'Pulse Techniques for Quantum Information Processing' in ...

Pulse Techniques for Quantum Information ProcessingGary Wolfowicz1 & John J.L. Morton2

1University of Chicago, Chicago, IL, USA2University College London, London, UK

The inherent properties of quantum systems, such as superposition and entanglement, offer a new paradigm to the treatment and storage of infor-mation in physical systems, giving rise to potentially faster computation, more secure communication, and better sensors. Electron spins in a rangeof host environments are ideal systems for representing quantum information, and such systems can be both characterized and controlled throughthe arsenal of pulse electron paramagnetic resonance (EPR) techniques. In this article, we introduce basic concepts behind quantum information pro-cessing with spins, including how spin decoherence (i.e., the corruption of quantum information) has been studied and mitigated in various systems;how EPR methods can be used to manipulate quantum information in spins with high fidelity; how individual spins can be used as high-resolutionsensors (e.g., of magnetic field); and how multiple spins can be coupled together to develop the building blocks of a larger scale quantum computer.Keywords: quantum bit (qubit), spin qubit, quantum information processing, quantum sensors, quantum memories, decoherence, gate fidelity,quantum state tomography

How to cite this article:eMagRes, 2016, Vol 5: 1515–1528. DOI 10.1002/9780470034590.emrstm1521

Since the early 1980s, it has been recognized1,2 that representinginformation within quantum systems offers the potential forprofound enhancements in information processing.3,4 Byinitializing, controlling, and measuring arrays of coupledquantum systems, it is possible, in principle, to build efficienttools for the simulation of quantum systems, ranging frompurpose-built analog models through to ‘digital’ quantumsimulators, which follow algorithmic approaches to solve theevolution of a Hamiltonian. So-called universal quantum com-puters possess a sufficient set of logical operations, or gates,to perform arbitrary quantum algorithms such as those forfactoring large numbers and searching datasets, and representa generalization of the concept of a Turing machine. Motivatedby these opportunities, the past 20 years has seen a great dealof experimental research into realizing quantum informationprocessors, based on a wide range of physical systems, includ-ing spins, atoms, ions, and superconducting circuits. Morerecently, applications of individual highly coherent quantumsystems as sensors have been identified, potentially yieldingnearer-term technologies. In this article, we will review pulsetechniques in electron paramagnetic resonance (EPR) relatedto research in quantum information processing and sensingusing electron spins.

Spin QubitsIn analogy to digital binary logic, the fundamental unit of quan-tum information is the quantum bit, or qubit, consisting of atwo-level quantum system labeled with the basis states |0⟩ and|1⟩. The ability of a quantum system to exist in a superpositionof states, with some phase relationship, provides the qubit with a

much richer state space to occupy than its classical counterpart.We can write the state of the qubit in general as

|𝜓⟩ = cos 𝜃2|0⟩ + ei𝜙 sin 𝜃

2|1⟩ (1)

which maps out the surface of the Bloch sphere (Figure 1a).A spin-1/2 in a magnetic field is a natural realization of

a qubit, with the eigenstates |||mS = − 12

⟩≡ |↓⟩ ≡ |0⟩ and|||mS = + 1

2

⟩≡ |↑⟩ ≡ |1⟩ (representing, respectively, the spin

projection, the spin orientation, and the qubit state— (seeSpin Dynamics). Much early progress was made in exploringquantum algorithms and control using liquid-state NMR,including the application of Shor’s factoring algorithm to thenumber 15 in Ref. 5. However, thermal initialization of NMRsystems becomes exponentially harder with increasing numberof qubits, limiting the scalability of the approach. Focus onspin qubits has therefore shifted to a large extent to electronspins, though nuclear spin qubits remain attractive for memoryapplications due to their long quantum state lifetimes of up to6 h.6 Electron spins being explored for quantum informationapplications include the nitrogen vacancy (NV) and other colorcenters in diamond7,8 and SiC,9 donors in silicon,10 and quan-tum dots in silicon,11 GaAs,12 and other semiconductors.13

Molecular electron spins14–16 have also been explored as use-ful systems to test concepts in quantum information, owingto their reproducible nature and the ability, in principle, toengineer coupled systems.

Single-qubit Gates

In classical binary logic, the only nontrivial operation, or ‘gate’,which can be applied to a single bit is the NOT gate (where

Volume 5, 2016 © 2016 John Wiley & Sons, Ltd. 1515

G Wolfowicz and JJL Morton

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z2π

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Echo

−θz

Figure 1. Single-qubit representation and control. (a) The Bloch sphere representation of a qubit state |𝜓⟩, defined by angles 𝜃 and 𝜙. Spin-down andspin-up eigenstates can be mapped onto the qubit states |0⟩ and |1⟩. (b) Rotations of an electron spin qubit about the x-axis can be performed using asimple resonant microwave pulse, observable (using a subsequent 2-pulse echo sequence) as Rabi oscillations where the duration of the pulse is swept. (c)Rotations about the z-axis can be performed by decomposing into x- and y-rotations. Experimental data taken using P donors in Si, at 8 K

0 → 1 and 1 → 0). Thanks to the much richer state space ofthe qubit (Figure 1a), there is a much greater set of single-qubitgates, which (neglecting a global phase) can be represented asrotations of the Bloch sphere around some axis and by someangle. In pulse EPR, and in the rotating frame, rotations aboutaxes in the x–y-plane can be trivially performed through sim-ple microwave pulses, where the pulse phase determines therotation axis, while the pulse power and duration set the angle(Figure 1b). Rotations about other axes (and thus any unitaryoperation) can be performed by decomposition into a seriesof x and y rotations (��x,y), or through more elaborate methodssuch as pulse shaping, as discussed in Shaped Pulses in EPR.For example, a z-rotation (Figure 1c) can be implemented as

��z = (π∕2)x(𝜃)y(π∕2)−x (2)

while the Hadamard gate, a common single-qubit gate used togenerate superposition states (and representing a rotation byπ about an axis in the direction x + z), can be decomposed asfollows:

UHadamard = 1√2

(1 11 −1

)= i(π)x

(π2

)y

(3)

An additional approach to single-qubit operations employsthe geometric phase acquired by spins driven through someclosed trajectory. In particular, when a spin qubit is driventhrough a cyclic evolution (i.e., the initial and final statescoincide on the Bloch sphere), it acquires a geometric, or‘Berry’, phase that is equal to half the solid angle enclosed by itsevolution on the Bloch sphere (Figure 2).17 Because this phaseis global (the same phase is acquired by both eigenstates), itserves no purpose for an isolated spin-1/2 qubit. However,given the availability of additional levels (e.g., thanks to hyper-fine coupling, or for S > 1∕2), it can be exploited to createthe so-called Aharonov–Anandan (AA) phase gate, which isan effective z-rotation of the qubit.18 AA phase gates havebeen performed on coupled electron–nuclear spin systems by

applying a selective 2π microwave pulse to an electron spintransition, resulting in a nuclear spin phase gate that is ordersof magnitude faster than that which could be achieved usingconventional ENDOR/NMR pulse sequences.14

Quantum State Tomography

The complete state of a qubit, taking into account effects such asdecoherence, must in general be described by the density matrix(see Spin Dynamics), ��, which can be decomposed as follows:

�� = 𝟏2+ rxSx + rySy + rzSz (4)

where 𝟏 is the identity matrix, Sx,y,z are the spin operators, andrx,y,z are the real coefficients that determine the state. In order tocharacterize the practical implementation of the gates describedabove, it is necessary to measure the resulting state of the qubit,determining its density matrix through a process known as statetomography. Owing to the collapsing effect of measurement onquantum systems, state tomography is achieved only by com-bining many repetitions of an experiment on a single qubit, orusing an ensemble of many such qubits. In conventional EPR,values proportional to rx and ry can be determined upon forma-tion of a spin echo (Figure 3). A corresponding value for rz can-not be directly measured simultaneously (at least not withoutsome additional measurement technique, such as a SQUID),but its value can be inferred if the echo is followed by an idealπ∕2-pulse, effectively mapping the desired Sz component ontothe Sx observable.

The measured values r′x,y,z(= 𝛼rx,y,z) must then be normal-ized to produce a physical density matrix–this can be donein various ways, but a common approach in ensemble EPRis to make the ‘pseudo-pure’ approximation,19 which worksas follows: At typical temperatures and magnetic fields, elec-tron spins are in a weakly polarized state such that rx,y,z ≪ 1.However, it is possible to neglect the large 𝟏 component of thedensity matrix, as it is invariant under unitary transformations,

1516 © 2016 John Wiley & Sons, Ltd. Volume 5, 2016

Pulse Techniques for Quantum Information Processing

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(2π)MW (2π)MW (2π)MW

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|1⟩

|2⟩

|1⟩

|2⟩

|0⟩

MW

Electron spin Nuclear spin

RF

(a)

|3⟩

Φ

ϕ = π

Figure 2. Aharonov–Anandan geometric phase gates using coupledelectron and nuclear spins. (a) A microwave pulse may be applied reso-nant with the |1⟩ ↔ |2⟩ transition to drive states |1⟩ and |2⟩ around a closedtrajectory on the corresponding Bloch sphere. The solid angle,𝛷, enclosedby this trajectory defines the geometric phase 𝜙 = 𝛷∕2 acquired by boththe states. Example closed trajectories are shown for the cases where thetransition is driven resonantly (white) or nonresonantly (yellow). (b) Theeffect of this phase shift on a qubit defined by states |0⟩ and |1⟩ can beobserved by driving nuclear spin Rabi oscillations without (gray) and with(black) fast phase gates. After each 2π microwave pulse applied on the elec-tron spin, the nuclear spin acquires a π phase and is prevented from makinga complete rotation around the Bloch sphere. Experiment performed usingN@C60. (Reprinted by permission from Macmillan Publishers Ltd: NaturePhysics, Ref. 14 © 2006.)

and treat the remaining part of the state as if it were pure. Inthis way, we can assume a given state to be ‘pseudo-pure’ andimpose the normalization condition for a pure state, ||r|| = 1,in order to extract a value of the proportionality constant, 𝛼.This assumption is usually applied to the initial state in thepulse sequence, such that subsequent states produced (whichin general will not be pure) are normalized using the samevalue of 𝛼.

As multiple qubits are coupled together, the state space expo-nentially expands. This gives an illustration of the increasedpower of quantum information processors, and also bringsnew challenges in performing state tomography due to thenumber of operations required (see section titled ‘MultipleQubits’). Having discussed methods to extract the full stateof a spin qubit, we next turn to ways in which this state canbe corrupted–first through ‘natural’ processes in the systemand its environment (decoherence) and second through thenonideality of the applied operations (pulse errors).

+x +y +z

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0

Image

Real

Identity

2π

Prepare

x, y

πEcho

Measure Sz

2πEcho

π

Measure Sx, y

(a)

(b)

Figure 3. Quantum state tomography using EPR pulses to obtain thedensity matrix of a qubit. (a) Two consecutive Hahn echo sequences pro-vide separately the x, y, and z components of the qubit state, necessaryto reconstruct the density matrix. (b) Extracted spin-1/2 density matrices(matrix elements plotted on a vertical scale), given different state prepara-tion pulses

DecoherenceOne of the primary criteria in choosing a particular physicalrepresentation of a qubit or quantum sensor is maximizingthe coherence lifetime: decoherence, defined as undesiredphase shifts in the qubit state, are errors in quantum com-putation, while a short coherence time prevents long-phaseaccumulation necessary for sensing applications.

Although electron spin (2-pulse echo) coherence times (T2)as long as several hundred microseconds have been observedin certain molecular systems,20 the longest coherence lifetimeshave emerged from defect spins in solid-state materials, in par-ticular those with a low natural abundance of nuclear spins suchas carbon- and silicon-based materials. The ability to grow purecrystals with low charge defect concentration and the possibil-ity of isotopically enriching samples to minimize the nuclearspin concentration have led to measured electron spin coher-ence times of up to 1.8 ms for NV centers in diamond at roomtemperature21 and up to 20 ms for P donors in silicon at 6 K.22

Notably, when measuring such long times in spin ensemblesby conventional EPR, instantaneous diffusion often becomesthe limiting decoherence mechanism23 (see Pulse EPR). Thiscan be confirmed by measuring the decoherence time as afunction of the refocusing pulse angle in the 2-pulse echosequence (Figure 4). Instantaneous diffusion23 can be avoidedusing alternative spin measurement techniques with greatersensitivity such as optically detected magnetic resonance,where more dilute ensembles,24 or indeed single spins,7,25 canbe measured.

The T2 values measured in bulk crystals can become ordersof magnitude shorter when the electron spins are brought nearinterfaces, as they often must for practical applications. Thereis good motivation therefore to develop methods for extend-ing coherence times and protecting the spin qubit from noisesources. In the following section, we describe two approaches:first, using a technique known as dynamical decoupling (DD),which is a natural extension of the 2-pulse (Hahn) echo to alarge number of refocusing pulses and second, by identifying



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Figure 4. Instantaneous diffusion. (a) Measuring echo decay curves fordifferent refocusing pulse angles (𝜃), as shown in the pulse sequence (inset),can be used extract the instantaneous diffusion contribution to T2. (Note:Curves are normalized to unit intensity (for 𝜏 ≈ 0) and the actual echointensity falls as 𝜃 decreases). (b) Extrapolating the measured decay con-stants (T2) for the case of 𝜃 → 0 provides an estimate for T2 in the absenceof instantaneous diffusion (i.e. in the dilute spin limit).22 Experiment per-formed at 5 K using Bi donors in silicon, at a concentration of 4 × 1015 cm−3

spin transitions that are inherently insensitive to external per-turbations such as magnetic field noise. These methods can beused to achieve gains in coherence time between one and twoorders of magnitude.

Dynamical Decoupling (DD)

The single refocusing π-pulse in the Hahn echo is the most basicexample of DD, able to protect the spin qubit from dephasingcaused by variations in the spin transition frequency that arestatic (at least, on the timescale of the experiment). More gen-eral DD sequences can be used to protect the spin from moreelaborate noise sources and are adaptable to various noise spec-tra and spin interactions.26

Decoherence from dynamic noise can be suppressed byrepeated application of refocusing π-pulses, with an inverseperiod 1∕𝜏 chosen to be less than the high-frequency cut-off of the noise spectrum.26 Such a sequence correspondsto a bandpass filtering effect centered around harmonics off = 1

2𝜏, and with a bandwidth narrowing with large pulse

number,27,28 as shown in Figure 5. The higher harmonics off have a lesser contribution to the decoherence as the noisefrequency becomes too fast compared to the spin evolution.

DD sequences have been applied with millions of pulses inNMR30 and up to 10 000 pulses in EPR29 (Figure 5), and in suchcases, careful consideration must be given to the effect of pulseerrors. In order for the DD pulses not to significantly intro-duce errors themselves, either pulses with very high fidelity arerequired or the DD sequence should be robust to pulse error.For example, when the basic DD sequence

(− 𝜏

2− πx −

𝜏

2−)n

Frequency × 2 τ0.1 1 10

Filt

er

fun

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n/a

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5 pulses

40 pulses

1

(b)

π π π π π

τ

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(c) Number of pulses n

T 2/m

s

100

300 K240 K190 K160 K120 K77 K

100

101

101

102

102

103

103

104

104 105

Figure 5. Dynamical decoupling (DD). (a) A typical DD sequence con-sists of repeated π refocusing pulses whose inverse period 1∕𝜏 must besmaller than the frequency of the noise (black curve). (b) DD acts as a filterfunction, where larger pulse numbers lead to narrower bandpass filters. (c)Electron spin coherence time T2 as a function of pulse number in an NVcenter (Reprinted by permission from Macmillan Publishers Ltd: NatureCommunications, Ref. 29 © 2013.) showing an increase of up to three ordersof magnitude using 10 000 DD pulses

is applied to an input state oriented along x, the pulse errors areadditive leading to rapid corruption of the state. On the otherhand, if the same sequence is applied to a ±y input state, it isrobust to pulse errors as they cancel out to first order–indeed,this is why Meiboom and Gill31 adapted the CP sequence32 toproduce what we now term CPMG:

(π2

)x

(− 𝜏

2− πy −

𝜏

2−)n

.DD sequences of most interest in quantum information pro-

cessing possess robustness to pulse errors for all input states,and examples include the XYXY sequence and higher orders33:

XY − 4 ≡ πyπxπyπx (5)

XY − 8 ≡ πyπxπyπxπxπyπxπy (6)

where the delay times 𝜏 between the pulses are omitted forclarity.

The above sequences are suitable for decoupling a ‘cen-tral’ spin from some noise source, such as a bath of different



spins, but they are not able to refocus interactions betweenidentical spins, which also evolve under the DD pulses.Sequences to decouple such interactions involve π∕2 rota-tions, and examples to decouple dipolar interactions includethe Waugh–Huber–Haeberlen sequence (WAHUHA) and itshigher order extension MREV.34

DD sequences have applications beyond simply preservingspin-qubit states–their filtering properties can also be used inspectroscopic applications as a ‘lock-in amplifier’ to pick outcertain interactions35 (see section titled ‘Sensing’), while theycan also be used to indirectly drive rotations of coupled spins.For example, given an electron spin (with operator S) and anuclear spin (with operator I) coupled by an anisotropic inter-action, the repeated π-pulses on the electron spin under a DDsequence can, as a result of the Sz Ix or Sz Iy terms in the spinHamiltonian, be seen by the nuclear spin as an oscillating fieldalong the x- or y-axis. If the decoupling period matches theLarmor precession of the nuclear spin, there is a resonant condi-tion, and the nuclear spin rotates while the electron spin coher-ence is simultaneously preserved.36

Parameter-insensitive Transitions

Decoherence can be directly related to the loss of phase causedby uncontrolled fluctuations in the qubit transition frequency,altering the Larmor precession rate. Such fluctuations furtheroriginate from variations in free (i.e., externally controllable)parameters present in the spin Hamiltonian, such as the exter-nal magnetic or electric field. An important characteristicis the frequency sensitivity of the relevant spin transition tothese parameters. A simple and obvious example here is thedifference in coherence times typically seen for electron andnuclear spins: the greater decoherence rate of the electron spinis attributed to its much higher gyromagnetic ratio, which(for a system with only the Zeeman interaction) is equal tod𝜔∕dB0, the first-order dependence of the transition frequency𝜔 with respect to the external magnetic field B0. In systemswhere various interactions compete, this dependence can beless than the free-spin gyromagnetic ratio, and even reach zero,to first order, for certain parameter values (e.g., orientation andmagnetic field strength). In the case of magnetic field insen-sitivity, such transitions are called ‘zero first-order Zeemanshift (ZEFOZ)’ or ‘clock transitions (CTs)’ and can be used tosuppress decoherence (Figure 6).

CTs at, or close to, EPR frequencies have been studied intwo separate electron–nuclear spin systems, first theoreticallyin rare-earth-doped crystals37 and later demonstrated for Bidonors in silicon.38,39 In the case of Bi donors, the hyperfineinteraction between the electron spin (S = 1∕2) and nuclearspin (I = 9∕2) is close to 1.5 GHz, resulting in the presence offour CTs at frequencies between 5.2 and 7.3 GHz (Figure 6a).For rare-earth ions (Er3+), the weaker hyperfine interaction(|A| < 700 MHz, anisotropic) provides CTs below 1 GHz. Inboth these systems, the limiting decoherence mechanism isspectral diffusion (see Relaxation mechanisms; Pulse EPR)from either nuclear or electron spins in the environment,arising from dipolar or hyperfine interactions, which can betreated as magnetic field fluctuations. At the clock transition,decoherence due to spectral diffusion is suppressed linearly

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Figure 6. EPR clock transitions. (a) The allowed spin transition frequen-cies for Bi donor spins in silicon (S = 1∕2, I = 9∕2, A ≈ 1.5 GHz) show four‘clock transitions’ (CTs) at microwave frequencies (open circles) at whichthe first-order dependence on magnetic field goes to zero. (b) Electronspin coherence time, T2, shown as a function of the first-order sensitiv-ity of the transition frequency to magnetic field (d𝜔∕dB0) in Bi donors insilicon, for 3 different Bi concentrations. Measurements for d𝜔∕dB0 ≈ 𝛾ewere obtained from the 10 X-band (9.7 GHz) transitions (100–600 mT),while the remaining points were taken in the vicinity of the clock transitionhighlighted in panel (a) (B0 ≈ 80 mT, 𝜔0∕2π = 7.03 GHz). Lines are fit toa model with a quadratic dependence in d𝜔∕dB0 due to instantaneous dif-fusion and a linear dependence due to spectral diffusion. As d𝜔∕dB0 → 0,the coherence time becomes limited by direct flip-flop processes that arenot suppressed at the clock transition and are spin concentration depen-dent. The inset shows a Hahn echo decay measured at 4.3 K using [Bi] =3.6 × 1014 cm−3

with decreasing d𝜔∕dB0, leading to coherence times in Bidonors as long as 2.7 s at 4.3 K, measured using a 2-pulseHahn echo (Figure 6b,38). In addition to being a strategy tomitigate decoherence, the ability to tune the sensitivity of aspin transition to various parameters can also be used as a wayto distinguish various decoherence mechanisms.

Measuring Gate FidelitiesAs described above, the decoherence times in many candidateelectron spin-qubit systems have reached such a level that natu-ral decoherence can cease to be a major error source in a quan-tum algorithm. In such cases, imperfections of the qubit controlplace a limit on the achievable error rate–for example, the decayin the Rabi oscillations of Figure 1(b) and (c) is due to the accu-mulation of pulse error, and not intrinsic decoherence.



Practical gate errors in EPR are typically systematic in nature,arising, for example, from nonidealities in the microwave com-ponents (switches, IQ mixers, etc.) used to form the pulses,or the finite bandwidth of the cavity. In the usual case in EPRwhere spin ensembles are addressed, the dominant source oferror often arises from the inhomogeneity of the microwavefield intensity across the sample, leading to a variation in theachieved rotation angle across the ensemble. In principle,errors arising from field inhomogeneity are not relevant at thelevel of single-spin control.25 However, owing to the challengesof confining microwave magnetic fields to small regions ofspace, many quantum computing schemes based on spinsemploy microwaves applied to large arrays of spins, each ofwhich can be individually tuned into resonance with the field.40

Assuming the experimentally applied operator B is uni-tary, its fidelity with respect to an ideal operator A can beexpressed as (for a spin-1/2)3

= 12

Tr(AB−1) (7)

In the following sections, first, we describe the methods thathave been used to characterize gate fidelities in EPR and thenwe discuss the strategies for maximizing the fidelity.

CP vs CPMG

Minimum gate fidelities for fault-tolerant quantum compu-tation are around 99% (based on the latest, most forgivingerror-correction schemes identified so far41) but are ideally atthe level of 99.99% and above. When the error per gate is solow, accurately measuring the gate fidelity requires repeatedapplication of the gate so the errors can build up and be morereadily measured. For example, we have already seen howdifferent DD sequences, such as CP and CPMG, have differenttolerances to rotation angle errors and this difference can beexploited to measure pulse errors associated with rotationangle.42,43 Measuring the decay of echo intensity in a train ofspin echoes can provide a basic estimate for rotation angleerrors (from which gate fidelity can be derived). However, fora more complete characterization of the possible gate errors,more elaborate techniques are required.

Quantum Process Tomography

An experimentally applied operation can in general bedescribed by a process matrix, 𝛘, which describes how a

given input state described by the density matrix, ��in, istransformed into an output state:

��out =3∑

m,n=0𝛘mnAm��inA†

n (8)

where Ai is the Pauli basis {��, ��x, ��y, ��z}. Quantum processtomography (QPT) aims to obtain this matrix, 𝛘, by per-forming state tomography of the output state, given a suitablyrepresentative set of input states.3 Owing to state preparationand measurement (SPAM) errors, or variations in the appliedoperation, the process matrix directly constructed by linearinversion may not be physical (Hermitian positive). In suchcases, the closest physical operation can be found by leastsquared fitting.44,45 Some example process matrixes are shownin Figure 7.

Randomized Benchmarking

The above methods to characterize gates have a number of lim-itations: QPT is sensitive to SPAM errors (although in principlethey can be separated from the applied process with appropri-ate calibration steps) and requires an exponentially increasingnumber of operations as the number of qubits grows. On theother hand, the method of comparing CP and CPMG assessesonly one kind of error and gives an unrealistic picture of howerrors may combine in a practical computation.

Randomized benchmarking is a technique that solves theseissues by applying a sequence with increasing number of ran-domly selected pulses that eventually depolarize the qubit. Thepulses are chosen from the Clifford group, a set of rotations thatspan uniformly the Hilbert space, thus providing an averagepulse error.47 For a single qubit, the Clifford gates are theidentity, π, π∕2, and 2π∕3 rotations.48 After each sequence, thequbit state is measured along some axis (e.g., x or y). The decayin fidelity as a function of pulse number gives the average pulseerror, with randomization ensuring the fidelity is not corre-lated to any particular pulse. In order to test a specific gate, thefidelity decay due to an increasing number of Clifford gates iscompared to the decay with this gate interleaved between theClifford gates.

Randomized benchmarking was first performed usingliquid-state NMR,49 followed by EPR experiments on single-electron spins of NV centers in diamond50 and donors insilicon,11 and in ensemble EPR using a loop-gap resonator.51

1.0

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(c)0.0

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Figure 7. Quantum process tomography. Experimentally obtained processes matrices, 𝜒 , for (a) the identity operator, (b) πy rotation, (c) a Hadamardgate, and (d) a combination of processes, such as T1, T2 and pulse errors, leading to a partially mixed state. Red outlines correspond to the ideal processes.Experiments performed using (a, b) Sb : Si,44 (c) P : Si; and (d) Nd3+ : YSO46



High-fidelity OperationsAs mentioned above, and discussed in more detail in sectiontitled ‘Spin-qubit Networks’, required gate fidelities for quan-tum computation are typically 99.9–99.99%, much higher thanis typically required for spectroscopic applications. In conven-tional dielectric and loop-gap cavities in EPR, inhomogeneitiesof 10–20% in the microwave field strength are common, leadingto fidelities of less than 95–99% for a simple π-pulse. In striplineresonators and microresonators, the fidelities can be substan-tially less.52

The fidelity of a gate achieved by a single rectangular pulsecan be increased using composite pulses (comprising severalpulses) or by various pulse shaping methods (see ShapedPulses in EPR) including adiabatic pulses and the use of geo-metric phases. In each case, common considerations whendesigning a pulse shape or sequence to yield a high-fidelityoperation include (i) minimizing the pulse duration, giventhe available power and bandwidth of cavity and microwavecomponents; (ii) achieving the desired frequency selectivity(e.g., to perform a conditional operation); (iii) maximizing therobustness to the dominant imperfections in the system (suchas microwave field amplitude, or resonant frequency in thecase of inhomogeneous broadening in the spin system); and(iv) independence on the initial spin state–in other words, thesequence should be correcting the operator, not the final state.

Many studies, often originating in NMR, have attemptedto fulfill these criteria, offering robustness to errors in fre-quency/detuning and pulse amplitude, at the cost of longerpulses. In the following section, we discuss various approachesto performing high-fidelity gates in EPR.

Composite Pulses

Composite pulses attempt to minimize errors by concatenatingmultiple (rectangular) pulses with various phases and rotationangles,53 based on the assumption that the error is systematic, atleast over the duration of the composite pulse. The overall evo-lution is equivalent to a single rotation whose dependence onvariations in parameters such as frequency detuning or pulseamplitude cancels up to a desired order (in average Hamilto-nian theory54). Composite pulses are an appealing solution asthey are relatively simple to apply, often universal, and for high

concatenation levels can theoretically achieve arbitrary robust-ness. In practice, however, higher levels are rarely applied asthey require exponentially more pulses.

BB1 (short for ‘broadband 1’) is a common composite pulseto tackle pulse amplitude error,54 where the sensitivity of fidelityto rotation angle errors in each pulse is reduced to sixth order.It is defined with four pulses:

(𝜃)ideal0 = (π)𝜙(2π)3𝜙(π)𝜙(𝜃)0 (9)

with 𝜙 = cos−1(−𝜃∕4π) (10)

where 𝜃 is the desired rotation angle, while the phases 𝜙 and3𝜙 are defined with respect to the phase of the desired idealrotation. Using BB1, fidelities above 99.9% have been measuredin conventional dielectric EPR resonators, despite amplitudeerrors up to 10%.43 Other variations exist such as NB1 or PB1(short for ‘narrowband 1’ and ‘passband 1’) that provide highfidelity but over a restricted deviation of the amplitude,54 whichcan be useful to spatially select spin ensembles in a sampleusing microwave field gradients.

For robustness against frequency detuning (e.g., given inho-mogeneous broadening), the CORPSE (compensation for offresonance with a pulse sequence) pulse can be used,55 defined,for an ideal rotation (𝜃)0, as(

𝜃

2− 𝜓

)0(2π − 2𝜓)π

(2π + 𝜃

2− 𝜓

)0

(11)

with 𝜓 = sin−1(sin(𝜃∕2)∕2) (12)

If the desired pulse is a π-pulse (as is commonly used in manyDD sequences), the so-called Knill pulse56,57 can be very effec-tive, with robustness to both detuning and pulse amplitudeerrors. The Knill pulse, which results in a π∕3 phase shift inadditional to the desired rotation, consists of a symmetricsequence of five π-pulses:

πideal0

(π3

)z= (π) π

6(π)0(π) π

2(π)0(π) π

6(13)

The performance of BB1, CORPSE, and Knill sequences is com-pared in Figure 8. Composite pulses can themselves be used asthe base pulses of other composite pulses, in order to combinedifferent type of robustness.58

BB10.6

−0.6

Δω1/ω

1

Δωs/ω1

0.4

0.2

0.0

−0.2

−0.4

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

CORPSE0.6

−0.6

Δω1/ω

1

0.4

0.2

0.0

−0.2

−0.4

Δωs/ω1

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

Knill0.6

−0.6

Δω1/ω

1

0.4

0.2

0.0

−0.2

−0.4

Δωs/ω1

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

99.999%

99.9%

99%

97%

99.7%

99.97%

Fidelity

Figure 8. Robustness of composite pulses for frequency detuning and pulse amplitude errors. The fidelity of BB1, CORPSE, and Knill sequences arecompared as a function of frequency detuning (𝛥𝜔S) and pulse amplitude detuning (𝛥𝜔1) expressed as a fraction of𝜔1(= 𝛾eB1), the microwave field strength(in angular frequency units). Both the BB1 and Knill sequences are five times the duration of a simple π-pulse (5t180), while the CORPSE sequence is 13

3t180

in duration



Adiabatic Pulses

As discussed in Spin Dynamics and Pulse EPR, an appliedmicrowave field resonant with a spin appears, in the rotatingframe, as a static magnetic field in the x–y-plane about whichthe spin then precesses. If the microwave field is off-resonant,the static field in the rotating frame acquires a residualz-component. As an alternative to abruptly turning on andoff such fields to drive spin rotations, the microwave fieldamplitude and frequency can be gradually adjusted to changethe direction of the effective magnetic field in the rotatingframe. In this way, thanks to the adiabatic theorem in quantummechanics,59 spin eigenstates can be moved along determinedtrajectories around the Bloch sphere, defining some unitaryoperation. (The adiabatic theorem states that a physical systemremains in its instantaneous eigenstate if a given perturbationis acting on it slowly enough and if there is a gap between theeigenvalue and the rest of the spectrum of the Hamiltonian.)

A prototypical adiabatic pulse is the adiabatic fast pas-sage (AFP), introduced by Bloch60: a chirp pulse where themicrowave frequency is swept across the spin resonancefrequency. In the rotating frame, all spins whose resonantfrequency lies within the range swept by the chirp pulse expe-rience an effective magnetic field that smoothly rotates fromup to down, creating an effective π-pulse (see Shaped Pulses inEPR).

AFPs are robust to variations in the microwave irradia-tion amplitude as well as detuning; however, care must betaken when applying them to coherent states–for example,the AFP includes a inhomogeneous phase shift due to fre-quency broadening. This phase shift is canceled when pairsof AFPs are applied, making them suitable for multipulse DDsequences.38,61

While an AFP can be used to achieve a π-pulse, other rota-tions can be performed with more complex adiabatic pulsessuch as the BIR-1 or BIR-4 pulse, which are composite AFPpulses.62 These have the disadvantage (in samples with shortT2) of being even longer than AFPs, which are already anorder of magnitude or more longer than a standard rectangularpulse, and are only robust to pulse amplitude errors and notfrequency. BIR pulses have been used, for example, for EPRusing coplanar waveguides in Ref. 52 to perform 2-pulse echoexperiments in the presence of over one order of magnitudevariation in pulse amplitude.

SensingResearch toward developing spin qubits has helped motivatethe identification and control of spins with long coherencelifetimes, as well as their measurement at the single-spinlevel (e.g., by optical63 or electrical25 means). These achieve-ments have opened up new possibilities for single spins toact as sensitive local probes for parameters such as mag-netic field,64 electric field,65 or even temperature.66 The useof an electron spin as a probe of local environment is wellknown in EPR spectroscopy, for example, through techniquessuch as ENDOR (see Hyperfine Spectroscopy—ENDOR)and DEER (double electron-electron resonance, see DipolarSpectroscopy—Double-Resonance Methods). However, the

applicability of such methods at the single-spin level holdsthe potential for high-spatial resolution studies, for example,using scanning probe techniques to create spatial magneticfield images.

The development of electron spin-based ‘sensors’ has pri-marily arisen out of research on NV centers in diamond,where the electron spin of a single center can be opticallymeasured (see ODMR) at room temperature with coherencetimes around a millisecond (Figure 9,21). Figure 9 (a, inset)shows the energy level diagram of the (S = 1) NV center, inthe low-field limit–the difference of the fluorescence inten-sities of the mS = ±1 and mS = 0 states form the basis forthe optical measurement. Such systems couple to magneticfields via the Zeeman interaction and to other spins via dipolarand hyperfine interactions, while, thanks to the crystal fieldsplitting term, the NV center spin Hamiltonian also dependson electric field, temperature, and strain. Energy shifts arisingfrom variations in these parameters impart phase shifts tothe electron spin, and thus the longer the T∗

2 and T2 times ofthe system is, the weaker will be the perturbation that can bemeasured. The NV centers are brought in close proximity tothe sample that is to be imaged, either by implanting them nearthe surface of a diamond substrate (Figure 9b,67) or withindiamond nanocrystals on the tip of an AFM (Figure 9a,61).Recent achievements and proposals include nanoscale imagingof living cells (Figure 9c,66), measurement of action potentialsin neurons,68 and NMR-like spectroscopy distinguishing mul-tiple nuclear species.69 Sensing techniques can be categorizedinto two approaches: DC and AC sensings.

DC Sensing

DC sensing aims at measuring parameters that are static, at leaston the timescale of the measurement. Measurements are real-ized either by detecting changes in the electron spin resonancefrequency in the CW EPR spectrum or in the time domain viathe free induction decay (π∕2 − 𝜏). In both cases, the sensi-tivity is determined by the inhomogeneous spin linewidth orT∗

2 – this includes the effective broadening caused by randomlyfluctuating fields (aside from that to be detected). Dependingon the source of inhomogeneous broadening and the parame-ter to be measured, this limit can be extended to T2 when therefocusing pulse in a Hahn echo sequence does not reverse theinteraction between the spin and the measured quantity. Forexample, using NV centers in diamond, DC sensing of the tem-perature is possible with measurement times limited only by T2(Figure 9c and Ref. 66), because while phase shifts caused bymagnetic field inhomogeneities are reversed following the refo-cusing π-pulse, those due to temperature-dependent shift in thezero field splitting are not.

AC Sensing

In AC sensing, the sensor spin is tuned to sense the localenvironmental noise at some defined frequency. This can beachieved, for example, by applying DD to the sensor spin andexploiting the bandpass filtering effect of repeated π-pulsesin DD, as was described in the section titled ‘DynamicalDecoupling (DD)’ and Figure 5(a, b). As the pulse periodicity



Scanning diamondplatform

Hydrocarbon

Detection volume

∼ (5 nm)3Sample 1H

13C

Sensor

MicrowaveNV center

15

π/2 π/22π

NV1

(×104 c.p.s)

NV2

Heat

50

10

15

20

25

15

10

10

5

5

0

0

−5

−5

−10

y/μm

x/μm

Excitation laser

MW coil Sensor NV

z

yx

Target spins

|1⟩

|0⟩

2γB|−1⟩

ωMW

νL

τ τ

Sensor:

Target:

(a) (b) (c)

ππ π π πx

2

πy

2

∼5 nm

2.0

−60.0

1H PMMA

2τ = 250 μsOil

Noise

−30.0 0.0 30.0 60.0

(ν − νL)/kHz

24.2 24.22 24.24 24.26 24.28

T/°C

4.0

6.0

0

0.5

1

8.0

10.0

Popula

tion

SB(ω

) (n

T/√

Hz)

Figure 9. Quantum sensing using single-electron spins of nitrogen-vacancy (NV) centers in diamond. (a) Magnetic sensing with an NV center embed-ded in a scanning diamond nanopillar, allowing for 2D imaging with nanometer resolution (Reprinted by permission from Macmillan Publishers Ltd:Nature Physics, Ref. 61 © 2013). Here, a single-electron spin within the sample, also diamond, is targeted using DD (bottom) on both sensor and targetspins (lock-in method). (b) (Top) Schematic of a shallow NV center as a magnetic sensor for molecules at the surface of the diamond67 (From T Staudacher,F Shi, S Pezzagna, J Meijer, J Du, C A Meriles, F Reinhard, and J Wrachtrup. Nuclear Magnetic Resonance Spectroscopy on a (5-Nanometer)3 SampleVolume. Science, 339(6119):561–563, feb 2013. Reprinted with permission from AAAS. ). The detection sensitivity can reach volumes down to 5 nm3,corresponding to about 104 protons. (Bottom) Frequency spectra obtained using DD on the NV electron spin and shown as sensitivity against detuningfrom the expected Larmor frequency (𝜈L) of spins in the test molecules: protons in oil (yellow) and 1H polymethyl methacrylate (PMMA) (green). Thespectrum from an external magnetic field noise source at a fixed frequency, simulating a precessing spin, is also shown (red) for reference. (c) (Top) Confocalmicroscopy measurement of a single human embryonic fibroblast cell, outlined by the dotted line, with injected nanodiamonds (NV) as local thermometers(Reprinted by permission from Macmillan Publishers Ltd: Nature, Ref. 66 © 2013). (Bottom) Spin-state projection as a function of temperature, using thegiven modified Ramsey sequence for the spin-1 NV system, where the 2π-pulse here refocuses phase shifts from static magnetic fields but not from tem-peratures. The projection amplitude oscillates with temperature, hence this technique is used only for relative temperature sensing. In addition, because thedelay 𝜏 determines the oscillation period, it can be varied to adjust the balance between greatest dynamic range (small 𝜏) and maximum sensitivity (large 𝜏)

is swept, so is the filter frequency (centered around 1∕2𝜏 inFigure 5b); and when the filter coincides with a dominant com-ponent in the noise spectrum, a reduction is observed in theecho intensity. Mapping out the noise spectrum experiencedby a spin can be useful to investigate sources of decoherence.For sensing, the technique has been described as the quantumanalog of a lock-in amplifier.35 The detection frequency ofthe sensor spin could be set, for example, to match the nat-ural Larmor precession of a nuclear spin.36 In Figure 9(b),1H spins in hydrocarbon molecules were detected using thismethod with NV electron spins. Alternatively, DD can beapplied simultaneously to some spin that is to be probed, aswell as to the sensor spin, in order to single out that particularinteraction61 (pulse sequence shown in Figure 9a, bottom), asan extension of multipulse DEER methods.

While measurements in the frequency domain are the mostcommon, time-resolved sensing is also possible. In this case,DD sequences are used to obtain the different components ina time series expansion (e.g. the polynomial or Walsh series)from which the signal can be reconstructed.68 This has beenproposed to monitor the magnetic field produced by actionpotentials in neurons.

For magnetic field sensing (or magnetometry), the sensitivity(in T∕

√Hz) of an ensemble of N independent electron spin-1/2

is approximately given by the relation64:

𝜂 ≈ π2|𝛾e|C√NT2

(14)

where C is the measurement efficiency, which is the fractionof spins that can be measured in a single experiment with asignal-to-noise ratio of 1. In EPR, C encompasses factors suchas the efficiency of the microwave detection, the resonatorquality and filling factors, and inhomogeneous broadening.The relation above assumes the sensed field is fully synchro-nized with the applied DD sequence (i.e., same frequency andphase, see bottom of Figure 9b).

Multiple QubitsUp to this point, we have been concerned primarily with singlequbits, the measurement and control of their states, and theirapplication as quantum sensors. Applications in quantum infor-mation processing rely on the ability to engineer interactionsbetween multiple spin qubits in order to perform useful com-putations.

Multiple Qubit Gates

In classical logic, the so-called universal gates such as NANDand NOR operations can be used to construct any combi-natorial logic function, and thus to perform any classicalalgorithm. Analogously, although there exist a vast set of pos-sible two-qubit operations that can be performed, we can focuson a subset that possesses this ‘universal’ property and thusprovides the necessary toolbox to perform universal quantumcomputation. One such universal set of gates is combination of



the single-qubit Hadamard gate (see section titled ‘Single-qubitGates’) and the two-qubit controlled-NOT, or C-NOT gate,expressed as

UC-NOT =⎛⎜⎜⎜⎝1 0 0 00 1 0 00 0 0 10 0 1 0

⎞⎟⎟⎟⎠(15)

in the basis {|00⟩, |01⟩, |10⟩, |11⟩}. In other words, the C-NOTcorresponds to an inversion (or πx,y rotation) of the second(‘target’) qubit if the first (‘control’) qubit is in state |1⟩, and nooperation (the �� process) if the control qubit is in state |0⟩. AC-NOT gate can be performed in EPR (modulo some phasefactor) as a selective πx-pulse, which can be readily imple-mented between pairs of spins if their coupling is large enoughto be resolved. Another universal gate (when combined withthe H-gate) is the controlled-phase (C-PHASE) gate:

UCPHASE =⎛⎜⎜⎜⎝1 0 0 00 1 0 00 0 1 00 0 0 −1

⎞⎟⎟⎟⎠(16)

which has the effect of applying a π phase shift to |11⟩ withrespect to the other three basis states. A C-PHASE can be con-verted to a C-NOT gate by applying a Hadamard operation tothe ‘target’ qubit before and after the C-PHASE, and both theC-NOT and C-PHASE gates can be used to generate entan-gled states.70,71 A fast C-PHASE gate between two nuclear spinqubits coupled to an electron spin can be performed using theprinciple of the AA phase gate described in the section titled‘Single-qubit Gates’ and has been demonstrated using opticallyexcited triplet states72 and NV centers.73

State Tomography of Multiple Qubits

Extracting the density matrix for a set of coupled qubitsrequires going beyond the basic approach outlined in thesection titled ‘Quantum State Tomography’ and carries somefundamental challenges. First, the number of independentelements increases exponentially with the number of qubits,making full density matrix tomography practical for only smallnumbers of qubits. Second, these elements must be mapped toobservable quantities in the system (in the same way that theSz component was mapped onto Sx in the single-qubit case).Given imperfect pulses, this mapping process can becomeincreasingly sensitive to pulse errors as the number of qubitsincreases.

While these challenges cannot be readily overcome, it isat least possible to reduce the sensitivity to pulse error, tosome extent, by ‘labeling’ elements of the density matrix usingphase shifts, before mapping them onto observables.70 In thismethod, illustrated in Figure 10, a phase shift 𝜙i is applied tostate |i⟩, such that coherences between such states |i⟩ and |j⟩acquire phases 𝜙i ± 𝜙j. The phases 𝜙i are each incremented bysome fixed amount 𝛿𝜙i from measurement to measurement,leading to phase oscillations, and the phase increments 𝛿𝜙iare selected so that each type of coherence acquires phase at adistinct effective frequency. Various approaches can be used toapply the phase shift, for example, using geometric phases anda (π)0(−π)𝜙 sequence.71

Quantum Memory

The ability to transfer a qubit state between different degrees offreedom (such as between electron spin and nuclear spin transi-tions) has numerous applications, from enhanced fidelity spin

||3⟩

|1⟩ |1⟩ |1⟩

|3⟩ |3⟩

π0

π0 −πi⋅δϕ−πi⋅δσ

ϕ = 0 ϕ = π/4 ϕ = π/2

−πϕπ0

|1⟩

|3⟩

|2⟩⟨3|

|1⟩⟨3|In-phaseQuadrature



|4⟩⟨3|

|1⟩⟨3|

|2⟩

|4⟩

(a)

FT

/a.u

.

(c)

Am

plit

ude/a

.u.

(b)

20 40 60 80 100 120

Increment number (i)

−0.25 −0.15 −0.05 0.05 0.15 0.25

Frequency (phase acquired / i )

Figure 10. State tomography of a coupled electron and nuclear spin. (a) A geometric phase 𝜙 can be imparted onto two states driven by a pair of pulses:(π)0(−π)𝜙. (b) Such phase gates are applied to both the |1⟩ ↔ |3⟩ and |3⟩ ↔ |4⟩ transitions, where the phase applied is incremented from shot to shot by𝛿𝜙 and 𝛿𝜎, respectively (thus, the phases applied on the ith shot are i𝛿𝜙 and i𝛿𝜎). The effect of these phase gates is shown for three different initial states:|1⟩⟨3| (an electron spin coherence); |3⟩⟨4| (a nuclear spin coherence); and |2⟩⟨3| (a zero quantum coherence)–in each case, the state is first prepared, thenthe phase gate is applied, and then the state is transformed to be observable in the electron spin echo. (c) The Fourier transform of the resulting phaseoscillations yields distinct frequencies for the different quantum coherences, equal to 𝛿𝜙 for the |1⟩⟨3| coherence, 𝛿𝜎 for |3⟩⟨4|, and −𝛿𝜙 − 𝛿𝜎 for |2⟩⟨3|.(Reprinted by permission from Macmillan Publishers Ltd: Nature, Ref. 71 © 2011.)



Transfer to N

+X, φ = 0°

+Y, φ = 90°

−X, φ = 180°

−Y, φ = 270°

Ele

ctr

on

sp

in e

ch

o

Time/μs

ππ π

ππ

GenerateE-coherenceof phase φφ

RefocusN-coherence

0 20

Transfer to E

ππ

(Nuclear echo)

τnτe1 τn

(Electron echo)τe1 τe2 τe2

π/2

Electron echo

Measure

Sx Sy

(a)

3

2

1

010 0.5

Storage time, 2τn/s

T2n = 1.75 s

Ech

o in

ten

sity/a

.u.

(b)

|ψe⟩

|ψn⟩ = |0⟩

‘Store’ ‘Retrieve’

≡ C-NOT

Figure 11. SWAP operations between electron and nuclear spin degrees of freedom. (a) (top) Two C-NOTs approximate a SWAP and can be used as thebasis for a quantum memory where a coherent state of the electron spin |𝜓e⟩ is transferred to a nuclear spin degree of freedom and subsequently retrieved.In the implementation of this scheme (bottom), C-NOT gates are achieved using selective π rotations, while additional refocusing pulses are needed toaccount for inhomogeneous broadening. The final electron spin echo is shown to vary its phase in accordance with the original π∕2-pulse applied at thebeginning of the sequence (50 ms earlier) (Reprinted by permission from Macmillan Publishers Ltd: Nature, Ref. 76 © 2008). (b) Measuring the recoveredecho intensity as a function of the total storage time 2𝜏n yields a nuclear spin T2 of almost 2 s, compared to the electron spin T2 of 5 ms under the sameconditions. In later experiments, nuclear spin coherence times as long as 3 min were measured for neutral P donors in silicon at 1.7 K.24

measurement,74 initialization and sensing, quantum repeatersfor quantum communication,75 and to preserve qubit states forlonger periods of time.76 This state transfer can be achieved bythe SWAP logical operation, which exchanges the states of twoqubits:

USWAP =⎛⎜⎜⎜⎝1 0 0 00 0 1 00 1 0 00 0 0 1

⎞⎟⎟⎟⎠(17)

and can be performed in general via three C-NOT operations(although typically in practice only two are used), as illustratedin Figure 11.

In NV centers in diamond, nuclear spin quantum memoryoperations have also been achieved using fast passages acrossLandau-Zener transitions,77 or exploiting anisotropic hyperfineinteractions in the spin system.8

Quantum Error Correction

Identifying and correcting errors is essential for any practi-cal realization of an information processor–this is especiallyimportant in quantum information processing where theexpected error rates due to decoherence and gate fidelity areexpected to be high, and also particularly demanding, as weare prohibited from directly measuring the state of any logicalqubit during the calculation (to avoid collapse of the qubitstate). Fortunately, strategies have been put forward since the

mid-1990s to identify and correct errors in qubits withoutlearning about their state.78,79 These strategies have the com-mon feature of redundancy, representing one logical qubitusing several physical qubits, and have different fault-tolerantthresholds, corresponding to the maximum intrinsic errorrates per gate that can be effectively corrected (see sectiontitled ‘High-fidelity Operations’).

One approach to quantum error correction is based on‘majority voting’ and requires at least three physical qubits perlogical qubit. This scheme has been implemented using NVcenters in diamond, employing the NV center electron spinand two nearby nuclear spins to represent the logical qubit.73,80

A qubit state is first mapped from the electron spin to all threespins via two C-NOT gates, at which point it becomes robust tospin-flip errors (Figure 12). After some period, further C-NOTgates are applied to distill any single spin-flip error onto thenuclear spins, leaving the electron spin qubit in its originalstate. The nuclear spins must then be refreshed into an initialstate before being reused.

Notably, among the assumptions in such error-correctionschemes is that of uncorrelated errors, and in the case of severalnuclear spins with significant coupling to an electron spin, thisassumption may not be justified. Nevertheless, such demon-strations highlight the potential benefit of small ‘registers’ ofnuclear spins coupled to an electron spin.

More recent approaches to quantum error correction basedon topological schemes such as the surface code41 have highthreshold error rates, although at the expense of even larger



|0⟩

|0⟩

|ψ⟩ E

0.15

Error on:

Electron

Electron (+Nucleus 1)

Nucleus 1

Nucleus 2

0.0 1.0 1.50.5

Error angle θ/π

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Pro

ce

ss fi

de

lity F

p‘Encode’ ‘Decode’

(a)

(b)

Figure 12. Quantum error correction using an electron spin coupled totwo nuclear spins.80 (a) Theoretical error-correction protocol for bit-fliperror (box labeled ‘E’). The initial ‘encoding’ step consists of entanglingthe qubit (in state |𝜓⟩) with two nuclear spin ancillae, while the decod-ing stage leaves the qubit in its original state, |𝜓⟩, regardless of whether asingle bit-flip error occurred. (b) Process fidelity of the protocol consider-ing a single or two flip errors among the three spins. The gray line gives theaverage fidelity without correction

resource overheads (e.g., in terms of the number of physicalqubits used to represented one logical qubit).

Spin-qubit Networks

Controlling scalable arrays of interacting spin qubits remainsone of the greatest challenges in realizing spin-based quantuminformation processors. Approaches such as those based onelectrically tuneable exchange couplings in nanoelectronicdevices12,81 and dipolar couplings between spins82,83 are beingexplored; however, the short-range nature of the spin–spininteractions, combined with the need to address individualspins for measurement and control, provide considerable tech-nical challenges. For these reasons, quantum ‘interconnects’to couple distant spins are an attractive ingredient to createscalable networks of spin qubits.

Candidates for such interconnects include optical photons,which can be entangled with the state of an electron spinand then interfered to entangle spins separated by as much

as 1.3 km.75 Such schemes require spins systems with strongcoupling to light, for example, quantum dots or certain defectcenters in solids. Another possibility, albeit over shorter lengthscales, is to use single microwave photons strongly coupled tospins in microwave cavities. This represents a new regime forEPR, but one in which there is increasingly progress and manyopportunities.84–86

ConclusionsPulse techniques from NMR and EPR have played a major rolein the development of spin qubits, from the characterizationof candidate systems and measurement of coherence lifetimes,to the high-fidelity control of spins to achieve logical gates.We are now at the stage that innovations arising from researchinto quantum information processing may feed back into EPRmethodology, whether in the form of instrumentation suchas quantum-limited microwave amplifiers,86 magnetic fieldsensing techniques to complement structural characterizationby DEER, and the use of DD methods as a spectroscopictechnique, and, ultimately, it may provide advanced simulatorsto model the dynamics of coupled spin systems at scales notpossible today.

AcknowledgmentsWe thank Richard Brown and Stephanie Simmons for the datashown in Figure 1(c), as well as Sofia Qvarfort, Padraic Calpin,David Wise, and Philipp Ross for the data shown in Figure 7.

Related ArticlesSpin Dynamics; Relaxation mechanisms; Pulse EPR

Biographical SketchesGary Wolfowicz. b 1987; B.E. 2007, M.E. 2011, AppliedPhysics and Electrical Engineering, Ecole Normale Superieurede Cachan, France. D.Phil (PhD) 2015, Materials, Universityof Oxford, UK. Postdoctoral scholar, 2015–present, Universityof Chicago, USA. Approximately 10 publications. Currentinterests: control of electron and nuclear spin qubits.

John J.L. Morton. b 1980; B.E. M.E. 2002, Electrical Engi-neering, University of Cambridge, UK. D.Phil (PhD) 2005,Materials, University of Oxford, UK. Junior Research Fellow2005–2009 and Science Research Fellow 2010–2012, St. John’sCollege, University of Oxford, UK. Royal Society UniversityResearch Fellow 2008–2016. Reader at UCL, UK 2012–2014and Professor of Nanoelectronics & Nanophotonics at UCL2014–present. Approximately 90 publications. Current inter-ests: electron and nuclear spin qubits in semiconductors andmolecular materials.

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