmann-Jozsa fidelity measure [136, 137]:

F2 =(Tr(√√

ρtheoryρexpt√ρtheory

))2

(2.32)

where ρtheory and ρexpt denote the theoretically expected and experimentally recon-structed density matrices, respectively.

51

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

000110

11 0001

1011

0001

1011 00

0110

11

00

01

1011 00

0110

11

00

0110

11 0001

1011

(a)

(b)

Standard QST

Using ML

Real Imaginary

Figure 2.2: Real (left) and imaginary (right) parts of the experimental tomographs of theentangled state 1√

2(|01〉 + |10〉) reconstructed (a) using standard QST and (b) using ML

estimation. The fidelities computed using standard QST and using ML method for stateestimation are 0.9933 and 0.9999 respectively. The rows and columns are labeled in thecomputational basis ordered from |00〉 to |11〉.

2.2.1 Comparison of separable states estimation

On a system of two qubits, we began by tomographing a pure state |00〉, as well asa superposition state 1√

2(|00〉 + |01〉) (which can be written as a tensor product of

the first qubit in the |0〉 state and the second qubit in a coherent superposition of the|0〉 and |1〉 states). The mean of ten to twenty-five experimentally determined datamatrices were considered, and the reconstructed density matrices using the ML es-timation method and using the standard QST method are shown as bar tomographsin Figure 2.1, with the states labeled in the computational basis in the order |00〉to |11〉. Using standard QST, the reconstructed |00〉 state had negative eigenvalues:0.994 ± 0.000, 0.073 ± 0.006,−0.001 ± 0.002,−0.066 ± 0.005, and state fidelitywas computed with measure F1 to be 0.9937 and with measure F2 to be 0.9940.Reconstructing the state using ML estimation, we obtained all positive eigenvalues:0.965± 0.001, 0.035± 0.001, 0.000± 0.000, 0.000± 0.000, while state fidelity wascomputed with measure F1 to be 0.9992 and with measure F2 to be 0.9652. For the su-perposition state 1√

2(|00〉+ |01〉), state reconstruction using standard QST led to some

52

2.2 Comparison of quantum state estimation via ML estimation and standardQST schemes

negative eigenvalues: 1.011± 0.002, 0.052± 0.005, 0.016± 0.002,−0.079± 0.004with the fidelity measures F1 and F2 are 0.9928 and 1.0110 respectively. Using MLestimation on the other hand, led to all positive eigenvalues: 0.975 ± 0.001, 0.025 ±0.001, 0.001 ± 0.000, 0.000 ± 0.000 with a state fidelity with measures F1 and F2

are 0.9991 and 0.9745. While state fidelities with F1 measure are nearly the same (orslightly better when calculated after ML reconstruction of the density matrix), we findthat by using the ML estimation method for state estimation, we always obtain a ρwhich is physically valid.

2.2.2 Comparison of entangled states estimation

It has been previously noted [119] that the standard QST protocol frequently leads tounphysical density matrices for entangled multiqubit states. Since entanglement hasbeen posited to lie at the heart of quantum computational speedup, their constructionand estimation is of prime importance. We used the ML estimation method to re-construct two-qubit and three-qubit entangled states and evaluated the efficacy of thisscheme to construct valid density matrices.

The state estimation of a two-qubit entangled Bell state 1√2(|01〉+ |10〉) is shown in

Figure 2.2, using both QST and ML methods for density matrix reconstruction. Usingthe QST protocol for tomography, we obtain the eigenvalues: 0.996± 0.002, 0.018±0.001, 0.005 ± 0.001,−0.019 ± 0.002 with the first eigenvalue being negative, andwith a computed fidelity with measure F1 and F2 are 0.9933 and 0.9964 respectively.Using ML method for state estimation leads to all positive eigenvalues: 0.993 ±0.002, 0.006±0.001, 0.001±0.000, 0.001±0.000 with a computed state fidelity withmeasures F1 and F2 are 0.9999 and 0.9926 respectively.

Recently, schemes to construct maximally entangled three-qubit states from a genericstate have been implemented on an NMR quantum information processor [138, 139].We used these schemes to construct the maximally entangled W state on a systemof three qubits |W 〉 = 1√

3(i|001〉 + |010〉 + |100〉), and thereafter performed state

estimation using both the standard QST and the ML methods. The experimentally re-constructed tomographs are depicted in Figure 2.3, with the states being labeled in thecomputational basis ordered from |000〉 to |111〉. After QST tomography on this three-qubit state, we obtained the eigenvalues: 0.939, 0.104, 0.078, 0.054, -0.002, -0.042,-0.061, -0.071, and a calculated state fidelity F1 is 0.9759 and F2 is 0.9399. Afterperforming state estimation using the ML method, the eigenvalues turned out to be allpositive: 0.919, 0.036, 0.027, 0.008, 0.006, 0.002, 0.002, 0.000, with a calculatedstate fidelity with F1 is 0.9968 and with F2 is 0.9191.

A topic of much research focus here is the accurate measurement of the decay ofmultiqubit entanglement with time. To study this, we performed state estimation of the

53

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

000001

010011

100101

110111

000001

010011

100101

110111

000001

010011

100101

110111

000001

010011

100101

110111

000

001010011100

101110

111

000

001010011100

101110

111

000

001010011100

101110

111

000

001010011100

101110

111

(a)

(b)

Standard QST

Real Imaginary

Using ML

Figure 2.3: Real (left) and imaginary (right) parts of the experimental tomographs of thethree-qubit maximally entangled state |W 〉 = 1√

3(i|001〉+ |010〉+ |100〉), reconstructed

(a) using standard QST with a computed fidelity of 0.9833 and (b) using ML estimationwith a computed fidelity of 0.9968. The rows and columns are labeled in the computationalbasis ordered from |000〉 to |111〉.

entangled two-qubit state 1√2(|00〉+ |11〉) using both QST and ML protocols. The bar

tomographs of the reconstructed density matrices at different times (T=0, 0.04, 0.08,0.12, 0.16 sec) are shown in Figure 2.4.

The amount of entanglement that remains in the state after a certain time can bequantified by an entanglement parameter denoted by η [63]. Since we are dealingwith mixed bipartite states of two-qubit, all entangled states will be negative underpartial transpose (NPT). For such NPT states, a reasonable measure of entanglementis the minimum eigenvalue of the partially transposed density operator. For a givenexperimentally tomographed density operator ρ, we obtain ρPT by taking a partialtranspose with respect to one of the qubits. The entanglement parameter η for the stateρ in terms of the smallest eigenvalue Eρ

Min of ρPT is defined as [63].

η =

−Eρ

Min if EρMin < 0

0 if EρMin > 0

(2.33)

For the calculation of the entanglement parameter η, the purity factor ε ≈ 1, since,

54

2.2 Comparison of quantum state estimation via ML estimation and standardQST schemes

H

000110

11

000110

11

0001

1011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

0001

1011

000110

11

000110

11

000110

11000110

11

0001

1011

0001

1011

0001

1011

0001

10

11

000110

11

000110

11

000110

1100

0110

11

0001

1011

00

0110

11

0001

1011

0001

1011

0001

1011

0001

1011

00011011

0001

10 11

0001

1011

0001

1011

0001

1011

0001

1011

00011011

000110

11

0001101100

0110

11

00

0110

11

0001

1011

00

011011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 0.04s

T = 0.08s

T = 0.12s

T = 0.16s

Standard QST Using ML

Real Imaginary Real Imaginary

Figure 2.4: Real (left) and imaginary (right) parts of the experimental tomographsof the (a) 1√

2(|00〉 + |11〉) state. Tomographs (b)-(e) depict the state at T =

0.04, 0.08, 0.12, 0.16s, with the tomographs on the left and the right representing thestate estimated using standard QST and using ML estimation, respectively. The rows andcolumns are labeled in the computational basis ordered from |00〉 to |11〉.

55

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

we are considering the subensemble of spins that have been prepared in the pseudop-ure state, and in which the spins are genuinely entangled as described in Chap. 1,Sec. 1.3.2.

A plot of the entanglement parameter η with time is depicted in Figure 2.5, for thetwo-qubit maximally entangled Bell state 1√

2(|00〉 + |11〉), estimated using both stan-

dard QST and the ML method. As can be seen from Figure 2.5, the QST method ledto negative eigenvalues in the reconstructed (unphysical) density matrix and hence anoverestimation of the entanglement parameter quantifying the residual entanglementin the state. The ML estimation method on the other hand, by virtue of its leading toa physical density matrix reconstruction every time, gives us a true measure of resid-ual entanglement, and hence can be used to quantitatively study the decoherence ofmultiqubit entanglement. The difference between the two methods is statistically sig-nificant, as is evident from the graph, where the error bars are shown on each datapoint. The error is in fact very small, compared to the anomaly in the estimation of theentanglement.

η(E

ntaglemen

t)

Time(s)

0

0.1

0.2

0.3

0.4

0.5 Using QSTUsing ML

0 0.1 0.2 0.3 0.4

Figure 2.5: Plot of the entanglement parameter η with time, using standard QST and MLprotocols for state reconstruction, computed for the 1√

2(|00〉+ |11〉) state.

2.3 ConclusionsWe used the maximum likelihood estimation method for state estimation on an NMRquantum information processor, to circumvent the problem of unphysical density ma-trices that occur due to statistical errors while using the standard QST protocol. Ithas been previously shown that state reconstruction using QST, of entangled statesand other fragile quantum states are particularly susceptible to errors, and can lead tounphysical density matrices for such states. We showed that the experimental den-sity matrices reconstructed for entangled states of two and three qubits using the ML

56

2.3 Conclusions

estimation method are always positive, definite and normalized. The state fidelitiescomputed using fidelity measure F1 are comparable for standard QST and using MLestimation. However if we use the fidelity measure F2, the obtained state fidelity withML estimation is less than the standard QST. The advantage of the ML estimationmethod is that it always leads to a valid density matrix and hence is a better estimatorof the state of the quantum system. In the rest of this thesis we will use the ML esti-mation method for quantum state tomography and F2 as the measure for state fidelity.

57

2. State tomography on an NMR quantum information processor via maximumlikelihood estimation

58

Chapter 3

Experimental protection of quantumstates via a super-Zeno scheme

3.1 Introduction

Unwanted changes occur in quantum systems due to their interaction with the envi-ronment which leads to state degradation. This process, where quantum coherenceis lost, is known as decoherence. Decoherence is a major obstacle in implementingquantum computing and quantum information processing schemes. In order to tacklethis problem a number of techniques have been developed. The dynamical decou-pling (DD) approach can be used to effectively decouple the system from its envi-ronment and thus decoherence can be suppressed [140, 141]. The idea that frequentmeasurements which project a quantum system back to its initial state can be used tonot let the state evolve is known as the quantum Zeno effect [142, 143, 144, 145]. Ifthe measurements project the system back into a finite-dimensional subspace that in-cludes the initial state, the state evolution remains confined within this subspace andthe subspace can be protected against leakage of population using a quantum Zenostrategy [146, 147]. Zeno-like schemes have been used for error prevention [148], andto enhance the entanglement of a state and bring it to a Bell state, even after entan-glement sudden death [149, 150]. It has been shown that under certain assumptions,the Zeno effect can be realized with weak measurements and can protect an unknownencoded state against environmental effects [151]. All these strategies are based onour knowledge of the system-environment interaction and the state that needs to bepreserved.

In a situation where the system state is known but there is no knowledge aboutits interaction with the environment, decoherence can be tackled by an interesting

59

3. Experimental protection of quantum states via a super-Zeno scheme

quantum Zeno-type strategy for state preservation. This is achieved by using a se-quence of non-periodic short duration pulses and is called the super-Zeno scheme [60].The super-Zeno scheme does not assume any Hamiltonian symmetry, does not involveprojective quantum measurements and achieves a significant reduction of the leakageprobability as compared to standard Zeno-based preservation schemes. The super-Zeno scheme for state protection is similar to universal dynamical decoupling schemesfor multi-qubit states [61]. The only difference in the super-Zeno scheme is that non-periodic short duration between the unitary kicks are optimized numerically by mini-mizing the leakage probability. Similar schemes involving dynamical decoupling havebeen devised to suppress qubit pure dephasing and relaxation [111, 152]. Anotherscheme to preserve entanglement in a two-qubit spin-coupled system has been con-structed, which unlike the super-Zeno scheme, is based on a sequence of operationsperformed periodically on the system in a given time interval [153].

There are several experimental implementations of the quantum Zeno phenomenon,including suppressing unitary evolution driven by external fields between the two statesof a trapped ion [154], in atomic systems [155] and suppressing failure events in a lin-ear optics quantum computing scheme [156]. Decoherence control in a superconduct-ing qubit system has been proposed using the quantum Zeno effect [157]. Unlike thesuper-Zeno and dynamical decoupling schemes that are based on unitary pulses, thequantum Zeno effect achieves suppression of state evolution using projective measure-ments. The quantum Zeno effect was first demonstrated in NMR by a set of symmetricπ pulses [158], wherein pulsed magnetic field gradients and controlled-NOT gateswere used to mimic projective measurements. The entanglement preservation of a Bellstate in a two-spin system in the presence of anisotropy was demonstrated using apreservation procedure involving free evolution and unitary operations [28]. An NMRscheme to preserve a separable state was constructed using the super-Zeno scheme andthe state preservation was found to be more efficient as compared to the standard Zenoscheme [62]. The quantum Zeno effect was used to stabilize superpositions of statesof NMR qubits against dephasing, using an ancilla to perform the measurement [159].Entanglement preservation based on a dynamic quantum Zeno effect was demonstratedusing NMR wherein frequent measurements were implemented through entangling thetarget and measuring qubits [160].

In this chapter we demonstrate the use of the super-Zeno scheme, while the im-plementation of dynamical decoupling schemes will be taken up in later chapters ofthe thesis. Two applications of the super-Zeno scheme are described in this chap-ter: (i) Preservation of a state by freezing state evolution (one-dimensional subspaceprotection) and (ii) Subspace preservation by preventing leakage of population to anorthogonal subspace (two-dimensional subspace protection). Both kinds of protectionschemes are experimentally demonstrated on separable as well as on maximally en-

60

3.2 The super-Zeno scheme

tangled two-qubit state. One-dimensional subspace protection is demonstrated on theseparable |11〉 state and on the maximally entangled 1√

2(|01〉 − |10〉) (singlet) state.

Two-dimensional subspace preservation is demonstrated by choosing the |01〉, |10〉subspace in the four-dimensional Hilbert space of two qubits, and implementing thesuper-Zeno subspace preservation protocol on three different states, namely |01〉, |10〉and 1√

2(|01〉 − |10〉) (singlet) states. Complete state tomography via maximally likeli-

hood estimation as described in Chapter 2 is utilized to compute experimental densitymatrices at several time points. State fidelities at these time points were computed toevaluate how closely the states resemble the initially prepared states, with and with-out super-Zeno protection. The success of the super-Zeno scheme in protecting statesin the two-dimensional subspace spanned by |01〉, |10〉 is evaluated by computinga leakage parameter, which computes leakage to the orthogonal subspace spanned by|00〉, |11〉. For entangled states, an additional entanglement parameter is constructedto quantify the residual entanglement in the state over time. State fidelities, the leakageparameter and the entanglement parameter are plotted as a function of time, to quantifythe performance of the super-Zeno scheme.

3.2 The super-Zeno schemeThe super-Zeno algorithm to preserve quantum states has been developed along linessimilar to bang-bang control schemes, and limits the quantum system’s evolution toa desired subspace using a series of unitary kicks [60]. A finite-dimensional Hilbertspace H can be written as a direct sum of two orthogonal subspaces P and Q. Thesuper-Zeno scheme involves a unitary kick J, which can be constructed as

J = Q−P (3.1)

where P,Q are the projection operators onto the subspaces P,Q respectively. Theaction of this specially crafted pulse J on a state |ψ〉 ∈ H is as follows:

J|ψ〉 = −|ψ〉, |ψ〉 ∈ P

J|ψ〉 = |ψ〉, |ψ〉 ∈ Q (3.2)

where P is the subspace being preserved [60].The basic aim of this scheme is to design a sequence of appropriately spaced in-

verting pulses, such that if the system is initially in a state |ψ〉 ∈ P, then the leakageof the system state over time to Q after this pulse sequence is minimum. The invert-ing pulse J produces destructive interference of quantum amplitudes and reduces thetransition rate from the P subspace to Q. Let the system be prepared in a general state

61

3. Experimental protection of quantum states via a super-Zeno scheme

|ψ(0)〉 = |p〉 ∈ P, the system Hamiltonian H is bounded and the unitary operatorcorresponding to the evolution for a time interval t is given by

U(t) = e−ιHt/~. (3.3)

For simplicity we use natural units and ~ = 1. Then, the state |ψ(0)〉 after time t is

|ψ(t)〉 = e−ιHt|ψ(0)〉 (3.4)

The amplitude of the system to be in a state |q〉 ∈ Q after a time interval t is given by

〈q|e−ιHt|ψ(0)〉 (3.5)

However, if we evolve the system for the half interval t/2, subject the system to aninverting pulse J, and then further evolve by a time t/2, the above amplitude turns outto be

〈q|e−ιHt/2Je−ιHt/2|ψ(0)〉 (3.6)

For a small time interval t and the unitary operator U(t) satisfying U(t) = I + O(t)

〈q|(I +−ιHt

2

)J

(I +−ιHt

2

)|ψ(0)〉 (3.7)

= 〈q|J|p〉+ 〈q|J−ιHt

2+−ιHt

2J|p〉+ O(t2)

= 〈q|J−ιHt

2|p〉+ 〈q|−ιHt

2J|p〉+ O(t2)

= 〈q|−ιHt

2|p〉 − 〈q|−ιHt

2|p〉+ O(t2)

Due to the action of J, destructive interference takes place between the two amplitudesof O(t) in Eq.(3.7) and we are left with terms of O(t2) term and higher, which arenegligible for a small time interval t. In general, if the matrix elements of U(t) i.eUqp and Upq are of O(t(r)) then the transition amplitude

〈q|U(t).J.U(t)|p〉 = −∑p′∈P

Uqp′Up′p +∑q′∈Q

Uqq′Uq′p −UqpUpp −UqqUqp

= O(tr+1)−UqpUpp −UqqUqp = O(tr+1) (3.8)

where a precise cancellation of the amplitude of O(tr) term occurs [60].

62

3.2 The super-Zeno scheme

It is important to note that V(t) = U(t).J.U(t) tends to J as t → 0 but V(t)2

tends to I and if V(t) = J + O(t) and V(t)qp = O(tr)

〈q|V(t)2|p〉 = −∑p′′∈P

V(t)qp′V(t)p′p +∑q′∈Q

V(t)qq′V(t)q′p

−V(t)qpV(t)pp −V(t)qqV(t)qp

= O(tr+1) + V(t)qp(V(t)qq + V(t)pp)

〈q|V(t)2|p〉 = O(tr+1) (3.9)

Hence, if for U0(t) the transition amplitude to an orthogonal state is proportionalto t, then for the operator U1(t) = U0(t).J.U0(t) from Eq. (3.8), the transition am-plitude is O(t2). For U2

1 using Eq. (3.9), the transition amplitude is O(t3) [60]. Usingthis result, it is straight forward to construct a pulse sequence where the transition am-plitude is O(tr) for any positive integer r by recursion. Defining operators Um by therecursion relations

Um+1(t) = Um(t/2) J Um(t/2), for m even

= Um(t/2) Um(t/2), for m odd (3.10)

with U0 = U0(t). Then, by induction, it follows that the transition amplitude 〈q|Um|p〉is of order O(tm+1).

Um(t) can be written explicitly as a product of U0(t/2m)’s and J’s. For example

U1(t) = U0(t/2)JU0(t/2),

U2(t) = [U0(t/4)JU0(t/4)]2,

U3(t) = [U0(t/8)JU0(t/8)]2J[U0(t/8)JU0(t/8)]2. (3.11)

If Nm are the number of pulses used in the sequence Um, then it is easily verifiedthat

Nm = (2m+1 − 2)/3, for m even,

= (2m+1 − 1)/3, for m odd. (3.12)

The leakage probability Lm from subspace P to subspace Q for operator Um ap-plied for a total time T is

Lm =∑q∈Q|〈q|Um|p〉|2 ≤ 2−m(m+1)[ET ]2m+2 (3.13)

63

3. Experimental protection of quantum states via a super-Zeno scheme

where E = |H| is the norm of the Hamiltonian as defined by

|H| = sup

ψ||H|ψ〉|| / || |ψ〉|| (3.14)

where the sup notation is defined as (for any function f and constraint set X)

sup

x ∈ Xf(x) ≥ f(v) ∀ v ∈ X (3.15)

The total number of inverting pulses Nm required to keep the leakage probabilityless than ε up to time T grows as 2

3ET 2log2(E2T 2/ε)

1/2

[60]. It is possible to decrease therequired number of pulses significantly by allowing the intervals between the invertingpulses to be varied continuously independent of each other.

For a givenN , further improvements are possible and one can express 〈q|WN(t)|p〉as a Taylor series in powers of t. The coefficients of different powers of t are sums ofmatrix-elements of the type 〈q|Hn1JHn2J..|p〉, with coefficients that are polynomialsof xj. Thus, the total super-Zeno sequence for N pulses is given by

WN(t) = U(xN+1t)J . . .JU(x2t)JU(x1t) (3.16)

where U denotes unitary evolution under the system Hamiltonian and xit is the timeinterval between the ith and (i + 1)th pulse. The sequence xit of time intervalsbetween pulses is optimized such that if the system starts out in the subspace P, aftermeasurement the probability of finding the system in the orthogonal subspace Q isminimized. In this work we used four inverting pulses interspersed with five unequaltime intervals in each repetition of the preserving super-Zeno sequence. The optimizedsequence is given by xi = β, 1/4, 1/2 − 2β, 1/4, β with β = (3 −

√5)/8, i =

1 . . . 5 and t is a fixed time interval (we used the xi as worked out in Ref. [60]).The explicit form of the unitary kick J depends on the subspace that needs to be

preserved, and in the following section, we implement several illustrative examples forboth separable and entangled states embedded in one- and two-dimensional subspacesof two qubits.

3.3 Experimental implementations of super-Zeno scheme

3.3.1 NMR system details

The two protons of the molecule cytosine encode the two qubits. The two-qubit molec-ular structure, system parameters and NMR spectra of the pseudopure and thermal ini-tial states are shown in Figs. 3.1(a)-(c). The Hamiltonian of a two-qubit system in the

64

3.3 Experimental implementations of super-Zeno scheme

Figure 3.1: (a) Molecular structure of cytosine with the two qubits labeled as H1 andH2 and tabulated system parameters with chemical shifts νi and scalar coupling J12 (inHz) and relaxation times T1 and T2 (in seconds) (b) NMR spectrum obtained after a π/2readout pulse on the thermal equilibrium state. The resonance lines of each qubit arelabeled by the corresponding logical states of the other qubit and (c) NMR spectrum of thepseudopure |00〉 state.

rotating frame is given by

H = −(ω1 − ωrf )I1z − (ω2 − ωrf )I2z + 2πJ12I1zI2z (3.17)

where ωi = 2πνi are the chemical shift of the spins in rad s−1, ωrf reference chemicalshift of rotating frame, and J12 is the spin-spin coupling constant. An average longitu-dinal T1 relaxation time of 7.4 s and an average transverse T2 relaxation time of 3.25 swas experimentally measured for both the qubits. The experiments were performed atan ambient temperature of 298 K on a Bruker Avance III 600 MHz NMR spectrometerequipped with a QXI probe. The two-qubit system was initialized into the pseudopurestate |00〉 using the spatial averaging technique [24], with the density operator givenby

ρ00 =1− ε

4I + ε|00〉〈00| (3.18)

with a thermal polarization ε ≈ 10−5 and I being a 4 × 4 identity operator. The ex-perimentally created pseudopure state |00〉 was tomographed with a fidelity of 0.99.The pulse propagators for selective excitation were constructed using the GRAPE al-

65

3. Experimental protection of quantum states via a super-Zeno scheme

gorithm [25] to design the amplitude and phase modulated rf profiles. Selective ex-citation was typically achieved with pulses of duration 1 ms. Numerically generatedGRAPE pulse profiles were optimized to be robust against rf inhomogeneity and hadan average fidelity of ≥ 0.99. All experimental density matrices were reconstructedusing a quantum state tomography via maximum likelihood protocol (Chapter 2). Thefidelity of an experimental density matrix was computed using Eq. (2.32).

3.3.2 Super-Zeno scheme for state preservationWhen the subspace P is a one-dimensional subspace, and hence consists of a singlestate, the super-Zeno scheme becomes a state preservation scheme.

3.3.2.1 Preservation of product states:

We begin by implementing the super-Zeno scheme on the product state |11〉 of twoqubits, where the Hilbert space can be decomposed as a direct sum of the subspacesP = |11〉 and Q = |00〉, |01〉, |10〉). The super-Zeno pulse J to protect the state|11〉 ∈ P is given by Eqn. (3.1):

J = I − 2|11〉〈11| (3.19)

with the corresponding matrix form

J =

1 0 0 00 1 0 00 0 1 00 0 0 −1

(3.20)

The super-Zeno circuit to preserve the |11〉 state, and the corresponding NMR pulsesequence is given in Fig. 3.2. The controlled-phase gate (Z) in Fig. 3.2(a) whichreplicates the unitary kick J for preservation of the |11〉 state is implemented using aset of three sequential gates: two Hadamard gates on the second qubit sandwichinga controlled-NOT gate (CNOT12), with the first qubit as the control and the secondqubit as the target. The ∆i time interval in Fig. 3.2(a) is given by ∆i = xit, with xi asdefined in Eq. (3.16). The five ∆i time intervals were worked to be 0.095 ms, 0.25 ms,0.3 ms, 0.25 ms, and 0.095 ms respectively, for t = 1 ms. One run of the super-Zenocircuit (with four inverting Js and five ∆i time evolution periods) takes approximately300 ms and the entire preserving sequence WN(t) in Eq. (3.16) was applied 30 times.The final state of the system was reconstructed using state tomography and the realand imaginary parts of the tomographed experimental density matrices without anypreservation and after applying the super-Zeno scheme, are shown in Fig. 3.3. The

66

3.3 Experimental implementations of super-Zeno scheme

|0〉

|0〉∆1 ∆2 ∆3 ∆4 ∆5

∆5∆1

|11〉

Z Z Z Z

s s s s N

N

|0〉 . . .

. . .|0〉

x x -x-x

-x

-x

y

x

-y

τ12

x -x-x

-x

-x

y

x

-y

τ12

G

(a)

(b)

T

O

M

O

G

R

A

P

H

Y

Figure 3.2: (a) Quantum circuit for preservation of the state |11〉 using the super-Zenoscheme. ∆i = xit, (i = 1...5) denote time intervals punctuating the unitary operationblocks. Each unitary operation block contains a controlled-phase gate (Z), with the first(top) qubit as the control and the second (bottom) qubit as the target. The entire schemeis repeated N times before measurement (for our experiments N = 30). (b) Block-wisedepiction of the corresponding NMR pulse sequence. A z-gradient is applied just beforethe super-Zeno pulses, to clean up undesired residual magnetization. The unfilled andblack rectangles represent hard 1800 and 900 pulses respectively, while the unfilled andgray-shaded conical shapes represent 1800 and 900 pulses (numerically optimized usingGRAPE) respectively; τ12 is the evolution period under the J12 coupling. Pulses are la-beled with their respective phases and unless explicitly labeled, the phase of the pulses onthe second (bottom) qubit are the same as those on the first (top) qubit.

67

3. Experimental protection of quantum states via a super-Zeno scheme

0001

1011

0001

1011

0001

1011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

00

0110

11

00

011011

00

011011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

00

0110

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0001

1011

00

011011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

00

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11

0001

1011

00

0110

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0110

11

0001

1011

0001

1011

0001

1011

00

01

10

11

00

01

10

110001

10

11

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 0.61s

T = 3.03s

T = 5.46s

T = 7.28s

No Super Zeno With Super Zeno

Figure 3.3: Real (left) and imaginary (right) parts of the experimental tomographs ofthe (a) |11〉 state, with a computed fidelity of 0.99. (b)-(e) depict the state at T =

0.61, 3.03, 5.46, 7.28 s, with the tomographs on the left and the right representing thestate without and after applying the super-Zeno preserving scheme, respectively. The rowsand columns are labeled in the computational basis ordered from |00〉 to |11〉.

68

3.3 Experimental implementations of super-Zeno scheme

initial |11〉 state (at time T = 0 s) was created (using the spatial averaging scheme)with a fidelity of 0.99. The tomographs (on the right in Fig. 3.3) clearly show that stateevolution has been frozen with the super-Zeno scheme.

3.3.2.2 Preservation of entangled states

We next apply the super-Zeno scheme to preserve an entangled state in our system oftwo qubits. We chose the singlet state 1√

2(|01〉 − |10〉) as the entangled state to be

preserved. It is well known that entanglement is an important but fragile resource forquantum information processing and constructing schemes to protect entangled statesfrom evolving into other states, is of considerable interest in quantum informationprocessing [1].

We again write the Hilbert space as a direct sum of two subspaces: the subspacebeing protected and the subspace orthogonal to it. In this case, the one-dimensionalsubspace P being protected is

P =

1√2

(|01〉 − |10〉)

(3.21)

and the orthogonal subspace Q into which one would like to prevent leakage is

Q =

1√2

(|01〉+ |10〉), |00〉, |11〉

(3.22)

The super-Zeno pulse to protect the singlet state as constructed using Eq. (3.1) is:

J = I − (|01〉〈01|+ |10〉〈10| − |01〉〈10| − |10〉〈01|) (3.23)

with the corresponding matrix form:

J =

1 0 0 00 0 1 00 1 0 00 0 0 1

(3.24)

The quantum circuit and the NMR pulse sequence for preservation of the singletstate using the super-Zeno scheme are given in Fig. 3.4. Each J inverting pulse inthe unitary block in the circuit is decomposed as a sequential operation of three non-commuting controlled-NOT gates: CNOT12-CNOT21-CNOT12, where CNOTij denotesa controlled-NOT with i as the control and j as the target qubit. The five ∆i time in-tervals were worked to be 0.95 ms, 2.5 ms, 3 ms, 2.5 ms, and 0.95 ms respectively, fort = 10 ms. One run of the super-Zeno circuit (with four inverting Js and five ∆i time

69

3. Experimental protection of quantum states via a super-Zeno scheme

|0〉

|0〉∆1 ∆2 ∆3 ∆4 ∆5

1 √2(|01〉−

|10〉) s

ss s

ss s

ss s

ss N

|0〉

G

τ12 τ12 τ12 τ12

|0〉

-y

y

x -xx x -x x x -y x -xy

(a)

(b)

Figure 3.4: (a) Quantum circuit for preservation of the singlet state using the super-Zenoscheme. ∆i, (i = 1...5) denote time intervals punctuating the unitary operation blocks.The entire scheme is repeatedN times before measurement (for our experimentsN = 10).(b) NMR pulse sequence corresponding to one unitary block of the circuit in (a). A z-gradient is applied just before the super-Zeno pulses, to clean up undesired residual mag-netization. The unfilled rectangles represent hard 1800 pulses, the black filled rectanglesrepresenting hard 900 pulses, while the shaded shapes represent numerically optimized(using GRAPE) pulses and the gray-shaded shapes representing 900 pulses respectively;τ12 is the evolution period under the J12 coupling. Pulses are labeled with their respectivephases and unless explicitly labeled, the phase of the pulses on the second (bottom) qubitare the same as those on the first (top) qubit.

70

3.3 Experimental implementations of super-Zeno scheme

0001

1011

000110

11

00

0110

11

00

0110

110001

1011

0001

10

11

0001

1011

0001

1011

0001

1011

000110

11

0001

1011

00

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110001

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1011

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011011

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11

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1011

0001

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1011

00

01

10

11

0001

1011

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 0.85s

T = 2.54s

T = 4.24s

T = 5.93s

No Super Zeno With Super Zeno

Figure 3.5: Real (left) and imaginary (right) parts of the experimental tomographs of the(a) 1√

2(|01〉−|10〉) (singlet) state, with a computed fidelity of 0.99. (b)-(e) depict the state

at T = 0.85, 2.54, 4.24, 5.93 s, with the tomographs on the left and the right representingthe state without and after applying the super-Zeno preserving scheme, respectively. Therows and columns are labeled in the computational basis ordered from |00〉 to |11〉.

71

3. Experimental protection of quantum states via a super-Zeno scheme

evolution periods) takes approximately 847 ms and the entire super-Zeno preservingsequence WN(t) in Eq. (3.16), is applied 10 times.

The singlet state was prepared from an initial pseudopure state |00〉 by a sequenceof three gates: a non-selective NOT gate (hard πx pulse) on both qubits, a Hadamardgate and a CNOT12 gate. The singlet state thus prepared was computed to have afidelity of 0.99. The effect of chemical shift evolution during the delays was compen-sated for with refocusing pulses. The final singlet state has been reconstructed usingstate tomography, and the real and imaginary parts of the tomographed experimentaldensity matrices without any preservation and after applying the super-Zeno scheme,are shown in Fig. 3.5. As can be seen from the experimental tomographs in Fig. 3.5, theevolution of the singlet state is almost completely frozen by the super-Zeno sequenceupto nearly 6 s, while without any preservation the state has leaked into the orthogonalsubspace within 2 s.

0 2 4 6 80

0.2

0.4

0.6

0.8

1(a)

Time(s)

F(F

idelity)

Without SZ

With SZ

0 2 4 6 80

0.2

0.4

0.6

0.8

1(b)

Time(s)

F(F

idelity)

Without SZ

With SZ

Figure 3.6: Plot of fidelity versus time of (a) the |11〉 state and (b) the 1√2(|01〉−|10〉) (sin-

glet) state, without any preserving scheme and after the super-Zeno preserving sequence.The fidelity of the state with the super-Zeno preservation remains close to 1.

3.3.2.3 Estimation of state fidelity

The plots of state fidelity versus time are shown in Fig. 3.6 for the state |11〉 and thesinglet state, with and without the super-Zeno preserving sequence. The signal attenu-ation has been ignored and the deviation density matrix is renormalized at every pointand the state fidelity is estimated using the definition in Eq. (2.32). Renormalization isperformed since our focus here is on the quantum state of the spins contributing to the

72

3.3 Experimental implementations of super-Zeno scheme

signal and not on the number per se of participating spins [161]. The norm of deviationdensity matrix is proportional to the intensity of the tomographed spectra and is pro-portional to the number of spins contributing to the signal intensity. Due to super-Zenopreserving pulses, an attenuation in signal intensity takes place and we renormalize theintensity of the of the tomographed spectra in order to compensate for this attenuation.The plot of signal intensity versus time is shown in Fig. 3.7 for the state |11〉 and thesinglet state, with the super-Zeno preserving sequence. The exponential intensity de-cay constant is 0.136±0.006 s−1 for the |11〉 state and 0.393±0.013 s−1 for the singletstate.

The plots in Fig. 3.6 and the tomographs in Fig. 3.3 and Fig. 3.5 show that withsuper-Zeno protection, the state remains confined to the |11〉 (singlet) part of the den-sity matrix, while without the protection scheme, the state leaks into the orthogonalsubspace. As seen from both plots in Fig. 3.6, the state evolution of specific states canbe arrested for quite a long time using the super-Zeno preservation scheme, while leak-age probability of the state to other states in the orthogonal subspace spanned by Q isminimized. A similar renormalization procedure is adopted in the subsequent sectionswhere we plot the leak fraction and entanglement parameters (Fig. 3.12 and Fig. 3.13).

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

I(Signalintensity)

Time(s)

|11〉1√2(|01〉 − |10〉)

Figure 3.7: Plot of signal intensity versus time of the |11〉 state and the 1√2(|01〉 − |10〉)

(singlet) state, after the super-Zeno preserving sequence.

3.3.3 Super-Zeno for subspace preservationWhile in the previous subsection, the super-Zeno scheme was shown to be effectivein arresting the evolution of a one-dimensional subspace (as applied to the cases of a

73

3. Experimental protection of quantum states via a super-Zeno scheme

|ψ〉ǫ

|01〉,|10〉

|0〉

|0〉∆1 ∆2 ∆3 ∆4 ∆5

∆5

|ψ〉ǫ

|01〉,|10〉

Uzz Uzz Uzz Uzz

N

N

|0〉

|0〉

x -x x -x x -x x -x

∆1 τ12 ∆2 τ12∆3τ12 ∆4 τ12

G

(a)

(b)

T

O

M

O

G

R

A

P

H

Y

Figure 3.8: (a) Quantum circuit for preservation of the 01, 10 subspace using the super-Zeno scheme. ∆i, (i = 1...5) denote time intervals punctuating the unitary operationblocks. The entire scheme is repeated N times before measurement (for our experimentsN = 30). (b) NMR pulse sequence corresponding to the circuit in (a). A z-gradient isapplied just before the super-Zeno pulses, to clean up undesired residual magnetization.The unfilled rectangles represent hard 1800 pulses; τ12 is the evolution period under theJ12 coupling. Pulses are labeled with their respective phases.

74

3.3 Experimental implementations of super-Zeno scheme

product and an entangled state), the scheme is in fact more general. For example, ifwe choose a two-dimensional subspace in the state space of two qubits and protect itby the super-Zeno scheme, then any state in this subspace is expected to remain withinthis subspace and not leak into the orthogonal subspace. While the state can meanderwithin this subspace, its evolution out of the subspace is frozen.

We now turn to implementing the super-Zeno scheme for subspace preservation, byconstructing the J operator to preserve a general state embedded in a two-dimensionalsubspace. We choose the subspace spanned by P = |01〉, |10〉 as the subspace to bepreserved, with its orthogonal subspace now being Q = |00〉, |11〉. It is worth notingthat within the subspace being protected, we have product as well as entangled states.

The super-Zeno pulse J to protect a general state |ψ〉 ∈ P can be constructed as:

J = I − 2 (|01〉〈01|+ |10〉〈10|) (3.25)

with the corresponding matrix form

J =

1 0 0 00 −1 0 00 0 −1 00 0 0 1

(3.26)

The quantum circuit and corresponding NMR pulse sequence to preserve a generalstate in the |01〉, |10〉 subspace is given in Fig. 3.8. The unitary kick (denoted as Uzzin the unitary operation block in Fig. 3.8(a)) is implemented by tailoring the gate timeto the J-coupling evolution interval of the system Hamiltonian, sandwiched by non-selective π pulses (NOT gates), to refocus undesired chemical shift evolution duringthe action of the gate. The five ∆i intervals were worked to be 0.95 ms, 2.5 ms, 3 ms,2.5 ms and 0.95 ms respectively, for t = 10 ms. One run of the super-Zeno circuit(with four inverting Js and five ∆i time evolution periods) takes approximately 288ms and the entire super-Zeno preserving sequence WN(t) in Eqn. (3.16), is applied 30times.

3.3.3.1 Preservation of product states in the subspace

We implemented the subspace-preserving scheme on two different (separable) states|01〉 and |10〉 in the subspace P. The efficacy of the preserving unitary is verified bytomographing the experimental density matrices at different time points and computingthe state fidelity. Both the |01〉 and |10〉 states remain within the subspace P and donot leak out to the orthogonal subspace Q = |00〉, |11〉.

The final |10〉 state has been reconstructed using state tomography, and the realand imaginary parts of the experimental density matrices without any preservation and

75

3. Experimental protection of quantum states via a super-Zeno scheme

00

0110

11

000110

11

0001

1011

0001

101100

01 10

11

0001

1011

0001

1011

0001

1011

0001

1011

000110

11

00

011011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

0001

1011

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0110

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1011

0001

1011

0001

1011

0001

1011

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1011

00

0110

11

00

0110

110001

1011

0001

1011

0001

1011

0001

1011

00

01

1011

0001

1011

0001

1011

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 1.15s

T = 3.45s

T = 5.75s

T = 7.48s

No Super Zeno With Super Zeno

Figure 3.9: Real (left) and imaginary (right) parts of the experimental tomographs of the(a) |10〉 state in the two-dimensional subspace 01, 10, with a computed fidelity of 0.98.(b)-(e) depict the state at T = 1.15, 3.45, 5.75, 7.48 s, with the tomographs on the leftand the right representing the state without and after applying the super-Zeno preservingscheme, respectively. The rows and columns are labeled in the computational basis orderedfrom |00〉 to |11〉.

76

3.3 Experimental implementations of super-Zeno scheme

00

0110

11

000110

11

0001

1011

0001

101100

01 10

11

0001

1011

0001

1011

0001

1011

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000110

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00

011011

0001

101100

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1011

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1011

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1011

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00

0110

11

00

0110

110001

1011

0001

1011

0001

1011

0001

1011

00

01

1011

0001

1011

0001

1011

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 1.15s

T = 3.45s

T = 5.75s

T = 7.48s

No Super Zeno With Super Zeno

Figure 3.10: Real (left) and imaginary (right) parts of the experimental tomographs of the(a) |01〉 state in the two-dimensional subspace 01, 10, with a computed fidelity of 0.99.(b)-(e) depict the state at T = 1.15, 3.45, 5.75, 7.48 s, with the tomographs on the leftand the right representing the state without and after applying the super-Zeno preservingscheme, respectively. The rows and columns are labeled in the computational basis orderedfrom |00〉 to |11〉.

77

3. Experimental protection of quantum states via a super-Zeno scheme

0001

1011

00011011

00

011011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

000110

11

0001

1011

00

011011

0001

101100

0110

11

0001

10 11

0001

1011

0001

1011

00

0110

11

00

011011

00

011011

0001

101100

0110

11

0001

10 11

0001

1011

000110

11

0001

1011

0001

1011

00

0110

11

00011011

0001

1011

0001

1011

0001

1011

0001

1011

0001

1011

00

0110

1100

0110

11

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 1.15s

T = 3.46s

T = 5.77s

T = 7.50s

No Super Zeno With Super Zeno

Figure 3.11: Real (left) and imaginary (right) parts of the experimental tomographs ofthe (a) 1√

2(|01〉 − |10〉) (singlet) state in the two-dimensional subspace 01, 10, with a

computed fidelity of 0.98. (b)-(e) depict the state at T = 1.15, 3.46, 5.77, 7.50 s, withthe tomographs on the left and the right representing the state without and after applyingthe super-Zeno preserving scheme, respectively. The rows and columns are labeled in thecomputational basis ordered from |00〉 to |11〉.

78

3.3 Experimental implementations of super-Zeno scheme

after applying the super-Zeno scheme, tomographed at different time points, are shownin Fig. 3.9. As can be seen from the experimental tomographs, the evolution of the |10〉state out of the subspace is almost completely frozen by the super-Zeno sequence uptonearly 7.5 s, while without any preservation the state has leaked into the orthogonalsubspace within 3.5 s. The tomographs showed in Fig. 3.10 for the |01〉 state, show asimilar level of preservation.

3.3.3.2 Preservation of an entangled state in the subspace

We now prepare an entangled state (the singlet state) embedded in the two-dimensionalP = |01〉, |10〉 subspace, and used the subspace-preserving scheme described inFig. 3.8 to protect P. The singlet state was reconstructed using state tomography, andthe real and imaginary parts of the tomographed experimental density matrices withoutany preservation and after applying the super-Zeno scheme, are shown in Fig. 3.11. Ascan be seen from the experimental tomographs, the state evolution remains within theP subspace but the state itself does not remain maximally entangled.

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5(a)

Time(s)

δ(Lea

kage)

Without SZ

With SZ

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5(b)

Time(s)

δ(Lea

kage)

Without SZ

With SZ

Figure 3.12: Plot of leakage fraction from the |01〉, |10〉 subspace to its orthogonalsubspace |00〉, |11〉 of (a) the |10〉 state and (b) the 1√

2(|01〉 − |10〉) (singlet) state,

without any preservation and after applying the super-Zeno sequence. The leakage tothe orthogonal subspace is minimal (remains close to zero) after applying the super-Zenoscheme.

79

3. Experimental protection of quantum states via a super-Zeno scheme

3.3.3.3 Estimating leakage outside subspace

An ensemble of spins initially prepared in a state belonging to the subspace P =|01〉, |10〉, can evolve to the orthogonal subspace Q = |00〉, |11〉 due to unwantedinteractions with its environment. This evolution of the state to the orthogonal sub-space is called leakage. We applied super-Zeno scheme to protect leakage of a statefrom subspace P to subspace Q. The subspace-preserving capability of the circuit givenin Fig. 3.8 was quantified by computing a leakage parameter that defines the amountof leakage of the state to the orthogonal Q = |00〉, |11〉 subspace.

For a given density operator ρ the “leak (fraction)” δ, into the subspace Q is definedas

δ = 〈00|ρ|00〉+ 〈11|ρ|11〉 (3.27)

The leak (fraction) δ represents the number of spins of the ensemble that have migratedto the subspace Q divided by the total number of spins in the ensemble; δ is equal to onewhen all the states remain in the subspace P and is equal to zero if all the states haveleaked to the subspace Q. The leak (fraction) δ versus time is plotted in Figs. 3.12(a)and (b), for the |10〉 and the singlet state respectively, with and without applying thesuper-Zeno subspace-preserving sequence. The leakage parameter remains close tozero for both kinds of states, proving the success and the generality of the super-Zenoscheme.

3.3.4 Preservation of entanglement

The amount of entanglement that remains in the state after a certain time is quanti-fied by an entanglement parameter denoted by η as described in Chap. 2, Sec. 2.2.2.We will use this entanglement parameter η to quantify the amount of entanglement atdifferent times. The maximally entangled singlet state was created and its evolutionstudied in two different scenarios. In the first scenario described in Sec. 3.3.2, the sin-glet state was protected against evolution by the application of the super-Zeno scheme.In the second scenario described in Sec. 3.3.3, a two-dimensional subspace contain-ing the singlet state was protected using the super-Zeno scheme. For the former case,one expects that the state will remain a singlet state, while in the latter case, it canevolve within the protected two-dimensional subspace. Since in the second case, theprotected subspace contains entangled as well as separable states, one does not expectpreservation of entanglement to the same extent as expected in the first case, where theone-dimensional subspace defined by the singlet state itself is protected. The experi-mental tomographs at different times and fidelity for the case of state protection andthe leakage fraction for the case of subspace protection have been discussed in detail inthe previous subsections. Here we focus our attention on the entanglement present in

80

3.4 Conclusions

0 2 4 6 8

0

0.1

0.2

0.3

0.4

0.5

(a)

Time(s)

η(E

ntanglemen

t)

Without SZ

With SZ

0 2 4 6 8

0

0.1

0.2

0.3

0.4

0.5

(b)

η(E

ntanglemen

t)Time(s)

Without SZ

With SZ

Figure 3.13: Plot of entanglement parameter η with time, with and without applying thesuper-Zeno sequence, computed for (a) the 1√

2(|01〉 − |10〉) (singlet) state, and (b) the

same singlet state when embedded in the subspace |01〉, |10〉 being preserved.

the state at different times. The entanglement parameter η for the evolved singlet stateis plotted as a function of time and is shown in Figs. 3.13(a) and (b), after applyingthe state-preserving and the subspace-preserving super-Zeno sequence respectively. Inboth cases, the state becomes disentangled very quickly (after approximately 2 s) ifno super-Zeno preservation is performed. After applying the state-preserving super-Zeno sequence (Fig. 3.13(a)), the amount of entanglement in the state remains closeto maximum for a long time (upto 8 s). After applying the subspace-preserving super-Zeno sequence (Fig. 3.13(b)), the state shows some residual entanglement over longtimes but it is clear that the state is no longer maximally entangled. This implies thatthe subspace-preserving sequence does not completely preserve the entanglement ofthe singlet state, as expected. However, while the singlet state becomes mixed overtime, its evolution remains confined to states within the two-dimensional subspace(P = |01〉, |10〉) being preserved as is shown in Fig. 3.12, where we calculate theleak (fraction).

3.4 Conclusions

In this chapter it was experimentally demonstrated that the super-Zeno scheme can ef-ficiently preserve states in one and two-dimensional subspaces, by preventing leakageto a subspace orthogonal to the subspace being preserved. The super-Zeno sequence

81

3. Experimental protection of quantum states via a super-Zeno scheme

was implemented on product as well as on entangled states, embedded in one- andtwo-dimensional subspaces of a two-qubit NMR quantum information processor. Theadvantage of the super-Zeno protection lies in its ability to preserve the state suchthat while the number of spins in that particular state reduces with time, the state re-mains the same. Without the super-Zeno protection, the number of spins in the statereduces with time and the state itself migrates towards a thermal state, reducing the fi-delity. This work adds to the arsenal of real-life attempts to protect against evolution ofstates in quantum computers and points the way to the possibility of developing hybridstrategies (combining the super-Zeno scheme with other schemes such as dynamicaldecoupling sequences) to tackle preservation of fragile computational resources suchas entangled states.

82

Chapter 4

Experimental protection of unknownstates using nested Uhrig dynamicaldecoupling sequences

4.1 Introduction

Different decoupling strategies can be used to decouple a quantum system from itsenvironment, thus controlling the effect of undesired changes. In Chapter 3, we pro-tected a known state of the system and also prevented leakage from a known subspaceof the system via the super-Zeno scheme. While protecting against leakage from thesubspace, the state was not leaking out to the orthogonal subspace but was evolvingwithin the subspace. Hence, the obvious question comes to mind: can one freeze thisevolution within the subspace? The answer is yes; an unknown state in a known sub-space can be frozen using nested Uhrig dynamical decoupling sequences. DD schemesrely on the repeated application of control pulses and delays to remove the unwantedcontributions due to system-environment interaction [105]. For a quantum system cou-pled to a bath, the DD sequence decouples the system and bath by adding a suitabledecoupling interaction, periodic with cycle time Tc to the overall system-bath Hamilto-nian [106]. AfterN applications of the cycle for a timeNTc, the system is governed bya stroboscopic evolution under an effective average Hamiltonian, in which system-bathinteraction terms are no longer present.

More sophisticated DD schemes are of the Uhrig dynamical decoupling (UDD)type, wherein the pulse timing in the DD sequence is tailored to produce higher-ordercancellations in the Magnus expansion of the effective average Hamiltonian, therebyachieving system-bath decoupling to a higher order and hence stronger noise protec-

83

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

tion [152, 162, 163, 164, 165]. A UDD sequence can suppress decoherence up toO(TN+1) with only N pulses. In a UDD sequence, the kth control pulse is applied atthe time

Tk = T sin2

(kπ

2N + 2

); k = 1, 2, . . . , N. (4.1)

UDD schemes are applicable when the control pulses can be considered as ideal(i.e. instantaneous) and when the environment noise has a sharp frequency cutoff [60,111, 166, 167]. These initial UDD schemes dealt with protecting a single qubit againstdifferent types of noise, and were later expanded to a whole host of optimized se-quences involving nonlocal control operators, to protect multi-qubit systems againstdecoherence [61, 168, 169, 170, 171, 172].

While UDD schemes can well protect states against single- and two-axis noise(i.e. pure dephasing and/or pure bit-flip), they are not able to protect against generalthree-axis decoherence [173]. Nested UDD (NUDD) schemes were hence proposedto protect multiqubit systems in generic quantum baths to arbitrary decoupling orders,by nesting several UDD layers. The structure of the Hamiltonian was exploited and anappropriate set of mutually orthogonal operation set (MOOS) was designed. On thebasis of commutation and anticommutation property of operators in an MOOS withHamiltonian, the NUDD scheme was constructed such that after nesting of operatorlayers only those terms remained in the generating algebra of Hamiltonian which didnot affect the state. It was shown that the NUDD scheme can preserve a set of uni-tary Hermitian system operators (and hence all operators in the Lie algebra generatedfrom this set of operators) that mutually either commute or anticommute [174, 175].Furthermore, it was proved that the NUDD scheme is universal i.e. it can preserve thecoherence ofm coupled qubits by suppressing decoherence upto orderN , independentof the nature of the system-environment coupling [176].

The efficiency of NUDD schemes in protecting unknown randomly generated two-qubit states was shown to be a powerful approach for protecting quantum states againstdecoherence [64]. Numerical simulations on a five-spin system were carried out in thiscontext. Two of the five spins were identified as a two-qubit system and the other threespins were regarded as the bath. To show the efficacy of the NUDD scheme, it wasapplied on ten arbitrary states belonging to a known subspace. For the correct nestingof UDD layers a remarkably high fidelity was achieved in locking the initial unknownsuperposition state [64].

In this chapter, the efficacy of protection of the NUDD scheme is first evaluated byapplying it on four specific states of the subspace P = |01〉, |10〉 i.e. two separa-ble states: |01〉 and |10〉, and two maximally entangled singlet and triplet Bell states:

1√2(|01〉 − |10〉) and 1√

2(|01〉 + |01〉) in a four-dimensional two-qubit Hilbert space.

Next, to evaluate the effectiveness of the NUDD scheme on the entire subspace, ran-

84

4.2 The NUDD scheme

domly states are generated in the subspace P (considered as a superposition of theknown basis states |01〉, |10〉) and protected them using NUDD scheme. Eight states inthe two-qubit subspace randomly generated and protected using a three-layer NUDDsequence. Full state tomography is used to compute the experimental density matri-ces. Each state is allowed to decohere, and the state fidelity is computed at each timepoint without protection and after NUDD protection. The results are presented as ahistogram and showed that while NUDD is always able to provide some protection,the degree of protection varies from state to state.

4.2 The NUDD schemeConsider a two-qubit quantum system with its state space spanned by the states |00〉,|01〉, |10〉, |11〉, the eigenstates of the Pauli operator σ1

z⊗σ2z . Our interest is in protect-

ing states in the subspace P spanned by states |01〉, |10〉, against decoherence. Thedensity matrix corresponding to an arbitrary pure state |ψ〉 = α|01〉+β|10〉 belongingto the subspace P is given by

ρ(t) =

0 0 0 00 |α|2 αβ∗ 00 βα∗ |β|2 00 0 0 0

(4.2)

with the coefficients α and β satisfying |α|2 + |β|2 = 1 at time t = 0. We describe herethe theoretical construction of a three-layer NUDD scheme to protect arbitrary statesin the two-qubit subspace P [61, 64].

The general total Hamiltonian of a two-qubit system interacting with an arbitrarybath can be written as

Htotal = HS +HB +HjB +H12 (4.3)

whereHS is the system Hamiltonian,HB is the bath Hamiltonian,HjB is qubit-bath in-teraction Hamiltonian andH12 is the qubit-qubit interaction Hamiltonian (which can bebath-dependent). Our interest here is in bath-dependent terms and their control, whichcan be expressed using a special basis set for the two-qubit system as follows [61, 64]:

H = HB +HjB +H12

=16∑j=1

WjYj (4.4)

where the coefficients Wj contain arbitrary bath operators. Y are the special ba-sis computed from the perspective of preserving the subspace spanned by the states

85

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

|01〉, |10〉 in the two-qubit space [61, 64]:

Y1 = I, Y2 = |01〉〈01|+ |10〉〈10|,Y3 = |00〉〈11|, Y4 = |00〉〈00| − |11〉〈11|,Y5 = |11〉〈00|, Y6 = |01〉〈01| − |10〉〈10|,Y7 = |10〉〈00|, Y8 = |00〉〈10|,Y9 = |10〉〈11|, Y10 = |11〉〈10|,Y11 = |01〉〈00|, Y12 = |00〉〈01|,Y13 = |01〉〈11|, Y14 = |11〉〈01|,Y15 = |01〉〈10|+ |10〉〈01|,Y16 = −i(|10〉〈01| − |01〉〈10|). (4.5)

To protect a general two-qubit state |ψ〉 ∈ P against decoherence using NUDD,we are required to protect diagonal populations |α|2, |β|2 and off-diagonal coherencesαβ∗. Hence the locking scheme requires the nesting of three layers of UDD sequences.• Innermost UDD layer: The diagonal populations Tr[ρ(t)|01〉〈01|] ≈ |α|2 are lockedby this UDD layer with the control operator

X0 = I − 2|01〉〈01|. (4.6)

We can write the total Hamiltonian,

H = H0 +H1,

H0 =10∑i=1

WiYi,

H1 =16∑i=11

WiYi, (4.7)

such that the X0 commute with H0, i.e. [X0, H0] = 0 and anti-commute with H1, i.e.X0, H1+ = 0. An inverting pulse with control Hamiltonian,

Hc =N∑j=1

πδ(t− Tj)X0

2, (4.8)

applied with the UDD timing Tj , defined as :

Tj = T sin2(jπ

2N + 2), j = 1, 2 · · · , N. (4.9)

86

4.2 The NUDD scheme

The unitary evolution operator of the system for time period t = 0 to t = T is givenby (~ = 1) :

UN(T ) = XN0 e−i[H0+H1](T−TN )(−iX0)

× e−i[H0+H1](TN−TN−1)(−iX0)

· · ·× e−i[H0+H1](T3−T2)(−iX0)

× e−i[H0+H1](T2−T1)(−iX0)

× e−i[H0+H1]T1 . (4.10)

Using the UDD universality proof [61] and the fact that H0 and H1 commute andanticommute respectively with X0, it can be shown that

UN(T ) = U evenN +O(TN+1), (4.11)

where

U evenN = exp(−iH0T )

+∞∑k=0

(−i)2k∆2k, (4.12)

with ∆2k containing only even powers of HI1 (t), defined by

HI1 (t) ≡ exp(iH0t)H1 exp(−iH0t).

∆2k can be expanded as a linear superposition of all possible basis operators that com-mute with X0. i.e.

∆2k =10∑i=1

AiYi (4.13)

where Ai are the expansion coefficients containing bath operators. The N th order,UN(T ) can be expressed as a combination of Y1, Y2, · · · , Y10 only. Using the closureof this set of operators, i.e.,(

10∑i=1

AiYi

)(10∑k=1

BkYk

)=

10∑l=1

ClYl (4.14)

we further obtain

UN(T ) = exp(−iHUDD-1eff T ) +O(TN+1), (4.15)

where

HUDD-1eff =

10∑i=1

D1,iYi, (4.16)

87

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

whereD1,i refer to the expansion coefficients of this first UDD layer. Terms containingbasis operators Y11 · · ·Y16 are efficiently decoupled.• Second UDD layer: The diagonal populations Tr[ρ(t)|10〉〈10|] ≈ |β|2 are locked bythis second UDD layer with the control operator

X1 = I − 2|10〉〈10| (4.17)

We can further decompose the effective Hamiltonian after the first layer HUDD-1eff into

HUDD-1eff = HUDD-1

eff,0 +HUDD-1eff,1 , (4.18)

such that [HUDD-1eff,0 , X1] = 0 and HUDD-1

eff,1 , X1+ = 0. where

HUDD-1eff,0 ≡

6∑i=1

D1,iYi;

HUDD-1eff,1 ≡

10∑i=7

D1,iYi. (4.19)

It is straightforward to see that the operators Yi, i = 1−6 form a closed algebra. Hencewhen a second layer of UDD sequence of X1 is applied to the N th order, the dynamicsof HUDD-1

eff reduce to

HUDD-2eff =

6∑i=1

D2,iYi, (4.20)

where D2,i refer to the expansion coefficients of this second UDD layer. Terms con-taining basis operators Y7 · · ·Y10 are efficiently decoupled.• Outermost UDD layer: The off-diagonal coherences Tr[ρ(t)|01〉〈10|] ≈ αβ∗ arelocked by this final UDD layer with the control operator

Xφ = I − [|01〉+ |10〉][〈01|+ 〈10|]. (4.21)

Again we can write the effective Hamiltonian after the second layer as

HUDD−2eff = HUDD−2

eff,0 +HUDD−2eff,1 , (4.22)

such that [HUDD-2eff,0 , Xφ] = 0 and HUDD-2

eff,1 , Xφ+ = 0. In the self-closed set of theoperators that form HUDD−2

eff , the only term D2,6Y6 = D2,6[|01〉〈01| − |10〉〈10|] canaffect |ψ(0)〉, which is effectively decoupled by this third layer. The final reducedeffective Hamiltonian after the three-layer NUDD contains five operators: HUDD−3

eff =∑5i=1 D3,iYi, where D3,i are the coefficients due to three UDD layers.

88

4.2 The NUDD scheme

The innermost UDD control X0 pulses are applied at the time intervals Tj,k,l, themiddle layer UDD control X1 pulses are applied at the time intervals Tj,k and the out-ermost UDD control Xφ pulses are applied at the time intervals Tj (j, k, l = 1, 2, ...N )given by:

Tj,k,l = Tj,k + (Tj,k+l − Tj,k) sin2

(lπ

2N + 2

)Tj,k = Tj + (Tj+1 − Tj) sin2

(kπ

2N + 2

)Tj = T sin2

(jπ

2N + 2

)(4.23)

The total time interval in the N th order sequence is (N + 1)3 with the total number ofpulses in one run being given by N((N + 1)2 +N + 2) for even N [64].Summary of the NUDD scheme:

The recipe to design UDD protection for a two-qubit state (say |χ〉) is given in thefollowing steps: (i) First a control operator Xc is constructed using Xc = I − 2|χ〉〈χ|such that X2

c = I , with the commuting relation [Xc, H0] = 0 and the anticommutingrelation Xc, H1 = 0; (ii) The control UDD Hamiltonian is then applied so that sys-tem evolution is now under a UDD-reduced effective Hamiltonian thus achieving stateprotection upto order N ; (iii) Depending on the explicit commuting or anticommutingrelations ofXc withH0 andH1, the UDD sequence efficiently removes a few operatorsYi from the initial generating algebra ofH and hence suppresses all couplings betweenthe state |χ〉 and all other states.

Yi, i = 1, 2, · · · , 16

⇓X0, UDD-1

Yi, i = 1, 2, · · · , 10

⇓X1,UDD-2

Yi, i = 1, 2, · · · , 6⇓Xφ,UDD-3

Yi, i = 1, 2, · · · , 5 .

89

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

|0〉

|0〉∆1 ∆9 ∆10 ∆18 ∆19 ∆27

|ψ〉ǫ

|01〉,|10〉

X1, X0 Xφ X1, X0 Xφ X1, X0

X0

∆2

X0

∆3

X1

∆4

X0

∆5

X0

∆6

X1

∆7

X0

∆8

X0

M(a)

xφ1-x x -x

yφ2

y x -x-y

τ12

(b)

τ12 τ12 τ12

x -x-x

x -x-x-y x -x y

x -xx

x -xx-y x -x y

(c)

Figure 4.1: (a) Circuit diagram for the three-layer NUDD sequence. The innermost UDDlayer consists of X0 control pulses, the middle layer comprises X1 control pulses and theoutermost layer consists of Xφ pulses. The entire NUDD sequence is repeated M times;∆i are time intervals. (b) NMR pulse sequence to implement the control pulses forX0 andX1 UDD sequences. The values of the rf pulse phases φ1 and φ2 are set to x and y for theX0 and to −x and −y for the X1 UDD sequence, respectively. (c) NMR pulse sequenceto implement the control pulses for the Xφ UDD sequence. The filled rectangles denoteπ/2 pulses while the unfilled rectangles denote π pulses, respectively. The time period τ12

is set to the value (2J12)−1, where J12 denotes the strength of the scalar coupling betweenthe two qubits.

90

4.3 Experimental protection of two qubits using NUDD

1H

13C

νC=11814.8Hz

νH=4783.0Hz

J12=214.9Hz

TC1 =16.6s

TH1 =7.9s

TC2 =0.3s

TH2 =2.9s

ωH(in ppm) ωC(in ppm)

Qubit 1 Qubit 2

|1〉 |0〉 |1〉 |0〉

|0〉 |0〉

(a) (b)

(c)

8.2 8.0 7.8 79.5 78.5 77.5

Figure 4.2: (a) Structure of isotopically enriched chloroform-13C molecule, with the 1Hspin labeling the first qubit and the 13C spin labeling the second qubit. The system pa-rameters are tabulated alongside with chemical shifts νi and scalar coupling J12 (in Hz)and NMR spin-lattice and spin-spin relaxation times T1 and T2 (in seconds). (b) NMRspectrum obtained after a π/2 readout pulse on the thermal equilibrium state and (c) NMRspectrum of the pseudopure |00〉 state. The resonance lines of each qubit in the spectra arelabeled by the corresponding logical states of the other qubit.

4.3 Experimental protection of two qubits using NUDD

4.3.1 Experimental implementation of the NUDD scheme

We now turn to the NUDD implementation for N = 2 on a two-qubit NMR sys-tem. The entire NUDD sequence can be written in terms of UDD control operatorsX0, X1, Xφ (defined in the previous section) and time evolution U(δit) under the gen-eral Hamiltonian for time interval fractions δi:

Xc(t) = U(δ1t)X0U(δ2t)X0U(δ3t)X1U(δ4t)X0U(δ5t)X0

U(δ6t)X1U(δ7t)X0U(δ8t)X0U(δ9t)XφU(δ10t)X0

U(δ11t)X0U(δ12t)X1U(δ13t)X0U(δ14t)X0U(δ15t)

X1U(δ16t)X0U(δ17t)X0U(δ18t)XφU(δ19t)

X0U(δ20t)X0U(δ21t)X1U(δ22t)X0U(δ23t)

X0U(δ24t)X1U(δ25t)X0U(δ26t)X0U(δ27t) (4.24)

In our implementation, the number of X0, X1 and Xφ control pulses used in one run ofthe three-layer NUDD sequence are 18, 6 and 2, respectively.

91

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

Using the UDD timing intervals defined above and applying the condition∑δi =

1, their values are computed to be

δi = β, 2β, β, 2β, 4β, 2β, β, 2β, β, 2β, 4β, 2β, 4β, 8β,4β, 2β, 4β, 2β, β, 2β, β, 2β, 4β, 2β, β, 2β, β (4.25)

where the intervals between the X0, X1 and Xφ control pulses turn out to be a multipleof β = 0.015625 .

The NUDD scheme for state protection and the corresponding NMR pulse se-quence is given in Fig. 4.1. The unitary gates X0, X1, and Xφ drawn in Fig. 4.1(a)correspond to the UDD control operators already defined in the previous section. The∆i time interval in the circuit given in Fig. 4.1(a) is defined by ∆i = δit, using the δigiven in Eqn. (4.25). The pulses on the top line in Figs. 4.1(b) and (c) are applied onthe first qubit (1H spin in Fig. 4.2), while those at the bottom are applied on the secondqubit (13C spin in Fig. 4.2), respectively. All the pulses are spin-selective pulses, withthe 90 pulse length being 7.6µs and 15.6µs for the proton and carbon rf channels, re-spectively. When applying pulses simultaneously on both the carbon and proton spins,care was taken to ensure that the pulses are centered properly and the delay betweentwo pulses was measured from the center of the pulse duration time. We note here thatthe NUDD schemes are experimentally demanding to implement as they contain longrepetitive cycles of rf pulses applied simultaneously on both qubits and the timings ofthe UDD control sequences were matched carefully with the duty cycle of the rf probebeing used.

We chose the chloroform-13C molecule as the two-qubit system to implement theNUDD sequence (Fig. 4.2 for details of system parameters and average NMR relax-ation times of both the qubits). The two-qubit system Hamiltonian in the rotating frame(which includes the Hamiltonians HS and H12 of Eqn. 4.3) is given by

Hrot = 2π[(νH − νrfH )IHz + (νC − νrfC )ICz + J12IHz I

Cz ] (4.26)

where νH (νC ) is the chemical shift of the 1H(13C) spin, νrfi is the rotating framefrequency (νrfi = νi for on-resonance), IHz (ICz ) is the z component of the spin angularmomentum operator for the 1H(13C) spin, and J12 is the spin-spin coupling constant.

The two qubits were initialized into the pseudopure state |00〉 using the spatialaveraging technique [177], with the corresponding density operator given by

ρ00 =1− ε

4I + ε|00〉〈00| (4.27)

with a thermal polarization ε ≈ 10−5 and I being a 4 × 4 identity operator. All ex-perimental density matrices were reconstructed using quantum state tomography via

92

4.3 Experimental protection of two qubits using NUDD

0 2 40

0.2

0.4

0.6

0.8

1(a)

Time(s)

F(F

idelity)

Without NUDD

With NUDD

0 2 40

0.2

0.4

0.6

0.8

1(b)

Time(s)F(F

idelity)

Without NUDD

With NUDD

Figure 4.3: Plot of fidelity versus time for (a) the |01〉 state and (b) the |10〉) state, withoutany protection and after applying NUDD protection. The fidelity of both the states remainsclose to 1 for upto long times, after NUDD protection.

a maximum likelihood protocol (Chapter 2). The experimentally created pseudopurestate |00〉 was tomographed with a fidelity of 0.99. The fidelity of an experimentaldensity matrix was computed using Eq. (2.32).

4.3.2 NUDD protection of known states in the subspaceWe begin evaluating the efficiency of the NUDD scheme by first applying it to protectfour known states in the two-dimensional subspace P, namely two separable and twomaximally entangled (Bell) states.Protecting two-qubit separable states: We experimentally created the two-qubit sep-arable states |01〉 and |10〉 from the initial state |00〉 by applying a πx on the secondqubit and on the first qubit, respectively. The states were prepared with a fidelity of0.98 and 0.97, respectively. One run of the NUDD sequence took 0.12756 s whichincluded the time taken to implement the control operators, and t = 0.05 s (as perEqn. (4.24)). The entire NUDD sequence was applied 40 times. The state fidelity wascomputed at different time instants, without any protection and after applying NUDDprotection. The state fidelity remains close to 0.9 for long times (upto 5 s) when NUDDis applied, whereas for no protection the |01〉 state loses its fidelity (fidelity approaches0.5) after 3 s and the |10〉 state loses its fidelity after 2 s. A plot of state fidelities ver-sus time is displayed in Fig. 4.3, demonstrating the remarkable efficacy of the NUDDsequence in protecting separable two-qubit states against decoherence. The signal at-tenuation has been ignored and the density matrix has been renormalized at every time

93

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

00

0110

11

000110

11

0001

1011

0001

101100

01 10

11

0001

1011

0001

1011

0001

1011

0001

1011

000110

11

00

011011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

0001

1011

0001

1011

0001

1011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

0001

1011

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1011

00

0110

11

00

0110

110001

1011

0001

1011

0001

1011

0001

1011

00

01

1011

0001

1011

0001

1011

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 1.02s

T = 2.04s

T = 3.06s

T = 4.08s

No NUDD With NUDD

Figure 4.4: Real (left) and imaginary (right) parts of the experimental tomographs ofthe (a) |01〉 state, with a computed fidelity of 0.98. (b)-(e) depict the state at T =

1.02, 2.04, 3.06, 4.08 s, with the tomographs on the left and the right representing thestate without any protection and after applying NUDD protection, respectively. The rowsand columns are labeled in the computational basis ordered from |00〉 to |11〉.

94

4.3 Experimental protection of two qubits using NUDD

00

0110

11

000110

11

0001

1011

0001

101100

01 10

11

0001

1011

0001

1011

0001

1011

0001

1011

000110

11

00

011011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

0001

1011

0001

1011

0001

1011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

0001

1011

0001

1011

00

0110

11

00

0110

110001

1011

0001

1011

0001

1011

0001

1011

00

01

1011

0001

1011

0001

1011

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 1.02s

T = 2.04s

T = 3.06s

T = 4.08s

No NUDD With NUDD

Figure 4.5: Real (left) and imaginary (right) parts of the experimental tomographs ofthe (a) |10〉 state, with a computed fidelity of 0.97. (b)-(e) depict the state at T =

1.02, 2.04, 3.06, 4.08 s, with the tomographs on the left and the right representing thestate without any protection and after applying NUDD protection, respectively. The rowsand columns are labeled in the computational basis ordered from |00〉 to |11〉.

95

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

0001

1011

000110

11

00

0110

11

00

0110

110001

1011

0001

1011

0001

1011

0001

1011

000110

11

000110

11

0001

1011

00

0110

110001

1011

0001

1011

0001

1011

0001

1011

0001

1011

00

011011

0001

1011

0001

1011

0001

1011

0001

1011

000110

11

0001

1011

0001

1011

0001

1011

0001

1011

0001

101100

0110

11

0001

1011

0001

1011

0001

1011

0001

1011

00

01

10

11

0001

1011

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 0.28s

T = 0.55s

T = 0.83s

T = 1.10s

No NUDD With NUDD

Figure 4.6: Real (left) and imaginary (right) parts of the experimental tomographs of the(a) 1√

2(|01〉 − |10〉) state, with a computed fidelity of 0.99. (b)-(e) depict the state at

T = 0.28, 0.55, 0.83, 1.10 s, with the tomographs on the left and the right representing thestate without any protection and after applying NUDD protection, respectively. The rowsand columns are labeled in the computational basis ordered from |00〉 to |11〉.

96

4.3 Experimental protection of two qubits using NUDD

00011011

000110

11

0001

1011

0001

101100

011011

0001

1011

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1011

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1011

000110

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00

011011

00

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1011

0001

1011

0001

1011

0001

1011

0001

1011

0001

1011

00

0110

11

00

0110

110001

1011

0001

1011

0001

1011

0001

1011

0001

1011

0001

101100

0110

11

0001

1011

(a)

(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 0.28s

T = 0.55s

T = 0.83s

T = 1.10s

No NUDD With NUDD

Figure 4.7: Real (left) and imaginary (right) parts of the experimental tomographs of the(a) 1√

2(|01〉 + |10〉) state, with a computed fidelity of 0.99. (b)-(e) depict the state at

T = 0.28, 0.55, 0.83, 1.10 s, with the tomographs on the left and the right representing thestate without any protection and after applying NUDD protection, respectively. The rowsand columns are labeled in the computational basis ordered from |00〉 to |11〉.

97

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

0 0.4 0.8 1.20

0.2

0.4

0.6

0.8

1(a)

Time(s)

F(F

idelity)

Without NUDD

With NUDD

0 0.4 0.8 1.20

0.2

0.4

0.6

0.8

1(b)

Time(s)

F(F

idelity)

Without NUDD

With NUDD

Figure 4.8: Plot of fidelity versus time for (a) the Bell singlet state and (b) the Bell tripletstate, without any protection and after applying NUDD protection.

point in Fig. 4.3, as described in Chap. 3,Sec. 3.3.2.3.Protecting two-qubit Bell states: We next implemented NUDD protection on themaximally entangled singlet state 1√

2(|01〉 − |10〉). We experimentally constructed the

singlet state from the initial |00〉 state via the pulse sequence given in Fig. 4.9 withvalues of θ = −π

2and φ = 0. The fidelity of the experimentally constructed singlet

state was computed to be 0.99.One run of the NUDD sequence took 0.27756 s and t was kept at t = 0.2 s. The

entire NUDD sequence was applied 4 times on the state. The singlet state fidelity atdifferent time points was computed without any protection and after applying NUDDprotection, and the state tomographs are displayed in Fig. 4.6. The signal attenuationhas been ignored and the density matrix has been renormalized at every time point inFig. 4.6. The fidelity of the singlet state remained close to 0.8 for 1 s when NUDDprotection was applied, whereas when no protection is applied the state decoheres(fidelity approaches 0.5) after 0.55 s. We also implemented NUDD protection on themaximally entangled triplet state 1√

2(|01〉 + |10〉). We experimentally constructed the

triplet state from the initial |00〉 state via the pulse sequence given in Fig. 4.9 withvalues of θ = π

2and φ = 0. The fidelity of the experimentally constructed triplet state

was computed to be 0.99. The total NUDD time was kept at t = 0.2 s and one runof the NUDD sequence took 0.27756 s. The entire NUDD sequence was repeated 4times on the state. The state fidelity at different time points was computed without anyprotection and after applying NUDD protection and the state tomographs are displayedin Fig. 4.7. The signal attenuation has been ignored and the density matrix has beenrenormalized at every time point in Fig. 4.7.

98

4.3 Experimental protection of two qubits using NUDD

π3

π4

π4

θ

x x x -x-y

x -x

τ12

G

1H

13C

x -y x -x

-y x -xφ2

τ12

|ψ〉ǫ

|01〉,|1

0〉

State Initialization

|00〉〈00|

Figure 4.9: NMR pulse sequence for the preparation of arbitrary states. The sequence ofpulses before the vertical dashed line achieve state initialization into the |00〉 state. Thevalues of flip angles θ and φ of the rf pulses are the same as the θ and φ values describinga general state in the two-dimensional subspace P = |01〉, |10〉. Filled and unfilledrectangles represent π2 and π pulses respectively, while all other rf pulses are labeled withtheir respective flip angles and phases; the interval τ12 is set to (2J12)−1 where J12 is thescalar coupling.

The fidelity of the triplet state remained close to 0.71 for 0.28 s when NUDDprotection was applied, whereas when no protection is applied the state decoheres quiterapidly (fidelity approaches 0.5) after 0.28 s. A plot of state fidelities of both Bell statesversus time is displayed in Fig. 4.8, In that the signal attenuation has been ignored andthe density matrix has been renormalized at every time point. While the NUDD schemewas able to protect the singlet state quite well (the time for which the state remainsprotected is double as compared to its natural decay time), it is not able to extendthe lifetime of the triplet state to any appreciable extent. What is worth noting hereis the fact that the state fidelity remains considerably higher under NUDD protectioncompared to no protection, implying that there is a reduction in the “leakage” to otherstates.

4.3.3 NUDD protection of unknown states in the subspace

We wanted to carry out an unbiased assessment of the efficacy of the NUDD schemefor state protection. To this end, we randomly generated several states in the two-

99

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

Figure 4.10: Geometrical representation of eight randomly generated states on a Blochsphere belonging to the two-qubit subspace P = |01〉, |10〉. Each vector makes anglesθ, φ with the z and x axes, respectively. The state labels RS-i (i = 1..8) are explained inthe text.

dimensional subspace P, and applied the NUDD sequence on each state. A generalstate in the two-qubit subspace P = |01〉, |10〉 can be written in the form

|ψ〉 = cosθ

2|01〉+ e−ιφ sin

θ

2|10〉 (4.28)

These states were experimentally created by using the values of θ and φ (Eqn. (4.28))as the flip angles of the rf pulses in the NMR pulse sequence (see Fig. 4.9 for visu-alization). The eight randomly generated representative two-qubit states are shown inFig. 4.10. The entire three-layered NUDD sequence was applied 10 times on each ofthe eight random states. The time t for the sequence was kept at t = 0.05 s and one runof the NUDD sequence took 0.12756 s. The plots of fidelity versus time are shown asbar graphs in Fig. 4.11, with the cross-hatched bars representing state fidelity withoutany protection and the solid bars representing state fidelity after NUDD protection.The signal attenuation has been ignored and the density matrix has been renormalizedat every time point in Fig. 4.11. The final bar plot in Fig. 4.11(i) shows the average fi-delity of all the randomly generated states at each time point. The results of protectingthese random states via three-layered NUDD are tabulated in Table 4.1. Each state hasbeen tagged by a label RS-i (RS denoting “Random State” and i = 1, ..8), with its θ, φvalues displayed in the next column. The fourth column displays the values of the nat-ural decoherence time (in seconds) of each state without NUDD protection, estimated

100

4.3 Experimental protection of two qubits using NUDD

0 0.5 10

0.5

1(a)

Time(s)

F

0 0.5 10

0.5

1(b)

Time(s)

F

0 0.5 10

0.5

1(c)

Time(s)

F

0 0.5 10

0.5

1(d)

Time(s)

F

0 0.5 10

0.5

1(e)

Time(s)

F

0 0.5 10

0.5

1(f)

Time(s)

F

0 0.5 10

0.5

1(g)

Time(s)

F

0 0.5 10

0.5

1(h)

Time(s)

F

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

(i)

Time(s)

Fav

Figure 4.11: Bar plots of fidelity versus time of eight randomly generated states (labeledRS-i, i = 1..8), without any protection (cross-hatched bars) and after applying NUDDprotection (red solid bars): (a) RS-1, (b) RS-2, (c) RS-3, (d) RS-4, (e) RS-5, (f) RS-6, (g)RS-7 and (h) RS-8. (i) Bar plot showing average fidelity of all eight randomly generatedstates, at each time point.

101

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

Table 4.1: Results of applying NUDD protection on eight randomly generated states inthe two-dimensional subspace. Each random state (RS) is tagged with a number for con-venience, and its corresponding (θ,φ) angles are given in the column alongside. The fourthcolumn displays the time at which the state fidelity approaches ≈ 0.8 without NUDD pro-tection and the last column displays the time for which state fidelity approaches ≈ 0.8

after applying NUDD protection.

State Label (θ, φ)(deg) Time (s) Time (s)(F > 0.8) (F > 0.8)

Without NUDD With NUDD0.29|01〉+ (0.94 + ι0.18)|10〉 RS-1 (147,57) 0.1s 1.0s0.15|01〉 − (0.76 + ι0.63)|10〉 RS-2 (163,349) 0.3s 1.1s0.98|01〉+ (0.11− ι0.17)|10〉 RS-3 (23,345) 0.1s 1.1s0.14|01〉+ (0.36− ι0.92)|10〉 RS-4 (164,175) 0.3s 1.1s0.99|01〉+ (0.10 + ι0.11)|10〉 RS-5 (18,51) 0.3s 1.1s0.91|01〉+ (0.22 + ι0.36)|10〉 RS-6 (50,152) 0.1s 0.9s

0.07|01〉+ (−0.77 + ι0.64)|10〉 RS-7 (172,285) 0.1s 1.1s0.06|01〉+ (0.99− ι0.16)|10〉 RS-8 (174,346) 0.3s 1.1s

by computing the time upto which state fidelity does not fall below 0.8. The last col-umn in the table displays the time for which the state remains protected after applyingNUDD, estimated by computing the time upto which state fidelity does not fall below0.8. While the NUDD scheme is able to protect specific states in the subspace withvarying degrees of success (as evidenced from the entries in the last column of in Ta-ble 4.1), on an average as seen from the bar plot of the average fidelity in Fig. 4.11(i),the scheme performs quite well.

4.4 Conclusions

In this chapter, a three-layer nested UDD sequence was experimentally implementedon an NMR quantum information processor and explored its efficiency in protectingarbitrary states in a two-dimensional subspace of two qubits. The nested UDD layerswere applied in a particular sequence, and the full NUDD scheme was able to achievesecond order decoupling of the system and bath. The scheme is sufficiently generalas it does not assume prior information about the explicit form of the system-bathcoupling. The experiments were highly demanding, with the control operations being

102

4.4 Conclusions

complicated and involving manipulations of both qubits simultaneously. However,our results demonstrate that such systematic NUDD schemes can be experimentallyimplemented, and are able to protect multiqubit states in systems that are arbitrarilycoupled to quantum baths.

In addition, to demonstrate that the NUDD scheme does not depend on the actualform of an initial superposition state. This scheme was applied to arbitrary states withrandomly sampled coefficients and the experimental results shows that advantage ofthe NUDD schemes lies in the fact that one is sure that some amount of state protec-tion will always be achieved. Furthermore, one need not know anything about the stateto be protected or the nature of the quantum channel responsible for its decoherence.All one needs to know is the subspace to which the state belongs. In summary, if theQIP experimentalist has full knowledge of the state to be protected, it is better to useUDD schemes that are not nested. However, if there is only partial knowledge of thestate, the QIP experimentalist would do better to use these “generic” NUDD schemes.If we do not know the subspace to which the state belongs, we need to consider the fullspace, which increases the number of nesting layers, and experimentally implementa-tion becomes a difficult task. The study in this chapter points the way to the realisticprotection of fragile quantum states up to high orders and against arbitrary noise.

103

4. Experimental protection of unknown states using nested Uhrig dynamicaldecoupling sequences

104

Chapter 5

Experimentally preservingtime-invariant discord usingdynamical decoupling

5.1 Introduction

Interaction of quantum systems with their environment causes the destruction of in-trinsic quantum properties such as quantum superposition, quantum entanglement andmore general quantum correlations [178]. It has been observed that quantum entan-glement may disappear completely but there still may exist quantum correlations ina quantum system. Quantum correlations are thus more fundamental than entangle-ment [179]. The quantification of quantum correlations, distinction from their classi-cal counterparts, and their behavior under decoherence, is of paramount importanceto quantum information processing [1]. Several measures of nonclassical correlationshave been developed [179] and their signatures experimentally measured on an NMRsetup [180, 181, 182]. Quantum discord is a measure of nonclassical correlations thatare not accounted for by quantum entanglement [183]. While the intimate connectionof quantum entanglement with quantum nonlocality is well understood and entangle-ment has long been considered a source of quantum computational speedup [184], theimportance of quantum discord, its intrinsic quantumness and its potential use in quan-tum information processing protocols, is being explored in a number of contemporarystudies [185].

A surprising recent finding suggests that for certain quantum states up to some timet, quantum correlations are not destroyed by decoherence whereas classical correla-tions decay. After time t the situation is reversed and the quantum correlations begin

105

5. Experimentally preserving time-invariant discord using dynamical decoupling

to decay [70, 186]. This inherent immunity of such quantum correlations to environ-mental noise throws up new possibilities for the characterization of quantum behaviorand its exploitation for quantum information processing. The peculiar “frozen” behav-ior of quantum discord in the presence of noise was experimentally investigated usingphotonic qubits [187] and NMR qubits [188, 189]. The class of initial quantum statesthat exhibit this sudden transition in their decay rates was theoretically studied underthe action of standard noise channels, and it was inferred that the type of states thatdisplay such behavior depends on the nature of the decohering channel being consid-ered [190]. Dynamical decoupling (DD) methods have been proposed to protect quan-tum discord from environment-induced errors [[191]] and a recent work showed thatinterestingly, DD schemes can also influence the timescale over which time-invariantquantum discord remains oblivious to decoherence [192].

In this chapter we demonstrate the remarkable preservation of time-invariant quan-tum discord upon applying time-symmetric DD schemes of the bang-bang variety, ona two-qubit NMR quantum information processor. We begin by looking at a measurefor quantum and classical correlations for a special class of two-qubit quantum states,namely, Bell-diagonal (BD) states. Then the noise affecting a spin system is charac-terized by experimentally measuring the relaxation of the spins. Considering the spinsystem used which is a heteronuclear and the experimental results of the relaxationparameters, a model is proposed that the decoherence channel acting on the two qubitsis mainly a phase damping channel acting independent on each qubit. According tothis model zero quantum coherences and double quantum coherences decay with samerates which is equal to the sum of single quantum coherences decay rates. Experi-mental results validate this model. The BD state experimentally prepared, its decayis observed and found the quantum correlations freeze for a time interval. The resultsobtained from the experiments and theoretical noise model are compared. In the finalsection of this chapter, several robust DD sequences are applied to the system in theBD state, resulting in a significantly prolonged freezing time of quantum correlations.

5.2 Measure for correlations of two qubitsConsider a bipartite system made of an system A and B into a Hilbert space HA ⊗HB with HA and HB is a Hilbert state of system A and B respectively. The totalcorrelations of a state in a bipartite quantum system are measured by the quantummutual information I(ρAB) defined as

I(ρAB) = S(ρA) + S(ρB)− S(ρAB) (5.1)

where S(ρ) = −Tr ρ log2 ρ is the von Neumann entropy.

106

5.2 Measure for correlations of two qubits

Then the quantum discord is defined as

D(ρAB) = I(ρAB)− C(ρAB) (5.2)

where C(ρAB) is a classical correlations of the state [70, 193]. Next we specify thequantity used for measuring the classical correlations. Such a quantity is based onthe generalization of the concept of conditional entropy. We know that performingmeasurements on system B affects our knowledge of system A. How much system Ais modified by a measurement of B depends on the type of measurement performedon B. Here the measurement is considered of von Neumann type. It is described bya complete set of orthonormal projectors Πk on subsystem B corresponding to theoutcome k. The classical correlations C(ρAB) are then defined as

C(ρAB) = maxΠk

[S(ρA)− S(ρρA|Πk)] (5.3)

where the maximum is taken over the set of the projective measurements Πk andS(ρρA|Πk) =

∑k pkS(ρk) is the conditional entropy of A, given the knowledge of

the state of B, with pk = TrAB(ρABΠk) and ρk = TrB(ΠkρABΠk)/pk.Consider a two-qubit quantum system with its state space spanned by the states

|00〉, |01〉, |10〉, |11〉 and the eigenstates of the Pauli operators σ1z ⊗σ2

z . Any state forsuch a system is locally equivalent to

ρ =1

4

(I + ~a.~σ ⊗ I + I ⊗~b.~σ +

3∑j=1

cjσj ⊗ σj)

(5.4)

where I is the identity operator, ~σ = (σx, σy, σz) with Pauli spin observables inthe x,y,z direction, ~u = (ux, uy, uz) ∈ R3, ~u~σ = uxσx + uyσy + uzσz (u = a, b, c) .Here we are only interested in the family of BD states which is described by a densitymatrix

ρAB =1

4

(I +

3∑j=1

cjσj ⊗ σj)

(5.5)

where cj are real constants such that ρAB is a valid density operator. The eigenvaluesof the density matrix ρAB are given by

107

5. Experimentally preserving time-invariant discord using dynamical decoupling

λ0 =1

4(1− c1 − c2 − c3),

λ1 =1

4(1− c1 + c2 + c3),

λ2 =1

4(1 + c1 − c2 + c3),

λ3 =1

4(1 + c1 + c2 − c3).

(5.6)

The reduced density matrices of ρAB as given in Eq. 5.5 are ρA = I2

and ρB = I2. The

total correlations I(ρ) are given by:

I(ρ) = 2 +3∑l=0

λl log2 λl (5.7)

The classical correlations for the BD states given as

C[ρAB] =2∑j=1

1 + (−1)jχ

2log2[1 + (−1)jχ], (5.8)

where χ = max|c1|, |c2|, |c3|. The maximization procedure with respect tothe projective measurements, present in the definition of the classical correlations ofEq.(5.3), can be performed explicitly for the system here considered noticing thatthe complete set of the orthogonal projectors is given by Πj = |θj〉〈θj|, with j=1,2,|θ1〉 = cosθ|0〉+ e−iφsinθ|1〉, |θ2〉 = sinθ|0〉+ e−iφcosθ|1〉, and the state of the systemalways remains of the form given by Eq.(5.5) during time evolution [70].

5.3 Characterization of noise channelsIn high field NMR, the Zeeman interaction causes a splitting of the energy levels ac-cording to the field direction and the difference between magnetic quantum numbers

∆mrs = mr −ms (5.9)

defines the order of the coherence. If ∆mrs = 0 the coherence is a zero quantum (ZQ)coherence, if ∆mrs = ±1 the coherence is a single quantum (SQ) coherence, and if∆mrs = ±2 the coherence is a double quantum (DQ) coherence. In general, a densitymatrix element ρrs represents p-quantum coherence (p = mr −ms) [23, 97]. For our

108

5.3 Characterization of noise channels

two-qubit system we use for the experiments, the experimentally measured longitudi-nal NMR spin relaxation times are TH

1 ≈ 7.9 s and TC1 ≈ 16.6 s, which are much longer

than the measured effective NMR transverse spin relaxation rates TH2 = 0.513± 0.01 s

and TC2 = 0.193±0.005 s. Since we study the decoherence of the state for an evolution

time of approximately 0.5 s, we neglect the effects of the amplitude-damping channelwhose effects are associated with T1 and assume that the main noise channel for oursystem is the phase-damping channel. The effective transverse spin relaxation rateswere measured by applying a 90 excitation pulse followed by a decay interval (withno refocusing pulse) and by fitting the resulting decay of the magnetization. The decayrate of SQ coherence of spin(i) is given by 1/T i2. Our experimental system is heteronu-clear, with two different nuclear species (proton and carbon), having very differentLarmor resonance frequencies and hence large chemical shift differences. We hencehypothesize that each spin decoheres in an independent phase damping channel whichis not correlated with that of the other spin. Therefore, we model the phase dampingchannel for this system as a homogeneous dephasing channel acting independently oneach qubit [194]. We use the operator-sum representation formalism and the associatedphase-damping Kraus operators [1].The Kraus operators for a two-qubit system underphase damping channel are:

E1(t) =1

2(1 + e−γ1t)

12 (1 + e−γ2t)

12 I ⊗ I,

E2(t) =1

2(1 + e−γ1t)

12 (1− e−γ2t) 1

2 I ⊗ σz,

E3(t) =1

2(1− e−γ1t) 1

2 (1 + e−γ2t)12σz ⊗ I,

E4(t) =1

2(1− e−γ1t) 1

2 (1− e−γ2t) 12σz ⊗ σz, (5.10)

where γ1 = 1/T i2 is the decay constant of the ith spin, I is the identity matrix and σz isa Pauli matrix. ∑

j

Ej(t)Ej(t)† = 1, (5.11)

ρBD(t) =∑j

Ej(t)ρAB(0)Ej(t)†. (5.12)

Once we fix the model, we can apply it to any state. One of the qualitative predic-tions of the model is that the DQ and ZQ decay rates are equal. We measured these ratesin two independent experiments and compared them with those predicted by the abovemodel (whose parameters are completely fixed). The experimentally measured DQ andZQ decay rates were γDQ = 6.395±0.23 s−1 and γZQ = 6.138±0.275 s−1 respectively,and the theoretically predicted common decay was γ1 +γ2 = 1

TH2+ 1

TC2= 7.21±0.173

s−1, showing the model works.

109

5. Experimentally preserving time-invariant discord using dynamical decoupling

5.3.1 Bell-diagonal states under a dephasing channelFor a two-qubit system where each qubit is affected by an independent local dephasingchannel, to see the evolution under the phase-damping channel acting independentlyon each qubit, we use the operator-sum representation formalism and the associatedphase-damping Kraus operators [1]. The evolution in a noisy environment is governedby the phase-damping channel in Eq. (5.10), and it turns out for Bell-diagonal statesρAB

c1(t) = c1(0)exp[−(γ1 + γ2)t],

c2(t) = c2(0)exp[−(γ1 + γ2)t],

c3(t) = c3(0). (5.13)

5.4 Time invariant quantum correlationsWe now consider a system in the special class of Bell-diagonal states for which c1(0) =±1 and c2(0) = ∓c3(0), with |c3| < 1 under the dephasing channel as define above inEq. (5.10). At time t the total correlations are given by

I[ρAB(t)] =2∑j=1

1 + (−1)jc3(t)

2log2[1 + (−1)jc3(t)] +

2∑j=1

1 + (−1)jc1(t)

2log2[1 + (−1)jc1(t)], (5.14)

the classical correlations are given by

C[ρAB(t)] =2∑j=1

1 + (−1)jχ(t)

2log2[1 + (−1)jχ(t)], (5.15)

where χ(t) = max|c1(t)|, |c2(t)|, |c3(t)|, and the quantum correlations are given by

D[ρAB(t)] = I[ρAB(t)]− C[ρAB(t)]. (5.16)

We can see that since our system is mainly affected by the phase damping channel,for

t < t = − ln(|c3|)2γ

(5.17)

quantum correlations remain constant

D[ρAB(t < t)] =2∑j=1

1 + (−1)jc3(0)

2log2[1 + (−1)jc3(0)], (5.18)

110

5.5 Experimental realization of time-invariant discord

and after t > t the quantum correlations start decreasing towards zero and the classicalcorrelations of system remain constant which is given by:

C[ρAB(t > t)] =2∑j=1

1 + (−1)jc3(0)

2log2[1 + (−1)jc3(0)]. (5.19)

5.5 Experimental realization of time-invariant discord

5.5.1 NMR SystemWe create and preserve time-invariant discord in a two-qubit NMR system of chloroform-13C, with the 1H and 13C nuclear spins encoding the two qubits (Fig. 4.2). The ensem-ble of nuclear spins is placed in a longitudinal strong static magnetic field (B0 ≈14.1T) oriented along the z direction. The 1H and 13C nuclear spins precess around B0

at Larmor frequencies of ≈ 600 MHz and ≈ 150 MHz, respectively. The evolution ofspin magnetization is controlled by applying rf-field pulses in the x and y directions.The internal Hamiltonian of the system in the rotating frame is given by Eq.(4.26)

The two-qubit system was initialized into the pseudopure state |00〉 by the spatialaveraging technique [24]. Density matrices were reconstructed from experimental databy using a reduced set of quantum state tomography (QST) operations combined withthe maximum likelihood method [59] as described in Chapter 2 to avoid any negativeeigen values. The fidelity F of all the experimental density matrices reconstructed inthis work was computed using the Eq.(2.32). The experimentally created pseudopurestate |00〉 was tomographed with a fidelity of 0.99, and the NMR signal of this statewas used as a reference for computation of state fidelity in all subsequent time-invariantdiscord experiments.

5.5.2 Observing time-invariant discordA class of Bell-diagonal (BD) states with maximally mixed marginals defined in termsof Pauli operators σi are

ρBD =1

4( I +

3∑i=1

ci σi ⊗ σi) (5.20)

where the coefficients ci with 0 ≤ |ci| ≤ 1 determine the state completely and can becomputed as ci = 〈σi ⊗ σi〉.

We aim to prepare an initial BD state with the parameters c1(0) = 1, c2(0) = 0.7,c3(0) = −0.7. The NMR pulse sequence for the preparation of this state from the |00〉

111

5. Experimentally preserving time-invariant discord using dynamical decoupling

tt

♥tst♦♥

②♠ ♣q

P❨

① ① ① ① ① ① ① ①

①① ① ① ①

❳

❳

❳

Figure 5.1: (a) Quantum circuit for the initial pseudopure state preparation, followed bythe block for BD state preparation. The next block depicts the DD scheme used to preservequantum discord. The entire DD sequence is repeated N times before measurement. (b)NMR pulse sequence corresponding to the quantum circuit. The rf pulse flip angles are setto α = 46 and β = 59.81, while all other pulses are labeled with their respective anglesand phases.

pseudopure state is given in Fig. 5.1(b). Preparing the BD state involves manipulationof NMR multiple-quantum coherences by applying rotations in the zero-quantum anddouble-quantum spin magnetization subspaces. Since the molecule is a heteronuclearspin system, high-power, short-duration rf pulses were used for gate implementation(with rf pulses of flip angles α = 45.57 and β = 59.81 in Fig. 5.1 having pulselengths of 6.85µs and 5.02µs, respectively). The experimentally achieved ρE

BD (re-constructed using the maximum likelihood method [59] had parameters c1(0) = 1.0,c2(0) = 0.680 and c3(0) = −0.680 and a computed fidelity of 0.99. The experimen-tally reconstructed density matrix (using quantum state tomography and maximumlikelihood) was found to be:

ρEBD ==

0.080 0.003 + 0.000i 0.003 + 0.000i 0.080 + 0.000i

0.003− 0.000i 0.420 0.420 + 0.001i 0.003 + 0.000i0.003− 0.000i 0.420− 0.001i 0.420 0.003 + 0.000i0.080− 0.000i 0.003− 0.000i 0.003− 0.000i 0.080

112

5.5 Experimental realization of time-invariant discord

Correlation functions for the BD states can be computed readily to give the classicalcorrelations (C[ρ(t)]), the quantum discord (D[ρ(t)]), and total correlations (I[ρ(t)] [70]:

C[ρ(t)] =2∑j=1

1 + (−1)jχ(t)

2log2[1 + (−1)jχ(t)]

I[ρ(t)] =2∑j=1

1 + (−1)jc1(t)

2log2[1 + (−1)jc1(t)]

+2∑j=1

1 + (−1)jc3

2log2[1 + (−1)jc3]

D(ρ) ≡ I(ρ)− C(ρ) (5.21)

where χ(t) = max|c1(t)|, |c2(t)|, |c3(t)|.For the class of BD states with coefficients c1 = ±1, c2 = ∓c3, |c3| < 1, the evo-

lution under a noisy environment is governed by the noise operators given in Eq.5.10,and it turns out that c1(t) = c1(0)exp[−(γ1 + γ2)t], c2(t) = c2(0)exp[−(γ1 + γ2)t],c3(0) ≡ c3, and the quantum discord does not decay up to some finite time t [70].Initially (t = 0), the computed correlations in the ρE

BD state (taken as an average offive density matrices initially prepared in the same state) turned out to be C[ρ(0)] =0.999± 0.169, D[ρ(0)] = 0.366± 0.024 and I[ρ(0)] = 1.366± 0.017. The simulatedand experimental plots of the dynamics of the quantum discord, classical correlationsand total correlations are shown in Fig.5.2 (a) and (b) respectively, and shows a dis-tinct transition from the classical to the quantum decoherence regimes at t = t. Thetransition time up to which quantum discord remains constant is t = 1

2γln∣∣∣ c1(0)c3(0)

∣∣∣.We allowed the experimentally prepared state ρE

BD, to evolve freely in time anddetermined the parameters ci at each time point. We used these experimentally de-termined coefficients to compute the classical correlations (C[ρ(t)]), the quantum dis-cord (D[ρ(t)]), and total correlations (I[ρ(t)]) at each time point. The transition timeup to which quantum discord remains constant was experimentally determined to bet = 0.052s. The experimental results correlate well with the theoretical noise modelwhich allows to calculate t and the value comes out to be 0.054 s. The tomographeddensity matrix, reconstructed at different time points, shows that state evolution re-mains confined to the subspace of BD states, as is evident from the experimentallyreconstructed density matrices of the state at different time points displayed in Fig.5.3.

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5. Experimentally preserving time-invariant discord using dynamical decoupling

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

(a) Simulation

Time(s)

Correlation

t

t = 0.05s

Total corr.

Classical corr.

Quantum discord

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

(b) Experiment

Time(s)

Correlation

t

t = 0.05s

Total corr.

Classical corr.

Quantum discord

Figure 5.2: Time evolution of total correlations (triangles), classical correlations (squares)and quantum discord (circles) of the BD state: (a) Simulation, (b) Experimental plot with-out applying any preservation.

5.6 Protection of time-invariant discord using dynami-cal decoupling schemes

DD schemes, consisting of repeated sets of π pulses with tailored inter-pulse delaysand phases, have played an important role in dealing with the debilitating effects ofdecoherence [166]. Several NMR QIP experiments have successfully used DD-typeschemes to preserve quantum states [63, 195]. These schemes are expected to providean advantage over traditional Carr-Purcell-Meiboom-Gill (CPMG) refocusing schemesand here we compared the two experimentally.

We first protect these states and their related quantum correlations using standardCPMG schemes of the type (τ − π − τ − π). The results are shown in Fig.5.4(a) and(b) for two different values of τ (the time interval between two consecutive π pulses).It turns out that the CPMG schemes which are based on the traditional Hahn spin echoare able to provide protection to some extent and the life time of the time invariantquantum discord grows from 0.05 s to 0.11 s as we increase the number of π pulses.However, as we try to increase the number of π pulses further the results are counter-productive as shown in Fig.5.4(b). Therefore, beyond a point we have to give up thestandard Hahn spin echo strategy.

Next we implemented more sophisticated DD sequences to explore if we can fur-ther protect the BD state, and hence the related quantum correlations typified via thesurvival time of the quantum discord t. If the π pulses in a DD sequence are non-ideal

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5.6 Protection of time-invariant discord using dynamical decoupling schemes

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(b)

(c)

(d)

(e)

Initial State (T = 0s)

T = 0.06s

T = 0.12s

T = 0.17s

T = 0.23s

Simulated Experimental

Figure 5.3: Real (left) and imaginary (right) parts of the experimental tomographs of the(a) Bell Diagonal (BD) state, with a computed fidelity of 0.99. (b)-(e) depict the state atT = 0.06, 0.12, 0.17, 0.23s, with the tomographs on the left and the right representing thesimulated and experimental state, respectively. The rows and columns are labeled in thecomputational basis ordered from |00〉 to |11〉.

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5. Experimentally preserving time-invariant discord using dynamical decoupling

0 0.1 0.2 0.30

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(a) CPMG(τ = 0.57ms)

Time(s)

Correlation

t

t = 0.11s

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Classical corr.

Quantum discord

0 0.1 0.2 0.30

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1

1.2

1.4

(b) CPMG(τ = 0.38ms)

Time(s)

Correlation

t

t = 0.07s

Total corr.

Classical corr.

Quantum discord

Figure 5.4: Time evolution of total correlations (triangles), classical correlations (squares)and quantum discord (circles) of the BD state: Experimental plots using CPMG preservingsequences (a) CPMG with τ = 0.57 ms and (b) CPMG with τ = 0.38 ms.

(either due to finite pulse lengths or flip angle and off-resonant driving errors), it leadsto imperfect system-bath decoupling. Several schemes have been designed to makeDD sequences robust against pulse imperfections by achieving ideal pulse rotations.Standard CPMG-based DD sequences used π pulses applied along the same rotationaxis, which then preserve coherence along only one spin component. The XY familyof DD sequences applies pulses along two orthogonal (x, y) axes, which preserves co-herences about both spin rotation axes. The basic XY4(s) DD sequence is a four-pulsesequence with phases x − y − x − y, with pulses applied at the center of each timeperiod such that the whole sequence is time-symmetric with respect to its center [68].The XY8(s) DD sequence uses the XY4(s) sequence as a building block, by combin-ing the XY4(s) sequence with its time-reversed version, so that the whole eight-pulsesequence is explicitly time-symmetric. Each cycle of the symmetrized DD sequencesis applied several times to achieve higher-order decoupling [172]. The Knill dynami-cal decoupling (KDD) sequence combines the rotation pattern of the XY4(s) sequencewith composite pulses, replacing each π pulse in the DD sequence with a compositesequence of five pulses with different phases [[172]]:

KDDφ = (π)π/6+φ − (π)φ − (π)π/2+φ − (π)φ − (π)π/6+φ. (5.22)

The additional phases in the KDDxy (xy in the subscript denoting pulses appliedalong both axes) lead to better compensation for pulse errors, combining two of thebasic five-pulse blocks given in Eq.(5.22) that are shifted in phase by π

2i.e [KDD0-

KDDπ2]. The four symmetrized DD schemes used in our experiments to preserve time-

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5.6 Protection of time-invariant discord using dynamical decoupling schemes

1H

13C

x yy x

τ42

τ4 τ4 τ4τ42

XY 4(s)(a)

1H

13C

x y x y y x y x

τ82

τ8 τ8 τ8 τ8 τ8 τ8 τ8 τ82

XY 8(s)(b)

1H

13C

x y x y y x y x -x -y -x -y -y -x -y -x

τ2

τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ2

XY 16(s)(c)

1H

13C

π6

0 π2

0 π6

2π3

π2

π π2

2π3

τk2

τk τk τk τk τk τk τk τk τk τk2

2KDDxy

(d)

Figure 5.5: NMR pulse sequence corresponding to DD schemes (a) XY4(s), (b) XY8(s),(c) XY16(s), and (d) KDDxy , with time delays between pulses denoted by τ4 , τ8, τ , τk ,respectively. All the pulses are of flip angle π and are labeled with their respective phases.The pulses are applied simultaneously on both qubits. The superscript ‘2’ in the KDD xysequence denotes that one unit cycle of this sequence contains two blocks of the ten-pulseblock represented schematically, i.e., a total of twenty pulses. The shorter duration protonpulses and the longer duration carbon pulses are centered on each other and the varioustime delays (τ4, τ8 , τ , τk) in all the DD schemes are tailored to the gap between twoconsecutive carbon pulses.

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5. Experimentally preserving time-invariant discord using dynamical decoupling

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

(a) XY4(s)

Time(s)

Correlation

t

t = 0.11s

Total corr.

Classical corr.

Quantum discord

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

(b) XY8(s)

Time(s)

Correlation

t

t = 0.13s

Total corr.

Classical corr.

Quantum discord

Figure 5.6: Time evolution of total correlations (triangles), classical correlations (squares)and quantum discord (circles) of the BD state: Experimental plots using (a) XY4(s) withτ4 = 0.58 ms and (b) XY8(s) with τ8 = 0.29 ms.

invariant discord, namely XY4(s), XY8(s), XY16(s) and KDDxy , are schematicallyrepresented in Fig.5.5. The ‘2’ in the superscript of the KDDxy scheme denotes thatone unit cycle of this scheme contains two of ten-pulse blocks, for a total of twentypulses. We applied π pulses simultaneously on both spins, with pulse lengths of 15.1µs and 26.8 µs for the proton and carbon spins, respectively. The proton and carbonpulses are centered on each other and the time delay between pulses was set at the gapbetween two consecutive carbon pulses. Each DD sequence was applied a repeatednumber of times (N being as large as experimentally possible), for good coherencepreservation.

We implemented the symmetrized XY4(s) DD scheme for τ4 = 0.58 ms and anexperimental time for one run of 2.43 ms. The time for which quantum discord persistsusing the XY4(s) scheme is t = 0.11 s, which is double the time as compared to ‘no-preservation’ (Fig. 5.6(a)). We implemented the XY8(s) DD scheme for τ8 = 0.29ms and an experimental time for one run of 2.52 ms. The time for which quantumdiscord persists using the XY8(s) scheme is t = 0.13s, nearly the same as the XY4(s)scheme (Fig. 5.6(b)). The XY16(s) DD scheme provides even better compensationthan the XY8(s) sequence. We implemented the XY16(s) scheme for τ = 0.145 msand an experimental time for one run of 2.75 ms. The time for which quantum discordpersists for the XY16(s) scheme is t = 0.24 s, which is four times the persistencetime of the discord when no preservation is applied (Fig.5.8(a)). We implementedthe KDDxy sequence with τ = 0.116 ms and an experimental time for one run of2.86 ms. In this case too, the time for which quantum discord persists t = 0.22 s

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5.6 Protection of time-invariant discord using dynamical decoupling schemes

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(a)

(b)

(c)

(d)

(e)

N = 20

N = 40

N = 60

N = 80

N = 100

XY4 XY8

Real Imaginary Real Imaginary

Figure 5.7: Real (left) and imaginary (right) parts of the experimental tomographs of the(a)-(e) depict the BD state at N = 20, 40, 60, 80, 100, with the tomographs on the left andthe right representing the BD state after applying the XY4 and XY8 scheme, respectively.The rows and columns are labeled in the computational basis ordered from |00〉 to |11〉.

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5. Experimentally preserving time-invariant discord using dynamical decoupling

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

(g) XY16(s)

Time(s)

Correlation

t

t = 0.24s

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Classical corr.

Quantum discord

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

(h) KDDXY

Time(s)

Correlation

t

t = 0.22s

Total corr.

Classical corr.

Quantum discord

Figure 5.8: Time evolution of total correlations (triangles), classical correlations (squares)and quantum discord (circles) of the BD state: Experimental plots using (a) XY16(s) withτ = 0.145 ms and (b) KDDxy with τk = 0.116 ms.

is quadrupled, as compared to no-preservation (Fig.5.8(b)). The τ delays are chosenin such a way that for a given sequence the protection is maximum. If we decreaseτ delay intervals further, errors due to π pulses dominates. The DD sequence waslooped for five times between each time point in Figs.5.4, 5.6 and 5.8, which typicallymeans repeating a DD sequence 175-200 times during an experiment covering all timepoints. The experimental tomographs of the BD state at different instances of time afterusing XY4(s), XY8(s), XY16(s) and KDDxy DD preservation schemes to protect time-invariant discord are given in Fig.5.7 and Fig.5.9. All the experimental tomographsdisplay excellent preservation of the state, with very little leakage.

In Fig.5.10, we plotted the protection of quantum entanglement in the two-qubitBD state via CPMG and DD schemes, using negativity as an entanglement measure. Itturns out that the results of entanglement protection are similar to that of discord pro-tection namely, that CPMG and XY4(s) schemes protect entanglement to some extentwhile the more involved DD schemes are able to protect entanglement for relativelylong times up to 0.3 s (as compared to its natural decay time of 0.12 s without any DDprotection). However, the phenomenon of freezing of entanglement is not observed,unlike the case for quantum discord.

5.6.1 Conclusions

The two-qubit system we used in our experiments is mainly affected by an independentphase damping channel on each qubit for the experimental time regime under consid-

120

5.6 Protection of time-invariant discord using dynamical decoupling schemes

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N = 100

XY16 KDDxy

Real Imaginary Real Imaginary

Figure 5.9: Real (left) and imaginary (right) parts of the experimental tomographs in (a)-(e) depict the Bell Diagonal (BD) state atN = 20, 40, 60, 80, 100, with the tomographs onthe left and the right representing the BD state after applying the XY16 and KDDxy pre-serving DD schemes, respectively. The rows and columns are labeled in the computationalbasis ordered from |00〉 to |11〉.

121

5. Experimentally preserving time-invariant discord using dynamical decoupling

0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

Time(s)

Entanglem

ent

Without DDCPMGXY4(s)XY8(s)XY16(s)KDDxy

Figure 5.10: Plot of time evolution of entanglement without applying any preservation(filled circle), CPMG with τ = 0.57 ms (empty circle), XY4(s) (filled rectangle), XY8(s)(empty rectangle), XY16(s) (filled triangle) and KDDxy (empty triangle),respectively.

eration. One of the main sources of relaxation in NMR is dipolar relaxation. Suchnoise can be ideally suppressed by the CPMG sequence. The suppression of noise isexpected to increase by reducing the delay between the π pulses, but due to imper-fect π pulses after a certain time point instead of suppressing noise, starts contributingto the noise. Also, the CPMG sequence is designed to protect magnetization alongone particular axis, not for a general axis. To mitigate all these problems we usedXY4(s), XY8(s), XY16(s) and KDDxy DD sequences, which are robust against theseimperfections and protect magnetization along the general axis. Our results show thattime-invariant quantum discord, which remains unaffected under certain decoherenceregimes, can be preserved for very long times using DD schemes. Our experimentshave important implications in situations where persistent quantum correlations haveto be maintained to carry out quantum information processing tasks.

122

Chapter 6

Dynamics of tripartite entanglementunder decoherence and protectionusing dynamical decoupling

6.1 Introduction

In Chapter 5, we considered the situation where we have knowledge of the state ofthe system as well as its interaction with the environment. In such situations, sincethe noise model is known, decoupling strategies can be designed to cancel this noise.Using these decoupling strategies, we experimentally extended the lifetime of time in-variant discord of two-qubit Bell-diagonal states. In this chapter we extend the sameidea for experimental preservation of three-qubit entangled states. Quantum entangle-ment is considered to lie at the crux of QIP [1] and while two-qubit entanglement canbe completely characterized, multipartite entanglement is more difficult to quantifyand is the subject of much recent research [184]. Entanglement can be rather frag-ile under decoherence and various multiparty entangled states behave very differentlyunder the same decohering channel [196]. It is hence of paramount importance tounderstand and control the dynamics of multipartite entangled states in multivariousnoisy environments [197, 198, 199]. A three-qubit system is a good model systemto study the diverse response of multipartite entangled states to decoherence and theentanglement dynamics of three-qubit GHZ and W states have been theoretically wellstudied [200, 201]. Under an arbitrary (Markovian) decohering environment, it wasshown that W states are more robust than GHZ states for certain kinds of channelswhile the reverse is true for other kinds of channels [202, 203, 204, 205].

On the experimental front, tripartite entanglement was generated using photonic

123

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

qubits and the robustness of W state entanglement was studied in optical systems [206,207, 208, 209]. The dynamics of multi-qubit entanglement under the influence ofdecoherence was experimentally characterized using a string of trapped ions [210]and in superconducting qubits [211]. In the context of NMR quantum informationprocessing, three-qubit entangled states were experimentally prepared [28, 138, 212,213], and their decay rates compared with bipartite entangled states [43].

With a view to protecting entanglement, dynamical decoupling (DD) schemes havebeen successfully applied to decouple a multiqubit system from both transverse de-phasing and longitudinal relaxation baths [65, 105, 152, 173]. UDD schemes havebeen used in the context of entanglement preservation [214, 215], and it was showntheoretically that Uhrig DD schemes are able to preserve the entanglement of two-qubit Bell states and three-qubit GHZ states for quite long times [216].

In this chapter, first the Lindblad master equation is solved analytically for the ro-bustness of three different tripartite entangled states, namely, the GHZ, W and WWstates with different noise channels. Then the robustness against decoherence of thesethree different tripartite entangled states are experimentally explored. The WW stateis a novel tripartite entangled state which belongs to the GHZ entanglement class inthe sense that it is SLOCC equivalent to the GHZ state, however stores its entangle-ment in ways very similar to that of the W state [139, 217]. Next, the experimentaldata are best modeled by considering the main noise channel to be an uncorrelatedphase damping channel acting independently on each qubit, along with a generalizedamplitude damping channel. Next, the entanglement of these states is protected us-ing two different DD sequences: the symmetrized XY-16(s) and the Knill dynamicaldecoupling (KDD) sequences, and evaluated their efficacy of protection.

6.2 Dynamics of tripartite entanglement

6.2.1 Tripartite entanglement under different noise channelsWe considered four different noise channels: phase damping (Pauli σz), amplitudedamping (Pauli σx), bit-phase flip (Pauli σy) and a uniform depolarizing channel alongthe lines suggested in Reference [218]. The master equation is given by [219]:

∂ρ

∂t= −i[Hs, ρ] +

∑i,α

[Li,αρL

†i,α −

1

2L†i,αLi,α, ρ

](6.1)

where Hs is the system Hamiltonian, Li,α ≡ √κi,ασ(i)α is the Lindblad operator acting

on the ith qubit and σ(i)α is the Pauli operator on the ith qubit, α = x, y, z; the con-

stant κi,α turns out to be the inverse of the decoherence time. This master equation

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6.2 Dynamics of tripartite entanglement

approach has been shown to be equivalent to the standard operator sum representationmethod for open quantum systems, where density operator evolution is given in termsof the standard Kraus operators for various noise channels [1]. For two qubits, all en-tangled states are negative under partial transpose (NPT) and for such NPT states, theminimum eigenvalues of the partially transposed density operator is a measure of en-tanglement [220]. This idea has been extended to three qubits, and entanglement canbe quantified for a three-qubit system using the well-known tripartite negativity N

(3)123

measure [201, 221]:N

(3)123 = [N1N2N3]1/3 (6.2)

where the negativity of a qubit Ni refers to the most negative eigenvalue of the partialtranspose of the density matrix with respect to the qubit i.

0 0.2 0.4 0.6 0.8 1.00

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0.4

0.6

0.8

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κt

N(3

)

GHZWW

W

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0.4

0.6

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κt

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)

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W

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0.2

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0.8

1.0(c) σz

κt

N(3

)

GHZWW

W

0 0.1 0.20

0.2

0.4

0.6

0.8

1.0(d) Depolarizing

κt

N(3

)

GHZWW

W

Figure 6.1: Simulation of decay of tripartite entanglement parameter negativity N(3) ofthe GHZ state (blue squares), the W state (red circles) and the WW state (green triangles)under the action of (a) amplitude damping (Pauli σx) channel, (b) bit-phase flip (Pauliσy) channel (c) phase damping (Pauli σz) channel and (d) isotropic noise (depolarizing)channel. The κ parameter denotes inverse of the decoherence time.

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6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

The analytical results of the Lindblad equation under the action of the differentnoise channels on tripartite entanglement are displayed in Fig. 6.1, where Figs. 6.1(a)-(d) show the decohering effects of the amplitude damping (Pauli σx) channel, the bit-phase flip (Pauli σy) channel, the phase damping (Pauli σz) channel and the depolariz-ing (isotropic noise) channel respectively, on the GHZ, W and WW states. The simu-lation indicates that the W state is more robust against the phase-damping channel ascompared to the GHZ state, while the reverse is true for the amplitude-damping chan-nel. Both states decohere to nearly the same extent under the action of the bit-phase flipchannel and the depolarizing channel. The WW state decoheres more rapidly underthe actions of the bit-phase flip channel and the depolarizing channel, as compared tothe other GHZ and the W states. For short timescales its decoherence behavior mim-ics the W state under the action of the amplitude damping channel while for longertimescales it decoheres similar to the GHZ state. On the other hand, under the actionof the phase-damping channel the WW state initially decoheres faster than the GHZstate and later closely follows the GHZ decay behavior.

6.2.2 NMR systemWe use the three 19F nuclear spins of the trifluoroiodoethylene (C2F3I) molecule toencode the three qubits. On an NMR spectrometer operating at 600 MHz, the fluorinespin resonates at a Larmor frequency of ≈ 564 MHz. The molecular structure ofthe three-qubit system with tabulated system parameters and the NMR spectra of thequbits at thermal equilibrium and prepared in the pseudopure state |000〉 are shown inFigs. 6.2(a), (b), and (c), respectively. The Hamiltonian of a weakly-coupled three-spin system in a frame rotating at ωrf (the frequency of the electromagnetic field B1(t)applied to manipulate spins in a static magnetic field B0) is given by [97]:

H = −3∑i=1

(ωi − ωrf)Iiz +3∑

i<j,j=1

2πJijIizIjz (6.3)

where Iiz is the spin angular momentum operator in the z direction for 19F; the firstterm in the Hamiltonian denotes the Zeeman interaction between the fluorine spins andthe static magnetic field B0 with ωi = 2πνi being the Larmor frequencies; the secondterm represents the spin-spin interaction with Jij being the scalar coupling constants.The three-qubit equilibrium density matrix (in the high temperature and high fieldapproximations) is in a highly mixed state given by:

ρeq = 18(I + ε ∆ρeq)

∆ρeq ∝3∑i=1

Iiz (6.4)

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6.2 Dynamics of tripartite entanglement

Figure 6.2: (a) Molecular structure of trifluoroiodoethylene molecule and tabulated sys-tem parameters with chemical shifts νi and scalar couplings Jij (in Hz), and spin-latticerelaxation times T1 and spin-spin relaxation times T2 (in seconds). (b) NMR spectrumobtained after a π/2 readout pulse on the thermal equilibrium state. and (c) NMR spec-trum of the pseudopure |000〉 state. The resonance lines of each qubit are labeled by thecorresponding logical states of the other qubit.

with a thermal polarization ε ∼ 10−5, I being the 8 × 8 identity operator and ∆ρeq

being the deviation part of the density matrix. The system was first initialized intothe |000〉 pseudopure state using the spatial averaging technique [24], with the densityoperator given by

ρ000 =1− ε

8I + ε|000〉〈000| (6.5)

The specific sequence of rf pulses, z gradient pulses and time evolution periods weused to prepare the pseudopure state ρ000 starting from thermal equilibrium is shownin Fig. 6.3. All the rf pulses used in the pseudopure state preparation scheme wereconstructed using the Gradient Ascent Pulse Engineering (GRAPE) technique [25]and were designed to be robust against rf inhomogeneity, with an average fidelity of

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6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

F 2

F 3

F 1

Gz

θ1

y y -y

y -y

y -y y -y

y -y y -y y -y

y -y y -y-

θ2

y

Up1 Up2 Up3 Up4

δ

y

δ

-y

δ

x

δ

-y

δ

x

δ

yτ13= 1

2J13τ12= 1

2J12τ23= 1

2J23

Figure 6.3: NMR pulse sequence used to prepare pseudopure state ρ000 starting fromthermal equilibrium.The pulses represented by black filled rectangles are of angle π. Theother rf flip angles are set to θ1 = 5π

12 , θ2 = π6 and δ = π

4 . The phase of each rf pulse iswritten below each pulse bar. The evolution interval τij is set to a multiple of the scalarcoupling strength (Jij).

≥ 0.99. Wherever possible, two independent spin-selective rf pulses were combinedusing a specially crafted single GRAPE pulse; for instance the first two rf pulses to beapplied before the first field gradient pulse, were combined into a single pulse speciallycrafted pulse (Up1 in Fig. 6.3), of duration 600µs. The combined pulses Up2 , Up3 andUp4 applied later in the sequence were of a total duration ≈ 20 ms.

All experimental density matrices were reconstructed using a reduced tomographicprotocol and by using maximum likelihood estimation [59, 95] as described in Chap-ter 2 with the set of operations III, IIY, IY Y, Y II,XY X,XXY, XXX; I is theidentity (do-nothing operation) and X(Y ) denotes a single spin operator implementedby a spin-selective π/2 pulse. We constructed these spin-selective pulses for tomogra-phy using GRAPE, with the length of each pulse≈ 600µs. The experimentally createdpseudopure state |000〉was tomographed with a fidelity of 0.99 and the total time takento prepare the state was ≈ 60 ms. The fidelity of an experimental density matrix wascomputed using Eq. (2.32).

128

6.2 Dynamics of tripartite entanglement

|0〉

|0〉

|0〉

s sR1(π2)−y

UG1 UG2 UG3

(a)

|0〉

|0〉

|0〉

ss

sR1(π)y

R2(0.39π)y

CR13(π2)y

(b)

UW1 UW2 UW3 UW4

|0〉

|0〉

|0〉

sss

sR1(π3)−y

CR12(2cos−1( 1√

3))y

CR21(π2)−y

R2(π2)y

R1(π2)y

R3(π2)y

(c)

UWW

Figure 6.4: (Quantum circuit showing the sequence of implementation of the single-qubitlocal rotation gates (labeled by R), two-qubit controlled-rotation gates (labeled by CR)and controlled-NOT gates required to construct the (a) GHZ state (b) W state and (c) WW

state.

6.2.3 Construction of tripartite entangled states

Tripartite entanglement has been well characterized and it is known that the two differ-ent classes of tripartite entanglement, namely GHZ-class and W-class, are inequivalent.While both classes are maximally entangled, there are differences in the their type ofentanglement: the W-class entanglement is more robust against particle loss than theGHZ-class (which becomes separable if one particle is lost) and it is also known thatthe W state has the maximum possible bipartite entanglement in its reduced two-qubitstates [222]. The entanglement in the WW state (which belongs to the GHZ-class

129

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

of entanglement) shows a surprising result, that it is reconstructible from its reducedtwo-qubit states (similar to the W-class of states). We now turn to the constructionof tripartite entangled states on the three-qubit NMR system. The quantum circuits toprepare the three qubits in a GHZ-type state, a W state and a WW state are shown inFigs. 6.4 (a), (b) and (c), respectively. Several of the quantum gates in these circuitswere optimized using the GRAPE algorithm and we were able to achieve a high gatefidelity and smaller pulse lengths.

000001

010011

100101

110111

000001

010011

100101

110111

000001

010011

100101

110111

000001

010011

100101

110111

000001010

011100101110

111

000001010011

100101

110111

000001010

011100

101110

111

000001

010011

100101110

111

000001

010011

100101

110111

000001

010011

100101

110111

000001010011100

101110

111

000001010

011100

101110111

(a)

(b)

(c)

Real Imaginary

Figure 6.5: The real (left) and imaginary (right) parts of the experimentally tomographed(a) GHZ-type state, with a fidelity of 0.97. (b) W state, with a fidelity of 0.96 and (c)WW state with a fidelity of 0.94. The rows and columns encode the computational basisin binary order from |000〉 to |111〉.

The GHZ-type 1√2(|000〉 − |111〉) state was prepared from the |000〉 pseudopure

state by a sequence of three quantum gates (labeled as UG1 , UG2 , UG3 in Fig. 6.4(a)):first a selective rotation of

[π2

]−y on the first qubit, followed by a CNOT12 gate, and

130

6.2 Dynamics of tripartite entanglement

finally a CNOT13 gate. The step-by-step sequential gate operation leads to:

|000〉R1(π2 )−y−→ 1√

2(|000〉 − |100〉)

CNOT12−→ 1√2

(|000〉 − |110〉)

CNOT13−→ 1√2

(|000〉 − |111〉) (6.6)

All the pulses for the three gates used for GHZ state construction were designed usingthe GRAPE algorithm and had a fidelity ≥ 0.995. The GRAPE pulse duration corre-sponding to the gate UG1 is 600 µs, while the UG2 and UG3 gates had pulse durationsof 24 ms. The GHZ-type state was prepared with a fidelity of 0.97. The W state wasprepared from the initial |000〉 by a sequence of four unitary operations (labeled asUW1 , UW2 , UW3 , UW4 in Fig. 6.4(b) and the sequential gate operation leads to:

|000〉R1(π)y−→ |100〉

R2(0.39π)y−→√

2

3|100〉+

1√3|110〉

CNOT21−→√

2

3|100〉+

1√3|010〉

CR13(π2 )y−→ 1√

3[|100〉+ |101〉+ |010〉]

CNOT31−→ 1√3

[|100〉+ |001〉+ |010〉] (6.7)

The different unitaries were individually optimized using GRAPE and the pulse du-ration for UW1 , UW2 , UW3 , and UW4 turned out to be 600µs, 24ms, 16ms, and 20ms,respectively and the fidelity of the final state was estimated to be 0.94.

The WW state was constructed by applying the following sequence of gate opera-

131

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

tions on the |000〉 state:

|000〉R1(π3 )−y−→

√3

2|000〉 − 1

2|100〉

CR12(0.61π)y−→√

3

2|000〉 − 1

2√

3|100〉 −

√1

6|110〉

CR21(π2 )−y−→ 1

2(√

3|000〉 − 1√3

(|100〉+ |110〉+

|010〉))CNOT13−→ 1

2(√

3|000〉 − 1√3

(|101〉+ |111〉+

|010〉))CNOT23−→ 1

2(√

3|000〉 − 1√3

(|101〉+ |110〉+

|011〉))R123(π2 )

y−→ 1√6

(|001〉+ |010〉+ |011〉+

|100〉+ |101〉+ |110〉) (6.8)

The unitary operator for the entire preparation sequence (labeled UWW in Fig. 6.4(c))comprising a spin-selective rotation operator: two controlled-rotation gates, two controlled-NOT gates and one non-selective rotation by π

2on all the three qubits, was created by a

specially crafted single GRAPE pulse (of pulse length 48ms) and applied to the initialstate |000〉. The final state had a computed fidelity of 0.95.

6.2.4 Decay of tripartite entanglementWe next turn to the dynamics of tripartite entanglement under decoherence channelsacting on the system. We studied the time evolution of the tripartite negativity N

(3)123

for the tripartite entangled states, as computed from the experimentally reconstructeddensity matrices at each time instant. The experimental results are depicted in Fig 6.6(a), (b) and (c) for the GHZ state, the WW state, and the W state, respectively. Ofthe three entangled states considered in this study, the GHZ and W states are maxi-mally entangled and hence contain the most amount of tripartite negativity, while theWW state is not maximally entangled and hence has a lower tripartite negativity value.The experimentally prepared GHZ state initially has a N

(3)123 of 0.96 (quite close to its

theoretically expected value of 1.0). The GHZ state decays rapidly, with its negativ-ity approaching zero in 0.55 s. The experimentally prepared WW state initially hasa N

(3)123 of 0.68 (close to its theoretically expected value of 0.74), with its negativity

132

6.2 Dynamics of tripartite entanglement

approaching zero at 0.67 s. The experimentally prepared W state initially has a N(3)123

of 0.90 (quite close to its theoretically expected value of 0.94). The W state is quitelong-lived, with its entanglement persisting up to 0.9 s. The tomographs of the exper-imentally reconstructed density matrices of the GHZ, W and WW states at the timeinstances when the tripartite negativity parameter N(3)

123 approaches zero for each state,are displayed in Fig. 6.7.

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0(a)

Time(s)

N(3

)

GHZcal

GHZexp

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0(b)

Time(s)

N(3

)

WWcal

WWexp

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0(c)

Time(s)

N(3

)

Wcal

Wexp

Figure 6.6: Time dependence of the tripartite negativity N(3) for the three-qubit systeminitially experimentally prepared in the (a) GHZ state (squares) (b) W state (circles) and (c)WW state (triangles) (the superscript exp denotes “experimental data”). The fits are thecalculated decay of negativity N(3) of the GHZ state (solid line), the WW state (dashedline) and the W state (dotted-dashed line), under the action of the modeled NMR noisechannel (the superscript cal denotes “calculated fit”). The W state is most robust againstthe NMR noise channel, whereas the GHZ state is most fragile.

We explored the noise channels acting on our three-qubit NMR entangled states

133

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

which best fit our experimental data, by analytically solving a master equation in theLindblad form, along the lines suggested in Reference [218]. The master equation isgiven by Eq. (6.1).

000001

010011

100101

110111

000001

010011

100101

110111

000001

010011

100101

110111

000001

010011

100101

110111

000001010

011100101110

111

000001010011

100101

110111

000001

010011

100101

110111

000001

010011

100101110

111

000001010011100

101110

111

000001010

011100

101110111000

001010

011100

101110

111

000001

010011

100101

110111

(a) (GHZ at t=0.55s)

(b) (W at t=0.90s)

(c) (WW at t=0.67s)

Real Imaginary

Figure 6.7: The real (left) and imaginary (right) parts of the experimentally tomographeddensity matrix of the state at the time instances when the tripartite negativity N

(3)123 ap-

proaches zero for the (a) GHZ state at t = 0.55 s (b) W state at t = 0.90 s and (c) WW

state at t = 0.67 s. The rows and columns encode the computational basis in binary order,from |000〉 to |111〉.

We consider a decoherence model wherein a nuclear spin is acted on by two noise

134

6.2 Dynamics of tripartite entanglement

channels namely a phase damping channel (described by the T2 relaxation in NMR)and a generalized amplitude damping channel (described by the T1 relaxation in NMR) [67].As the fluorine spins in our three-qubit system have widely differing chemical shifts,we assume that each qubit interacts independently with its own environment. The ex-perimentally determined T1 NMR relaxation rates are T1F

1 = 5.42 ± 0.07 s, T2F1 =

5.65 ± 0.05 s and T3F1 = 4.36 ± 0.05 s, respectively. The T2 relaxation rates were

experimentally measured by first rotating the spin magnetization into the transverseplane by a 90 rf pulse followed by a delay and fitting the resulting magnetization de-cay. The experimentally determined T2 NMR relaxation rates are T1F

2 = 0.53± 0.02s,T2F

2 = 0.55 ± 0.02 s, and T3F2 = 0.52 ± 0.02 s, respectively. We solved the mas-

ter equation (Eq. (6.1)) for the GHZ, W and WW states with the Lindblad operators

Li,x ≡√

κi,x2σ

(i)x and Li,z ≡

√κi,z

2σ

(i)z , where κi,x = 1

T i1and κi,z = 1

T i2.

Under the simultaneous action of all the NMR noise channels, the GHZ state deco-heres as:

ρGHZ =

α1 0 0 0 0 0 0 β1

0 α2 0 0 0 0 β2 00 0 α3 0 0 β3 0 00 0 0 α4 β4 0 0 00 0 0 β4 α4 0 0 00 0 β3 0 0 α3 0 00 β2 0 0 0 0 α2 0β1 0 0 0 0 0 0 α1

(6.9)

where

α1 =1

8(1 + e−(κx,1+κx,2)t + e−(κx,1+κx,3)t + e−(κx,2+κx,3)t)

α2 =1

8(1 + e−(κx,1+κx,2)t − e−(κx,1+κx,3)t − e−(κx,2+κx,3)t)

α3 =1

8(1− e−(κx,1+κx,2)t + e−(κx,1+κx,3)t − e−(κx,2+κx,3)t)

α4 =1

8(1− e−(κx,1+κx,2)t − e−(κx,1+κx,3)t + e−(κx,2+κx,3)t)

β1 =1

8(e−(κ1,x+κ2,x+κ3,x+κ1,z+κ2,z+κ3,z)t

(eκ1,xt + eκ2,xt + eκ3,xt + e(κ1,x+κ2,x+κ3,x)t)

β2 =1

8(e−(κ1,x+κ2,x+κ3,x+κ1,z+κ2,z+κ3,z)t

(−eκ1,xt − eκ2,xt + eκ3,xt + e(κ1,x+κ2,x+κ3,x)t)

(6.10)

135

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

β3 =1

8(e−(κ1,x+κ2,x+κ3,x+κ1,z+κ2,z+κ3,z)t

(−eκ1,xt + eκ2,xt − eκ3,xt + e(κ1,x+κ2,x+κ3,x)t)

β4 =1

8(e−(κ1,x+κ2,x+κ3,x+κ1,z+κ2,z+κ3,z)t

(eκ1,xt − eκ2,xt − eκ3,xt + e(κ1,x+κ2,x+κ3,x)t)

(6.11)

Under the simultaneous action of all the NMR noise channels, the W state deco-heres as:

ρW =

α1 0 0 β1 0 β5 β1 00 α2 β2 0 β6 0 0 β10

0 β2 α3 0 β11 0 0 β7

β1 0 0 α4 0 β12 β8 00 β6 β11 0 α5 0 0 β3

β5 0 0 β12 0 α6 β4 0β1 0 0 β8 0 β4 α7 00 β10 β7 0 β3 0 0 α8

Where

α1 =1

8− 1

24e−(κx,1+κx,2+κx,3)t(3 + eκx,1t + eκx,2t −

e(κx,1+κx,2)t + eκx,3t − e(κx,1+κx,3)t − e(κx,2+κx,3)t)

α2 =1

8+

1

24e−(κx,1+κx,2+κx,3)t(3 + eκx,1t + eκx,2t −

e(κx,1+κx,2)t − eκx,3t + e(κx,1+κx,3)t + e(κx,2+κx,3)t)

α3 =1

8+

1

24e−(κx,1+κx,2+κx,3)t(3 + eκx,1t − eκx,2t

+e(κx,1+κx,2)t + eκx,3t − e(κx,1+κx,3)t + e(κx,2+κx,3)t)

α4 =1

8− 1

24e−(κx,1+κx,2+κx,3)t(3 + eκx,1t − eκx,2t

+e(κx,1+κx,2)t − eκx,3t + e(κx,1+κx,3)t − e(κx,2+κx,3)t)

α5 =1

8+

1

24e−(κx,1+κx,2+κx,3)t(3− eκx,1t + eκx,2t

+e(κx,1+κx,2)t + eκx,3t + e(κx,1+κx,3)t − e(κx,2+κx,3)t)

(6.12)

136

6.2 Dynamics of tripartite entanglement

α6 =1

8+

1

24e−(κx,1+κx,2+κx,3)t(−3 + eκx,1t − eκx,2t

−e(κx,1+κx,2)t + eκx,3t + e(κx,1+κx,3)t − e(κx,2+κx,3)t)

α7 =1

8+

1

24e−(κx,1+κx,2+κx,3)t(−3 + eκx,1t + eκx,2t

+e(κx,1+κx,2)t − eκx,3t − e(κx,1+κx,3)t − e(κx,2+κx,3)t)

α8 =1

8− 1

24e−(κx,1+κx,2+κx,3)t(−3 + eκx,1t + eκx,2t

+e(κx,1+κx,2)t + eκx,3t + e(κx,1+κx,3)t + e(κx,2+κx,3)t)

β1 =1

12(e−(κx,1+κx,2+κx,3+κz,2+κz,3)t

(1 + e(κx,1)t)(−1 + e(κx,2+κx,3)t))

β2 =1

12(e−(κx,1+κx,2+κx,3+κz,2+κz,3)t

(1 + e(κx,1)t)(1 + e(κx,2+κx,3)t))

β3 =1

12(e−(κx,1+κx,2+κx,3+κz,2+κz,3)t

(−1 + e(κx,1)t)(−1 + e(κx,2+κx,3)t))

β4 =1

12(e−(κx,1+κx,2+κx,3+κz,2+κz,3)t

(−1 + e(κx,1)t)(1 + e(κx,2+κx,3)t))

β5 =1

12(e−(κx,1+κx,2+κx,3+κz,1+κz,3)t

(1 + e(κx,2)t)(−1 + e(κx,1+κx,3)t))

β6 =1

12(e−(κx,1+κx,2+κx,3+κz,1+κz,3)t

(1 + e(κx,2)t)(1 + e(κx,1+κx,3)t))

β7 =1

12(e−(κx,1+κx,2+κx,3+κz,1+κz,3)t

(−1 + e(κx,2)t)(−1 + e(κx,1+κx,3)t))

β8 =1

12(e−(κx,1+κx,2+κx,3+κz,1+κz,3)t

(−1 + e(κx,2)t)(1 + e(κx,1+κx,3)t))

β9 =1

12(e−(κx,1+κx,2+κx,3+κz,1+κz,2)t

(−1 + e(κx,1+κx,2)t)(1 + e(κx,3)t))

(6.13)

137

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

β10 =1

12(e−(κx,1+κx,2+κx,3+κz,1+κz,2)t

(−1 + e(κx,1+κx,2)t)(−1 + e(κx,3)t))

β11 =1

12(e−(κx,1+κx,2+κx,3+κz,1+κz,2)t

(1 + e(κx,1+κx,2)t)(1 + e(κx,3)t))

β12 =1

12(e−(κx,1+κx,2+κx,3+κz,1+κz,2)t

(1 + e(κx,1+κx,2)t)(−1 + e(κx,3)t)) (6.14)

Under the simultaneous action of all the NMR noise channels, the WW state de-coheres as:

ρWW =

α1 β1 β2 β3 β4 β5 β6 β7

β1 α2 β8 β9 β10 β11 β12 β13

β2 β8 α3 β14 β15 β16 β11 β5

β3 β9 β14 α4 β17 β15 β10 β4

β4 β10 β15 β17 α4 β15 β9 β18

β5 β11 β16 β15 β15 α3 β8 β2

β6 β12 β11 β10 β9 β8 α2 β1

β7 β13 β5 β4 β18 β2 β1 α1

(6.15)

where

α1 =1

24(3− e−(κx,1+κx,2)t − e−(κx,1+κx,3)t

−e−(κx,2+κx,3)t)

α2 =1

24(3− e−(κx,1+κx,2)t + e−(κx,1+κx,3)t

+e−(κx,2+κx,3)t)

α3 =1

24(3 + e−(κx,1+κx,2)t − e−(κx,1+κx,3)t

+e−(κx,2+κx,3)t)

α4 =1

24(3 + e−(κx,1+κx,2)t + e−(κx,1+κx,3)t

(6.16)

138

6.2 Dynamics of tripartite entanglement

−e−(κx,2+κx,3)t)

β1 =1

12e−(κx,1+κx,2+2κz,3)t

(e(κx,1+κx,2+κz,3)t − eκz,3t)β2 =

1

12e−(κx,1+κx,3+2κz,2)t

(e(κx,1+κx,3+κz,2)t − eκz,2t)β3 =

1

12e−(κx,2+κx,3+2(κz,2+κz,3))t

(e(κx,2+κx,3+κz,2+κz,3)t − e(κz,2+κz,3)t)

β4 =1

12e−(κx,2+κx,3+2κz,1)t

(−eκz,1t + e(κx,2+κx,3+κz,1)t)

β5 =1

12e−(κx,1+κx,3+2(κz,1+κz,3))t

(−e(κz,1+κz,3)t + e(κx,1+κx,3+κz,1+κz,3)t)

(6.17)

β6 =1

12e−(κx,1+κx,2+2(κz,1+κz,2))t

(−e(κz,1+κz,2)t + e(κx,1+κx,2+κz,1+κz,2)t)

β7 = − 1

24e−(κx,1+κx,2+κx,3+κz,1+κz,2+κz,3)t

(eκx,1t + eκx,2t + eκx,3t − 3e(κx,1+κx,2+κx,3)t)

β8 =1

12e−(κx,2+κx,3+2(κz,2+κz,3))t

(e(κz,2+κz,3)t + e(κx,2+κx,3+κz,2+κz,3)t)

β9 =1

12e−(κx,1+κx,3+2κz,2)t

(eκz,2t + e(κx,1+κx,3+κz,2)t)

β10 =1

12e−(κx,1+κx,3+2(κz,1+κz,3))t

(e(κz,1+κz,3)t + e(κx,1+κx,3+κz,1+κz,3)t)

β11 =1

12e−(κx,2+κx,3+2κz,1)t(eκz,1t + e(κx,2+κx,3+κz,1)t)

β12 =1

24e−(κx,1+κx,2+κx,3+κz,1+κz,2+κz,3)t

(6.18)

139

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

(eκx,1t + eκx,2t − eκx,3t + 3e(κx,1+κx,2+κx,3)t)

β13 =1

12e−(κx,1+κx,2+2(κz,1+κz,2))t

(−e(κz,1+κz,2)t + e(κx,1+κx,2+κz,1+κz,2)t)

β14 =1

12e−(κx,1+κx,2+2κz,3)t

(eκz,3t + e(κx,1+κx,2+κz,3)t)

β15 =1

12e−(κx,1+κx,2+2(κz,1+κz,2))t

(e(κz,1+κz,2)t + e(κx,1+κx,2+κz,1+κz,2)t)

β16 =1

24e−(κx,1+κx,2+κx,3+κz,1+κz,2+κz,3)t

(eκx,1t − eκx,2t + eκx,3t + 3e(κx,1+κx,2+κx,3)t)

β17 =1

24e−(κx,1+κx,2+κx,3+κz,1+κz,2+κz,3)t

(−eκx,1t + eκx,2t + eκx,3t + 3e(κx,1+κx,2+κx,3)t)

β18 =1

12e−(κx,2+κx,3+2(κz,2+κz,3))t

(e(κz,2+κz,3)t + e(κx,2+κx,3+κz,2+κz,3)t) (6.19)

With this model, the GHZ state decays at the rate γalGHZ = 6.33 ± 0.06s−1, and itsentanglement approaches zero in 0.53 s. The WW state decays at the rate γal

WW=

5.90 ± 0.10s−1, and its entanglement approaches zero in 0.50 s. The W state decaysat the rate γalW = 4.84 ± 0.07s−1, and its entanglement approaches zero in 0.62 s.Solving the master equation (Eq. (6.1)) ensures that the off-diagonal elements of thecorresponding ρ matrices satisfy a set of coupled equations, from which the explicitvalues of αs and βs can be computed. The equations are solved in the high-temperaturelimit. For an ensemble of NMR spins at room temperature this implies that the energyE << kBT where kB is the Boltzmann constant and T refers to the temperature,ensuring a Boltzmann distribution of spin populations at thermal equilibrium. Theresults of the analytical calculation and the experimental data match well, as shown inFig. 6.6.

6.3 Protecting three-qubit entanglement via dynamicaldecoupling

As the tripartite entangled states under investigation are robust against noise to varyingextents, we wanted to discover if either the amount of entanglement in these states

140

6.3 Protecting three-qubit entanglement via dynamical decoupling

could be protected or their entanglement could be preserved for longer times, usingdynamical decoupling (DD) protection schemes. While DD sequences are effective indecoupling system-environment interactions, often errors in their implementation ariseeither due to errors in the pulses or errors due to off-resonant driving [54].

F 2

F 3

F 1

N

x y x y y x y x -x -y -x -y -y -x -y -x

τ2 τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ

τ2

XY 16(s)(a)

F 2

F 3

F 12N

π6

0 π2

0 π6

2π3

π2

π π2

2π3

τk2

τk τk τk τk τk τk τk τk τkτk2

KDDxy(b)

Figure 6.8: NMR pulse sequence corresponding to (a) XY-16(s) and (b) KDDxy DDschemes (the superscript 2 implies that the set of pulses inside the bracket is applied twice,to form one cycle of the DD scheme). The pulses represented by black filled rectangles (inboth schemes) are of angle π, and are applied simultaneously on all three qubits (denotedby F i, i = 1, 2, 3). The angle below each pulse denotes the phase with which it is applied.Each DD cycle is repeatedN times, withN large to achieve good system-bath decoupling.

Two approaches have been used to design robust DD sequences which are imper-vious to pulse imperfections: the first approach replaces the π rotation pulses withcomposite pulses inside the DD sequence, while the second approach focuses on opti-mizing phases of the pulses in the DD sequence. In this work, we use DD sequencesthat use pulses with phases applied along different rotation axes: the XY-16(s) and theKnill Dynamical Decoupling (KDD) schemes [69]. In conventional DD schemes theπ pulses are applied along one axis (typically x) and as a consequence, only the coher-

141

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

ence along that axis is well protected. The XY family of DD schemes applies pulsesalong two perpendicular (x, y) axes, which protects coherence equally along both theseaxes [223]. The XY-16(s) sequence is constructed by combining an XY-8(s) cycle withits phase-shifted copy, where the (s) denotes the “symmetric” version i.e. the cycle istime-symmetric with respect to its center. The XY-8 cycle is itself created by combin-ing a basic XY-4 cycle with its time-reversed copy. One full unit cycle of the XY-16(s)sequence comprises sixteen π pulses interspersed with free evolution time periods, andeach cycle is repeated N times for better decoupling. The KDD sequence has addi-tional phases which further symmetrize pulses in the x − y plane and compensate forpulse errors; each π pulse in a basic XY DD sequence is replaced by five π pulses,each of a different phase [68, 224]:

KDDφ ≡ (π)π6

+φ − (π)φ − (π)π2

+φ − (π)φ − (π)π6

+φ (6.20)

where φ denotes the phase of the pulse; we set φ = 0 in our experiments. The KDDφ

sequence of five pulses given in Eq. (6.20) protects coherence along only one axis.To protect coherences along both the (x, y) axes, we use the KDDxy sequence, whichcombines two basic five-pulse blocks shifted in phase by π/2 i.e [KDDφ−KDDφ+π/2].One unit cycle of the KDDxy sequence contains two of these pulse-blocks shifted inphase, for a total of twenty π pulses. The XY-16(s) and KDDxy DD sequences aregiven in Figs. 6.8(a) and (b) respectively, where the black filled rectangles represent πpulses on all three qubits and τ (τk) indicates a free evolution time period. We notehere that the chemical shifts of the three fluorine qubits in our particular moleculecover a very large frequency bandwidth, making it difficult to implement an accuratenon-selective pulse simultaneously on all the qubits. To circumvent this problem, wecrafted a special excitation pulse of duration ≈ 400µs consisting of a set of threeGaussian shaped pulses that are applied at different spin frequency offsets and arefrequency modulated to achieve simultaneous excitation [139].

Figs.6.9(a),(b) and (c) show the results of protecting the GHZ, W and WW statesrespectively, using the XY-16(s) and the KDDxy DD sequences.GHZ state protection: The XY-16(s) protection scheme was implemented on theGHZ state with an inter-pulse delay of τ = 0.25 ms and one run of the sequence took10.40 ms (including the length of the sixteen π pulses). The value of the negativity N3

123

remained close to 0.80 and 0.52 for up to 80ms and 240 ms respectively when XY-16protection was applied, while for the unprotected state the state fidelity is quite lowand N3

123 decayed to a low value of 0.58 and 0.09 at 80ms and 240 ms, respectively(Fig. 6.9(a)). The KDDxy protection scheme on this state was implemented with aninter-pulse delay τk = 0.20 ms and one run of the sequence took 12 ms (includingthe length of the twenty π pulses). The value of the negativity N3

123 remained closeto 0.80 and 0.72 for up to 140ms and 240 ms when KDDxy protection was applied(Fig. 6.9(a)).

142

6.3 Protecting three-qubit entanglement via dynamical decoupling

0 0.1 0.20

0.2

0.4

0.6

0.8

1.0(a) GHZ

Time(s)

N(3

)

Without DD

With KDDxy

With XY16

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0(b) W

Time(s)

N(3

)

Without DD

With KDDxy

With XY16

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0(c) WW

Time(s)

N(3

)

Without DD

With KDDxy

With XY16

Figure 6.9: Plot of the tripartite negativity (N(3)123) with time, computed for the (a) GHZ-

type state, (b) W state and (c) WW state. The negativity was computed for each statewithout applying any protection and after applying the XY-16(s) and KDDxy dynamicaldecoupling sequences.Note that the time scale for part (a) is different from (b) and (c)

W state protection: The XY-16(s) protection scheme was implemented on the Wstate with an inter-pulse delay τ = 3.12 ms and one run of the sequence took 56.40ms (including the length of the sixteen π pulses). The value of the negativity N3

123

remained close to 0.30 for up to 0.68 s when XY-16 protection was applied, whereasN3

123 reduced to 0.1 at 0.68 s when no state protection is applied (Fig. 6.9(b)). TheKDDxy protection scheme was implemented on the W state with an inter-pulse delayτk = 2.5 ms and one run of the sequence took 58 ms (including the length of the twentyπ pulses). The value of the negativity N3

123 remained close to 0.21 for upto 0.70 s whenKDDxy protection was applied (Fig. 6.9(b)).WW state protection: The XY-16(s) protection sequence was implemented on the

143

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

WW state with an inter-pulse delay of τ = 3.12 ms and one run of the sequence took56.40 ms (including the length of the sixteen π pulses). The value of the negativityN3

123 remained close to 0.5 for upto 0.45 s when XY-16(s) protection was applied,whereas N3

123 reduced almost to zero (≈ 0.02) at 0.45 s when no protection was applied(Fig. 6.9(c)). The KDDxy protection sequence was applied with an inter-pulse delayof τk = 2.5 ms and one run of the sequence took 58 ms (including the length of thetwenty π pulses). The value of the negativity N3

123 remained close to 0.52 for upto 0.46s when KDDxy protection was applied (Fig. 6.9(c)).

The results of UDD-type of protection summarized above demonstrate that stateprotection worked to varying degrees and protected the entanglement of the tripartiteentangled states to different extents, depending on the type of state to be protected. TheGHZ state showed maximum protection and the WW state also showed a significantamount of protection, while the W state showed a marginal improvement under protec-tion. We note here that the lifetime of the GHZ state is not significantly enhanced byusing DD state protection; what is noteworthy is that state fidelity remains high (closeto 0.8) under DD protection, whereas the state quickly gets disentangled (fidelity dropsto 0.4) when no protection is applied. This implies that under DD protection, there isno leakage from the state to other states in the Hilbert space of the three qubits.

6.4 Conclusions

An experimental study of the dynamics of tripartite entangled states in a three-qubitNMR system was undertaken. The results are relevant in the context of other stud-ies which showed that different entangled states exhibit varying degrees of robustnessagainst diverse noise channels. The W state was found to be the most robust againstthe decoherence channel acting on the three NMR qubits, the GHZ state was the mostfragile and decayed very quickly, while the WW state was more robust than the GHZstate but less robust than the W state. For entanglement protection dynamical decou-pling sequences were implemented on these states. Both DD schemes were able toachieve a good degree of entanglement protection. The GHZ state was dramaticallyprotected, with its entanglement persisting for nearly double the time. The W stateshowed a marginal improvement, which was to be expected since these DD schemesare designed to protect mainly against dephasing noise, and the experimental resultsindicated that the W state is already robust against this type of decohering channel.Interestingly, although the WW state belongs to the GHZ entanglement-class, our ex-periments revealed that its entanglement persists for a longer time than the GHZ state,while the DD schemes are able to preserve its entanglement to a reasonable extent.The decoherence characteristics of the WW state hence suggest a way of protecting

144

6.4 Conclusions

fragile GHZ-type states against noise by transforming the type of entanglement (sincea GHZ-class state can be transformed via local operations to a WW state). These as-pects of the entanglement dynamics of the WW state require more detailed studies fora better understanding.

This work has provided new insights into the way entanglement behaves underdecoherence and can be protected from the same. This kind of protection strategies willprovide a benchmark for extension to higher qubit systems and to the study of differentclasses of entangled states. The WW state is in the GHZ class, however its dynamicsunder decoherence resembles the W state, thereby providing it a certain amount ofimmunity under dephasing. This aspect, coupled with the protection schemes, canlead to new ways of entanglement protection.

145

6. Dynamics of tripartite entanglement under decoherence and protection usingdynamical decoupling

146

Chapter 7

Summary and future outlook

The first project in this thesis focused on the reconstruction of a density matrix. Thedensity matrix which was reconstructed using the standard quantum state tomography(QST) method had a few negative eigenvalues, which makes it an invalid density ma-trix. The maximum likelihood (ML) estimation method estimates the quantum statedensity matrix by determining the parameters that are most likely to match the ex-perimentally generated data. The evaluation starts with initial parameters based on apriori knowledge of experimentally reconstructed density matrix. By putting a con-straint at every stage of the estimation process, the density matrix was obtained to bepositive and normalized. The standard QST method led to negative eigenvalues in thereconstructed density matrix and hence an overestimation of the entanglement param-eter quantifying the residual entanglement in the state. The ML estimation methodon the other hand, by virtue of its leading to a physical density matrix reconstructionevery time, gives us a true measure of residual entanglement, and hence can be usedto quantitatively study the decoherence of multiqubit entanglement. In this thesis MLestimation method is applied on two- and three-qubit systems. The ML estimation canextended further for higher qubit, but this kind of characterization of quantum statesbecomes too costly as the number of qubits increase. To solve this problem one canlook for the use of graphics processing units (GPUs) by incorporating them in the op-timization process, as it is recently used for speedup in quantum optimal control [225].

The later part of the thesis focused on mitigating the unwanted effects of system-environment interaction. The efficacy of different system-environment decouplingstrategies were explored experimentally. These strategies were built depending on ourknowledge about the system-environment interactions and the state to be preserved.

The first situation considered was one where the state of the system was known,and the system-environment interaction was unknown. The state was protected againstevolution using the super-Zeno scheme. The next situation considered was one where

147

7. Summary and future outlook

only the subspace to which the state belongs was known and an arbitrary state belong-ing to known subspace was protected using the nested Uhrig dynamical decoupling(NUDD) scheme. The advantage of the NUDD schemes lies in the fact that one is surethat some amount of state protection will always be achieved.

Next, a system was considered whose state as well as interaction with the environ-ment is known. A Bell-diagonal (BD) state was experimentally prepared on a two-qubit system. The noise model considered was that of each qubit was mainly affectedby an independent phase damping channel, and quantum discord of the system wasobserved. Quantum discord remained constant for some time t, after which it began todecay. The Carr-Purcell-Meiboom-Gill (CPMG) preserving sequence was applied onthe BD state and it was found that reducing the delay between the π pulses increasedthe lifetime of the time-invariant discord, up to a certain point. After that, the errorsdue to pulse imperfections dominated due to which the lifetime of the time-invariantdiscord started decreasing. To overcome this problem XY16 and KDDxy dynamical de-coupling sequences were used which are robust against such errors. These sequencessubstantially extended the lifetime of time-invariant discord. These experiments haveimportant implications in situations where persistent quantum correlations have to bemaintained in order to carry out quantum information processing tasks. Such situa-tions usually arises when the number of gates are large or quantum gates are compareto coherence time.

Finally, the decay of tripartite entangled states was studied by experimentally prepar-ing GHZ, W and WW states on a three-qubit NMR quantum information processor.The natural decoherence present in the three-spin spin system was best modeled byconsidering the main noise channel to be an uncorrelated phase damping channel act-ing independently on each qubit, with a small contribution from the generalized am-plitude damping channel. The W state was found to be the most robust against thistype of noise, whereas the GHZ state was the most fragile; the WW state was morerobust than the GHZ state but less robust than the W state. The dynamical decouplingsequences XY16 and KDDxy were applied on these states and a significant protectionof entanglement for the GHZ and WW states was observed. These quantum state pro-tection strategies can be explored for higher qubit systems and can be used to optimizegates such that the effect of the decoherence is minimized.

148

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