Grover's search algorithm

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National University of Singapore

Thesis Submitted in Partial Fulfilment of

The Requirements for The Degree of

Bachelor of Science,

Honours in Physics

Effects of Noise On

Grovers Quantum Search Algorithm

Author :

Angeline Shu Sze YiU074599X

Supervisor :

Dr. Yeo Ye

2010-2011


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Preface

For some years now, quantum mechanics has been avidly researched and explored. With its uniquely

quantum features, it has brought the application of physics in information transfer to lengths never

before reached. Quantum algorithms like Shors factoring algorithm and Grovers Search algorithm are

merely a few examples of the attempts to apply quantum mechanics in information processing. Our

study shall focus on Grovers Search.

This thesis will largely be focusing on the numerical studies of Grovers Quantum Search Algorithm

in the presence of dephasing noises. The physical system that our computation is based on is the solid-

state quantum dot system. Thus, we explore rstly, how a two-qubit Grovers Search is carried out

on this particular physical system via two different computational models the circuit model and the

one-way quantum computing model. We shall then study the effects of noise on the algorithm by means

of a mathematical model that we shall introduce. The application of this model requires one to solve

the Lindblad equations. We solved them for three different noise models, the bit ip, bit-phase ip and

phase ip models. Our rst and foremost aim in this project is to obtain practical advice in improving

this algorithm when in the presence of noise if the algorithm is at all a plausible solution to the search

problem in a noisy environment. We are also interested in the more lofty goal of understanding how the

environment affects the system. The mathematical model introduced does indeed allow us to probe into

this, as it allows us to take the search apart to observe its evolution in time. However, as this task is

tough and tedious, we achieve it but in part.

I would, at this point, like to thank Dr. Yeo Ye. If not for his patience and forbearing, wisdom and

constant interest in my progress, I would not have found as much joy as I did in studying how Grovers

Search actually works, how to model noise and the many other ancilliary principles and concepts that

accompanied this project. His encouragements when I nd myself discouraged and drowning in the mass

of information surrounding us has helped me steadily plough through, come what may. I would also

like to extend my gratitude to Setiawan who, despite being busy with his own research and under no

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obligation to assist me, has provided extremely helpful feedback. He has not only helped me side-step

many technical difficulties in my computation, but has also helped deepen my theoretical understanding

of quantum mechanics. To all my other friends who had to bear through my constant moaning and

worrying, who have been around for me to bounce ideas off and who cared enough to sit with me and

help me clarify my thoughts, to those who accompanied me in my late night attempts to work through

my project, I am sincerely, sincerely grateful. I am especially thankful for Vino, who never gives up on

encouraging me when I am down, and continually tells me that things will work out okay even after the

n th time I repeat a complaint. I am thankful for Andrew, who has made this whole process of thesis

writing a whole lot more bearable by just... being around. I am grateful also to my family, who, though

are far away, supports me and reaches out to grab and steady me in a way that no one else can...

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Abstract

As of now, there has yet to be a detailed study of noise effects on Grovers Search that allows us to

actually peek into the dynamics of the search. Thus, every interaction applied in a noisy environment

is like a black box no one knows what really happens throughout the interaction, we know only its

end result. In this project however, we have developed a mathematical model of noise that allows us to

probe within the interaction to know explicitly how the state evolves with time.

Before we dive into the noise model however, we rst aim to understand the ideal search. Executing

the search on two different computation models, we nd the following impositions useful. For the circuit

model, we rst direct our efforts into nding the most efficient circuit that realizes Grovers algorithm

for two-qubits. As the study is on noise effects, it seems to make little sense to conduct a study on a

circuit that is not rstly made efficient, for there would be unnecessary steps in such searches. This leads

to unnecessary and unwanted reduction in delity. For the one-way computing model, we shall assume

that measurements are idealized. This simplies the study, allowing us to direct our focus to examining

the effects of noise only in the generation of the entangled resource.

However, due to the complexity of the mathematics involved, it still is highly inefficient to solve

the Lindblad equation for the output density matrix of the rst gate, plug that back into the Lindblad

equation to solve for the second, and so on so forth until the last gate few though those gates might

be. Obtaining the nal output state of Grovers Search in this manner yields extremely complicated

states. Given the computational power afforded to us by our current technology, it still is difficult to

solve these successive Lindblad equations. Thus, we propose a mathematical model of noise which de-

composes everything into noisy basis maps. Instead of plugging the density matrix of the system itself

into the Lindblad equation, we solve Lindblad equations for the all the basis states . Hence, in this

project we will obtain the time-evolution of each basis under a specic Hamiltonian in the presence of a

specic type of noise. To obtain the time-evolution of the input state, we extract the coefficient of each

basis and multiply it with the respective noisy basis evolution.

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With this, we analyze the search in the presence of the three different types of dephasing noise. We

then manipulate the different parameters available to study the system and to obtain the combinations

that would return us the state with the best delity.

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

1 Introduction 1

2 Grovers Ideal Algorithm 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Computational Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Two-Qubit Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 One-Way-Computing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Grovers Algorithm In Noise 20

3.1 Lindblad Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Solutions to the Lindblad Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Bit Flip Noise ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Bit-Phase Flip Noise ( y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.3 Phase Flip Noise ( z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Extending To A Larger Number of Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Analysis 34

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 One Gate Noisy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 All Gates Noisy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.1 Same Speed Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.2 Different Speed Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Maximizing Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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4.4.1 Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4.2 Same Speed Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4.3 Different Speed Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Conclusion 66

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Chapter 1

IntroductionQuantum information and quantum computing is and has been an area of immense interest and

intense study in recent years. Both classical and quantum computation requires information to be

encoded in physical bits two-state systems that, like a switch can be either switched on (0) or off

(1). These on and off states are called basis states. Quantum computation poses benets over classical

computation because a qubit (quantum bit) is able to be in both the on and the off states at the same

time with certain probabilities. This ability to be in superposition states, when extended to two or more

qubits leads naturally into entanglement between these qubits, allowing information to be stored not

only in the physical qubits, but in the correlation between them. The explicit expression of a quantum

state takes the following form,

| =i

ci |i (1.1)

where ci are the probability amplitudes of each basis state |i . These probability amplitudes are complexnumbers with both a magnitude and a phase, the square of which gives us the probability of | beingin that particular basis state when measured. It is this probability amplitude that allows us to speak of

states in the language of waves, alluding to superposition, interference and so on so forth. This language

will be extremely helpful in understanding Grovers Algorithm.

However, for all its benets over classical computing, quantum computing runs into problems of its

own. The most signicant difficulty that confronts quantum computing is the phenomenon of decoher-

ence [1]. As quantum states interact with the environment they decohere into classical states, losing their

quantum-ness their ability to superpose and be in entangled states and thus, losing their quantum

edge. Yet, interaction with the environment is necessary for us to both manipulate the states as well as to

encode and retrieve information from them; all of which are integral parts of our computation processes.

Thus, a quantum information theorist is often in the difficult dilemma of searching for a physical system

that is robust, and yet, one that allows him a measure of control over the qubits. Furthermore, as larger


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INTRODUCTION

and larger numbers of qubits interact together to form the framework of our computational system, this

becomes more serious a problem, making scalability of quantum computers a very challenging project.

For this reason, much time and effort have been devoted to studying how dephasing noise affects

quantum systems and quantum computation. Dephasing noise causes off-diagonal terms in our density

matrix to decay to 0, effectively reducing our system to a classical system. This project does a similar

sort of analysis with a signicantly new contribution: the model we introduce to study noise allows us

to study the dynamics of the system as it evolves with time. Thus, we can peek into the system itself

as it evolves with time, rather than treating the noisy time-evolution like a blackbox, the way most

of the currently prevailing studies have been done. We look not merely at the output state after each

interaction is over, but the evolution of the state throughout the interaction.

The model introduced can be applied over a wide range of quantum computation processes. In this

project, we apply this analysis to Grovers Search [2, 3, 4]. Grovers Search is an algorithm that pro-

vides a polynomial speedup to the classical solution of the following problem: Searching for a desired

entry in an unsorted database. Classically, unlike binary searches in a sorted database, searching an

unsorted database can only be executed by examining each item in the database until the desired item

is located. Such a search has a time complexity of O(N ), with the worse-case scenario being N accesses

to the database where N = 2 n is the number of possible states available in an n-bit or n-qubit system.

Grovers Search, on the other hand, requires at most N accesses to the database. For databases withN = 100, we would only need to access the database 10 times the effect of the speedup all but increas-

ing as N increases. In a realistic computation problem, N would be large indeed which unfortunately

brings us right back to the problem of scalability of quantum algorithms.

This project shall focus on the dynamics of a two-qubit Grovers Search in the presence of dephasing

noise. We shall rely especially on two quantities throughout our analysis: negativity and delity. The

former is a quantication of entanglement, the latter, of the success of the search. Fidelity essentially

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INTRODUCTION

measures how close the output density matrix obtained from our calculations is to the ideal density

matrix or the desired output of the search, while negativity measures the entanglement of two qubits.

Both these measures are in the scale of 0 to 1 with 1 being an exact match and maximum entanglement

respectively.

Grovers algorithm in itself is a probabilistic algorithm, returning us the desired state with a certain

probability. In his paper, Grover claims that the probability of obtaining the desired state shall never

go below a 0.5 delity, returning us the correct state at least half the time. Thus, in our project, only

searches returning a delity of 0.5 and above shall be considered successful. We shall run the search on

two types of computational models and study these under three different noise models. Our nal aim

is to provide practical advice on the steps one could take to improve Grovers Search. Along the way,

we hope to understand a little more about the relationship between entanglement and delity, as well

as the relationship between delity and all the other parameters available to us. This shall, hopefully,

motivating some level of understanding behind any advice given.

Before we reach the analysis section, we rst need a deeper understanding of Grovers Search as well

as the ways to realize this search. Following that, there will be a section on the formulation of our noise

model. The conclusion wraps up this thesis and the Mathematica programs involved in the computations

are attached in the Appendix.

Unfortunately, due to an initial misunderstanding of Grovers Search as executed on one-way com-puting models, as well as a misjudgment on the computational time required, we could not complete the

analysis of that model on time. Thus, for this model, we shall merely discuss the realization of Grovers

Search, the noisy generation of the cluster state, as well as attach the Mathematica program that was

written for the analysis without actually analyzing it.

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INTRODUCTION

g

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Chapter 2

Grovers Ideal Algorithm

2.1 Introduction

Grovers algorithm starts off by uniformly distributing the probability amplitude over the various

basis states, creating a superposition of the form

| =N

i=1

1 N |i

where |i are the encoded qubits. In our study, we shall assume the database to already be in thisdistribution, with no defects a perfect database. An iteration of Grovers algorithm involves two main

stages: (a) inverting the phase of the desired basis state, and (b) performing an inversion about the av-

erage operation. The rst stage of the search marks the state being searched for and the second enforces

its probability.

The essence of the algorithm is captured in the interference of probability amplitudes, where the

probability amplitude of the desired state undergoes constructive interference whilst those of the rest

undergo destructive. Thus, stage one of the search is crucial as it inverts the phase of the desired state,

singling it out from the rest of the states. With the phase of the marked basis state inverted, the

constructive and destructive interferences can be realized by doing an inversion about the average, as

can be better visualized in the picture below taken from Ref. [3]. The lines in the picture represent the

probability amplitudes of each basis state.


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Computational Models GROVERS IDEAL ALGORITHM

groverA.jpg

(a) First Stage

groverB.jpg

(b) Second Stage

Figure 2.1: Probability Amplitude Distribution of Basis States

As the average (the dotted line in the picture) will be towards the tip of the probability amplitude

of all the states except the marked state, inverting all phases with respect to this average causes

constructive interference for the desired state, and destructive interference for the rest. With enough

iterations, we can easily see how the probability of the desired basis state will far outweigh the other basis

states. The specic number of iterations necessary is worked out by Grover himself [4] and discussed

further by Boyer et al. [5]. For a two-qubit system however, which is the focus of our project, we require

only one iteration, as shall be shown in the upcoming subsections.

2.2 Computational Models

The two computational models through which our search is executed is the familiar circuit model and

the measurement-based one-way-computing model. The circuit model, like most classical models that

we are familiar with, is executed by applying the necessary unitary interaction (gates) on each qubit to

realized the desired effects, such as rotation.

The one-way-computing model [6] however, having no classical analogue or example, is more unique.

This model requires the entire resource for the computation to be a highly entangled cluster state [7]

involving a large number of qubits. Doing projective measurements in different basis realizes different

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GROVERS IDEAL ALGORITHM Two-Qubit Search

effects on the encoded qubits. Cluster states are prepared by initializing all qubits into the |+ state,followed by applying CPhase (controlled-phase) gates onto pairs of neighbouring qubits. A CPhase

operation yields the following: |i | j (1)ij |i | j where (i, j 0, 1), that is, it does a phase ipoperation on the second qubit when the rst qubit or control-qubit is in the |1 state. Our study, withrespect to this model, is directed towards the effects of noise on the generation of the entangled resource

rather than the execution of Grovers Search via measurements. Thus for the one-way-computing model,

we shall be looking at the delity of two different states: the delity of the generated cluster state under

noise; and the delity of the output of the search after the relevant idealized measurements are executed

on the cluster state generated in noise.

2.3 Two-Qubit Search

2.3.1 Circuit Model

The circuit model works by application of gates. The application of a particular sequence of gates

will achieve a particular effect on the qubits. This subsection shall attempt explain the effect we intend

to achieve (as shall be denoted in the matrix forms seen) as well as the sequence of gates necessary to

attain these effects.

Generic Two-Qubit Search

The rst stage of Grovers Search can be achieved by applying a selective phase inversion matrix,

which for one that inverts the |11 state, has the following form,

I |11 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 -1

(2.1)

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Two-Qubit Search GROVERS IDEAL ALGORITHM

Applying I |11 to the input state of a two-qubit Grovers Search yields,

vin = |++ (2.2)vm = I |11 vin

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 -1

12

1

1

1

1

= 1

2

1

1

1

-1

= 1

2 (|00 + |01 + |10 |11 ) (2.3)

where v in is the input state and v m is the marked state the state that the system will be in after stage

1 of the search has been completed. To invert any of the other basis states, just switch the sign on the

relevant diagonal element. This matrix can be expanded to higher qubits by just adding 0 entries to the

off-diagonal and 1 entries to the diagonal.

The second stage of the search is then realized through the application of the diffusion transform

matrix, D , where the componenets take the form

D ij = 2N ij (2.4)8


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For two qubits, it has the following effect,

vout = D

vm

=

12 12 12 1212 12 12 1212

12 12 12

12

12

12 12

121212

-12

=

0

0

0

1

= |11 (2.5)

where vout is the output of the search. One can easily see how the effect would be similar for the other

three marked states. Thus for a two-qubit system, we need only one iteration to obtain the desired state.

Furthermore, we obtain it with 100% probability. Again, this treatment can easily be expanded to deal

with higher qubits by just expanding the matrix D according to Eq. (2.4).

We can achieve the effects of matrices I |ij and D through the following equation

I |ij = R12z [

2

, 2

] USWAP R1z [] USWAP (2.6)D = W W I |00 W W (2.7)

where W is the Walsh Hadamard Transformation acting on one qubit, which shall be further explained in

a moment. Notice that the equtions is read from right to left, where the rightmost operation is the rst

gate. The R gates are rotation gates, rotating, in our case, the electron spin along the direction specied

by the subscript by the angle as specied inside the square brackets and acting on the qubits specied

by the superscript. The USWAP gate comes from the SWAP operation, which basically swaps the rstqubit with the second, |01 |10 and vice versa. The USWAP gate stops this process at midpoint,causing the resultant state to be a maximally entangled state, some combination of |01 + i |10 withappropriate phase factors. The gates that we will be using through out this project take the following

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explicit forms:

Rx [] =cos[

2]

i sin[

2]

i sin[ 2 ] cos[2 ](2.8)

Rz [] =cos[2 ] i sin[ 2 ] 0

0 cos[2 ]i sin[ 2 ](2.9)

USWAP =1 0 0 0

0 1+ i21i2 0

0 1i2 1+ i2 0

0 0 0 1

(2.10)

A = rotation of R x and R z leaves a bit ip and phase ip effect respectively. Working in the Dirac

notation (bras and kets) and understanding the gate operations in more phenomenological terms such as

these affords us a much deeper insights and appreciation of all that takes place. Thus, we will attempt, as

far as we can, to switch between the purely mathematical matrices and the more elegant Dirac notation,

explaining the phenomenological effects of the system as it evolves through time.

Referring back to Eq. (2.6), one can control the basis state marked by I |ij by choosing the appropriate

rotation angle in the last z-rotation gate. The following table summarizes the necessary rotation angle

to mark the appropriate basis states.

|00 |01 |10 |11R12

z [

2,

2] R12

z [

2,

2] R12

z [

2 ,

2 ] R12

z [

2,

2]

Table 2.1: Gates Marking the Appropriate Basis States

For simplicity, we shall denote the search for the |00 as GS00, |01 as GS01 and so on so forth.

We now return to the Walsh Hadamard Transformation, W . This transformation creates and destroys

superposition. The one qubit W gate can be exactly reproduced with two single qubit rotation, iRz [] 10


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Ry [2 ], and has the following matrix form

W = 1

21 1

1 -1

From the explicit expression of W above, we can see that it is its own inverse. This explains how W can

both create and destroy superposition

|0 1 2 (|0 + |1 ) (2.11)

|1 1 2 (|0 |1 ) (2.12)

1 2 (|0 + |1 ) |0 (2.13)1 2 (|0 |1 ) |1 (2.14)

Thus, substituting the relevant gates in for I |00 and W , we get,

D = R 12z [, ] R12y [2

, 2

] R12z [2

, 2

] USWAPR1z []

USWAP R12z [, ] R12y [

2

, 2

] (2.15)

Most Efficient Search

As we are studying the effects of noise, the sensible thing to do before we begin our analysis is to

ensure that our circuit has achieved maximum efficiency. Thus, in this subsection, we check if there

are redundant gates, asking if we could have achieved a similar effect (where the only system under

consideration is a two-qubit system) with a smaller number of gates. This is extremely important as it

is unwise to study the effects of noise on a search that does not accurately reect the least possible noise

one could have. Unfortunately, this inevitably limits the generality of our most efficient search to only

a search that works on two qubits.

This portion was dealt with through a lot of trial and error, as well as educated guessing and some

background researching. Bodoky and Blaauboers paper [8] was of much assistance in this portion of the

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project. In this most efficient search, we only have two USWAP gates, one to entangle the two qubits,and the other to disentangle them. We denote the most efficient selective phase inversion and diffusion

transform matrix by I |ij and D respectively.

I |ij = R12z [

2 ,

2 ] USWAP R

1z [] (2.16)

Da = R 1x [] USWAP R12x [2

, 2

] (2.17)

Db = USWAP R12x [2

, 2

] (2.18)

where Da inverts about the average for GS00 and GS11, and Db, for GS01 and GS10. Although wehave sacriced the generality of the search for its efficiency the different searches now require different

application of gates this sacrice seems reasonable for the following reason: In most cases, one is well

aware of the entry that is being searched for. And since we know which state is being searched for, we can

apply the appropriate algorithm accordingly. Do note that although I |ij and D achieve, in a two-qubitdatabase, the exact same effect as I |ij and D, they do not take the exact same explicit mathematicalform. The matrix components of I and D are different from those of I and D.

As we trudged through this project, we found the explicit form of each input state helpful. Thus the

table below is produced for future reference.

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Gate GS00 GS01 GS10 GS11

R1z [] |++ USWAP |+

R12z [2 , 2 ] 1+ i2 (|+ + i |+ )1i2 |y y 1+ i2 |y y 1+ i2 |y y 1+ i2 |y yR12x [2 , 2 ] i |y y + i |y y i |y y + i |y y

USWAP 1+ i2 (|01 + i |10 ) 1+ i2 (|00 + i |11 ) 1+ i2 (|00 i |11 ) 1i2 (|01 i |10 )R2x [] |01 - - |10

OUTPUT |00 |01 |10 |11

Table 2.2: Explicit Form of States Involved in Grovers Search

Note that the states listed are the input state of each gate, and recall that GS01 and GS10 require

one less gate operation compared to GS00 and GS11. As we shall be referring to the specic states and

gates of the search in later sections, we impose the following nomenclature. Each input state and each

gate shall be indexed with two numbers. The rst number establishes the stage of Grovers Search being

executed, whilst the second number details the n th gate operation of that stage. Thus G11 refers to the

rst gate, R 1z [] and v11 refers to state as it evolves through R 1z []. Any properties describing specic

states can be indexed in the same way, as shall be seen later. The gates to which each index refers to is

tabulated below.

G11 G12 G13 G21 G22 G23

R1z [] USWAP R12z [2 , 2 ] R12x [2 , 2 ] USWAP R1x []

2.3.2 One-Way-Computing Model

Two-Qubit Grovers Search

The cluster state needed for Grover Search has the form as pictured in the image below [9]. The line

connecting the qubits denotes entanglement.

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boxclusterstate.jpg

Figure 2.2: Cluster State

Computation on cluster states proceed via measurements on particular qubits in the appropriate

bases. When we speak of measurement in a particular basis, {|+ , | } for example, the 0 outcomeof that measurement refers to a measurement outcome of |+ , and the 1 to a measurement outcomeof |. If the basis of the measurement was {|, |+ } instead, the 0 outcome will refer to | and1 to |+ . For simplicity, a basis measurement shall be referred to as B j [ ] {|+ j , | j }, where

| j = 1 2 |0 j + ei |1 j and the subscript j refers to the qubit under consideration. After eachmeasurement is done, we trace those qubits out before proceeding to the next measurement for the

cluster state reduces in size as one proceeds with computation.

In cluster state computations, only resulting clusters where previous measurement outcomes were 0

will be allowed to proceed to the next measurement or the next step of computation. A feedforward will

have to be applied to cluster states resulting from measurements where the outcome was 1 before they

can be allowed to proceed with computation. Feedforward is basically a process in which we reinterpret

the resulting cluster by either changing the basis in which future measurements are to be done or by

performing rotation operations on certain qubits. For Grovers Search, the feedforward required is {s2s4, s3 s1, s5 s2, s6 s3}, meaning that if the measurement on qubit 4 (or 1) returns us 1 rather than0, we should measure qubit 2 (or 3) in a different basis [9]. If however, the measurement on qubit 2 (or

3) returns us an outcome of 1, we would need to apply a bit ip operation on qubit 5 (or 6) to correctly

interpret the resulting state.

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For our particular conguration of qubits in Fig. 2.2, measuring qubits 1 and 4 in the B1[ ] and B4[ ]

basis respectively, followed by qubits 2 and 3 in the B2[ ] and B3[] basis, will effect of the following

gate operation on the two encoded qubits:

Gates.jpg

Figure 2.3: Gate Operations

where the two with a line connecting them denotes that the two encoded qubits have undergonea CPhase operation. The R and W operations represent the rotation gate and the Walsh-Hadamard

transform respectively. It would be helpful to realize that the gate operations in the gure above are

read from left to right, with the CPhase gate on the left being the rst operation on the two encoded

qubits, unlike how gate operations are often read off from an equation.

The rst three gate operations (CPhase, R 12z [, ] and W W ) on the two encoded qubits completes

the rst stage of Grovers Search. As the measurements on the rst and fourth qubits are responsible

for these three gates, these two measuremetns mark the desired state in the search. The next two

15


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measurements complete the second stage, leaving the output on qubits 5 and 6. The table below

summarizes the basis in which measurements must be done to obtain the desired results.

State desired Basis measurement

|00 B14[, ] B23[, ]

|01 B14[, 0] B23[0, 0]

|10 B14[0, ] B23[0, 0]

|11 B14[0, 0] B23[, ]

Table 2.3: Basis in Which Measurements Should Be Made

where B [] = {|, |+ } and B [0] = {|+ , | }.

This amounts to the following operations for

1. GS00

vout = W W R12z [, ] CPhase W W R

12z [, ] CPhase |++

= |00 (2.19)

2. GS01

vout = W W R12z [0, 0] CPhase W W R12z [, 0] CPhase |++= |01 (2.20)

3. GS10

vout = W W R12z [0, 0] CPhase W W R12z [0, ] CPhase |++= |10 (2.21)

4. GS11

vout = W W R12z [, ] CPhase W W R12z [0, 0] CPhase |++

= |11 (2.22)16


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Recalling the role of CPhase, R z [] and W and their effects on a particular qubit, one can easily see how

these gates return us the desired state.

Running cluster state computation on Mathematica is slightly more challenging than running the

usual circuit model. We ran it on Mathematica by applying the appropriate two-qubit projector (applying

the |++ ++ | operator, for instance) before using a program (supplied by Wolfram website) to trace outthe relevant qubits. This is done twice, to simulate the measurements done on the rst 4 qubits. Then

we summed up over all the possible cases, applying the proper feedforward when necessary. That sum

yields the probabilities of the states that qubits 5 and 6 would be in the output of Grovers Search.

The program for performing the search via one-way-computing is attached in the appendix.

Obtaining Cluster State

To obtain our cluster state, we apply the CPhase gate to the appropriate pair of qubits.

boxclusterstate2.jpg

Figure 2.4: Cluster State

The letters in the picture above show the order in which the entanglements were generated, where

a denotes that it was the rst entanglement generated, b second and so on so forth. We did it in this

specic order because entangling two previously unentangled qubits require one less gate compared to

entangling two qubits which are already entangled to a third and fourth party. An entanglement between

the n th and ( n +1) th qubits which are both disentangled can be achieved by applying the following gates:-

vent = R n,n +1z [2 , 2 ] USWAP n,n +1 Rnz [] vin (2.23)17


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whilst to generate an entanglement between the n th and ( n +1) th qubit where either one or both of them

are already entangled to another qubit can be achieved by:-

vent = R n,n +1z [2

, 2

] USWAP n,n +1 Rnz [] USWAP n,n +1 vin (2.24)where the superscript of USWAP denotes which qubit the gate is acting on. Note that these equationsare almost entirely the same as the equation of I |11 because I |11 is indeed the CPhase operation. A

CPhase operation is, after all, a selective phase inversion. Thus, referring back to Eq. (2.22), we notice

that the second and third gates after the rst CPhase operation (R 12z [0, 0] and W W ) leave no net

effect on the state, for the CPhase operation is sufficient to produce the marked state.

Beginning with a disentangled six-qubit state,

vin = |+ + + + ++ (2.25)

we apply the relevant gate operations to end up with,

vout = 12 2 |

+0 + 0 + 0 + |+0 0 + 1 + |+1 + 1 0 + |+1 1 1+ |0 + 1 + 1 + |0 1 + 0 + |1 + 0 1 + |1 0 0

Recall that the main aim in this part of the project was to determine how noise would affect the

generation of this state and to determine what happens when this noisy state undergoes the relevant

idealized measurements necessary for Grovers Search. We thus would have been able to determine how

the delity of the entangled resource differs from the delity of the nal output of Grovers Search which

utilizes this state as its entangled resource.

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g

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Chapter 3

Grovers Algorithm In Noise3.1 Lindblad Equation

The noise models used in this project to characterize environmental effects are phenomenological

models, i.e. they characterize noise by the possible effects the environment could have had on the sys-

tem. We consider models where the only effect of the environment is a dephasing effect that is, the

only effect the environment leaves on our quantum system is the effect of decohering it into a classical

system. However, environmental effects could also be dissipative, where the environmental effects cause

energy to be lost from the system.

The phenomenological effects of the environment that shall be under consideration in this project

are bit ip, bit-phase ip and phase ip effects. Thus, we shall study how the system evolves under noise

that causes the following effects:-

bit ip bit-phase ip phase ip

Operator x = 0 11 0

y = 0 ii 0

z = 1 00 1

|0 |1 |0 |1 |0 |0Effects|1 |0 |1 |0 |1 |1

Table 3.1: Summary of Phenomenological Noise Models Studied

where x , y and z are the usual Pauli matrices, which shall also be denoted as 1, 2 and 3 for ease

of computation.

Other noise models that could be of interest are amplitude damping and depolarizing models, the

former being a dissipative noise on top of being dephasing. This whole study can be easily repeated for


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GROVERS ALGORITHM IN NOISE Lindblad Equation

these noise models with appropriate but simple tweaks made to the current Mathematica les.

To determine the evolution of a particular state in a noisy environment, we solved the Lindblad

form of the Markovian master equation. A Markovian master equation posits a Markovian environment,

where it assumes that the environment has no memory of the past, or that self-correlations within the

environment decays rapidly as compared to the time taken for the system to vary noticeably [1, 10]. The

Lindblad equation takes the following form:

ddt

= i [H, ]

a2

[L, [L, ]] (3.1)

where L are Lindblad operators; a, the decay rate which captures the strength of the coupling between

the system and the environment; and H is the Hamiltonian of the interaction under consideration. Thetilde on species that this is a reduced density matrix.

At this point, we are required to specify a physical system upon which these algorithms shall take

place. Having already determined the unitary evolution necessary to both execute Grovers Search and

generate the cluster state, we need the specic Hamiltonian operators that will generate these unitary

evolutions. Different physical systems utilize different interactions to achieve the same unitary opera-

tions. In our project, we deal specically with the quantum dot system, which exploits electron spin as

their fundamental quantum unit [11].

From the discussions in section 2, the only gate operations necessary in this project are one-qubit

local rotation gates and two-qubit USWAP gates. In fact, generally, these two gates are sufficient toform a universal set [12, 13] of transformations required to perform any necessary computation. In

quantum dot systems, rotation gates rely on an external magnetic eld to produce its effect whilst the

USWAP gate relies on the Heisenberg exchange interaction.

HR i = 1

2 i (3.2)

H USWAP n,n +1 =3

i=1

14

2

Jn

i n +1i (3.3)

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Noise Model GROVERS ALGORITHM IN NOISE

Here and J capture the strength of interaction, and thus, the speed of the operation. To obtain the

unitary evolution which are explicitly dependent on these parameters, one can do a spectral decomposi-

tion of the relevant H in terms of its normalized eigenvectors and eigenvalues before exponentiating it.The generic equation relating a unitary operator to a Hamiltonian is,

U (t) = e i Ht (3.4)

whilst the unitary operator for a rotation and USWAP gate (in a solid state quantum dot system) takesthe form of the equations below

U R i (t) = e i2 t i

= e i2 i (3.5)

U USWAP (t) = e i4

J t 3i =1 ni

n +1i

= e i4 3i =1

ni

n +1i (3.6)

where = t and = Jt . shall determine the extent or angle of rotation, whilst determines the

extent to which the SWAP operation is executed. A of returns us the complete SWAP operation

whilst a of 2 returns us the USWAP operation we desire. Thus, throughout this project, shall beset to 2 .

3.2 Noise Model

Our noise model differs from any as far as we know models presented before in this respect: the

input density matrix into the Lindblad equation is not the original input density matrix, v in = |++ forinstance, or any of the other input states listed in Table 2.2. Instead, it is the basis states that we solve

for in the Lindblad equation |0 0|, |0 1|, |1 0| and |1 1| for a one-qubit system. Thus, to obtain howvin evolves under a Hamiltonian HR z in the presence of a bit-ip noise L = x , we solve not one Lindbladequation with = |++ ++ |, but 16 Lindblad equations where the 16 are the 16 basis states. Whilstthere are now more equations to solve, the advantages of this formulation is two-fold:-

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GROVERS ALGORITHM IN NOISE Noise Model

1. The input density matrices of the partial differential equations are simple. In a noisy search,

the output of the rst noisy gate in itself is a relatively complicated density matrix. We would

then have to input this complicated density matrix into the Lindblad equation again, doing this

repeatedly with the input state becoming more and more complicated until the search is nished

or the cluster state is achieved. This formulation however, solves the partial differential equations

for very simple basis states.

2. Determining how the basis states of a two-qubit system evolves with a particular noise under

a particular Hamiltonian effectively solves for all such evolutions, regardless of the form of the

input states. This is true as the basis states span a complete basis. Any two-qubit state can be

represented in terms of the 16 basis states. Thus, we have potentially solved the z-rotation under

bit-ip noise for virtually any possible superposition of two-qubit states possible by just solving

the 16 relatively simple partial differential equations stated above.

From the Lindblad equation, we then have the following map of a particular basis state for one and

two qubits:

|i j | ij (3.7)

|ij kl| = |i k| | j l| ijkl (3.8)

where |i j | and |ij kl| are the original basis states and is the reduced density matrix that the originalbasis state is mapped to. It is now a reduced density matrix due to environmental interaction for we

have no access to the environment. Take note of the subscripts of and how they relate to the original

basis state, for this could be a source of confusion. In the following sections, shall refer to the solutions

of the Lindblad equations.

Upon solving the Lindblad equation for all the basis states and obtaining the relevant , we would

need a way to compile these solutions, allowing us to simulate the effect of a noisy gate operation acting

on any input state. This is done in accordance to the following equations for the two-qubit (circuit

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Solutions to the Lindblad Equation GROVERS ALGORITHM IN NOISE

model) and six-qubit (one-way computing model) gates respectively,

out =1

i,j,k,l =0

Tr in

|kl ij

|ijkl (3.9)

out =1

i,j,k,l,m,n,p,q,r,s,u,v =0

Tr in | pqrsuv ijklmn | ijklmnpqrsuv (3.10)

Basically, we multiply the coefficient of a particular basis |ij kl| of in (as obtained from the trace opera-tion) with the corresponding noisy basis evolution ijkl . By summing over all the possible basis, we obtain

the effect of a particular Hunder the inuence of a particular noise model of a particular input state, in .

With this simplication, we are able to obtain the explicit expression of how each density matrix

involved in the search evolves with time. Thus, we can trace the evolution of the density matrix through

time, studying its dynamics.

3.3 Solutions to the Lindblad Equation

In this section, we shall list the solutions of the Lindblad equation for the different noise models.

We ask that the readers keep in mind that the Lindblad equation returns us a map of how a particular

state evolves, as can be seen in Eq. (3.8). In our list, we use equality signs because we are referring

only to the right side of the map. The notations used for the right side is sufficient to identify the

original state on the left-hand side of the map, and thus, we leave that out of our equations. For the

one-qubit maps, each reduced density matrix is written in the complete basis spanned by I , x , y and

z . For the two-qubit solutions, each reduced density matrix is written in terms of its matrix elements, S ij

As superoperators preserve hermiticity, an off-diagonal matrix element will obey the following re-

lation: S ij = S ji . Recalling that we are dealing with time-evolution of basis states , which essentially

are components of a matrix, each basis evolution obeys that relation that is it obeys ij = ji and

ijkl = klij . Thus, for the one-qubit evolutions, we list down 3 rather than 4 solutions and for two-qubit

evolutions, we list down 10 rather than 16 solutions. For all the evolutions under the inuence of HR ,24


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GROVERS ALGORITHM IN NOISE Solutions to the Lindblad Equation

2 a2, whilst for those under the inuence of H USWAP , J 2 4a2.

3.3.1 Bit Flip Noise ( x )

Under HR x inuence

00 = 12

0 e2at (sin[t]2 cos[t]3) (3.11)01 =

12

1 + ie2at (cos[t]2 + sin[t]3) (3.12)

11 = 12 0 + e

2at

(sin[t]2 cos[t]3) (3.13)

Under HR z inuence

00 = 12

0 + e2at 3 (3.14)

01 = eat

2cos[t](1 + i2) +

sin[t]

(a i)1 (a + i)i2 (3.15)

11 =

1

2 0 e2at

3 (3.16)

Under H USWAP inuence, for the expressions with or signs, the top signs apply for the S ij aslisted before the semicolon. For the S ij after the semicolon, the bottom sign applies.

0000 =

S ij = e2at cosh2[at ] if S ij = S 11

S ij = 1 e4at

4 if S ij = S 22 , S 33

S ij = e2at sinh2[at ] if S ij = S 44

S ij = 0 otherwise

(3.17)

1111 =

S ij = e2at sinh2[at ] if S ij = S 11

S ij = 1 e4at

4 if S ij = S 22 , S 33


S ij = 0 otherwise

(3.18)

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0101 =

S ij = 1 e4at

4 if S ij = S 11 , S 44

S ij = 1 + e4at

4 cos[Jt ]

2e2at if S ij = S 22; S 33

S ij = i sin[Jt ]

2e2at if S ij = S 23; S 32

S ij = 0 otherwise

(3.19)

1010 =

S ij = 1 e4at

4 if S ij = S 11 , S 44

S ij = 1 + e4at

4 cos[Jt ]

2e2at if S ij = S 22; S 33

S ij =i sin[Jt ]

2e2at if S ij = S 23; S 32

S ij = 0 otherwise

(3.20)

0001 =

S ij = e2at

42 cosh2[at ](eiJ t + cos[t]

iJ

sin[t]) if S ij = S 12 , S 13

S ij = 18

1 e4at 4ae2at

sin[t] if S ij = S 21 , S 34 ; S 24 , S 31

S ij = e2at

42 sinh2[at ] (eiJt cos[t] +

iJ

sin[t]) if S ij = S 42 , S 43

S ij = 0 otherwise

(3.21)

0010 =

S ij = e2at

42 cosh2[at ] (eiJ t + cos[t]

iJ

sin[t]) if S ij = S 12 ; S 13

S ij = 18

1 e4at 4ae2at

sin[t] if S ij = S 21 , S 34 ; S 24 , S 31

S ij = e2at

42 sinh2[at ] (eiJt cos[t] +

iJ

sin[t]) if S ij = S 42 ; S 43

S ij = 0 otherwise

(3.22)

0011 =


S ij = 1 e4at4 if S ij = S 23 , S 32S ij = e2at sinh2[at ] if S ij = S 41

S ij = 0 otherwise

(3.23)

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0110 =

S ij = 1 e4at

4 if S ij = S 14 , S 41

S ij =

i sin[Jt ]

2e2at if S ij = S 22; S 33

S ij = 1 + e4at

4 cos[Jt ]

2e2at if S ij = S 23; S 32

S ij = 0 otherwise

(3.24)

0111 =

S ij = 18

1 e4at 4ae2at

sin[t] if S ij = S 12 , S 43; S 13 , S 42

S ij = e2at

42 sinh2[at ] (eiJt cos[t]

iJ

sin[t]) if S ij = S 21; S 31

S ij = e2at

42 cosh2[at ]

(eiJ t + cos[t] +

iJ

sin[t]) if S ij = S 24; S 34

S ij = 0 otherwise

(3.25)

1011 =

S ij = 18

1 e4at 4ae2at

sin[t] if S ij = S 12 , S 43; S 13 , S 42

S ij = e2at

42 sinh2[at ] (eiJt cos[t]

iJ

sin[t]) if S ij = S 21; S 31

S ij = e2at

42 cosh2[at ] (eiJ t + cos[t] +

iJ

sin[t]) if S ij = S 24; S 34

S ij = 0 otherwise

(3.26)

3.3.2 Bit-Phase Flip Noise ( y )

Under HR x inuence

00 = 12

0 + eat

cos[t]3 sin[t](2 + a 3) (3.27)

01 = 12

e2at 1 + ieat

cos[t]2 + sin[t](a 2 + 3) (3.28)

11 = 12

0 eat

cos[t]3 sin[t](2 + a 3) (3.29)

Under HR z inuence

00 = 12

0 + e2at 3 (3.30)

01 = eat

2cos[t](1 + i2)

sin[t]

(a + i)1 + ( a + i)i2 (3.31)

11 = 12 0 e

2at

3 (3.32)

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Under H USWAP inuence

0000 =


S ij = 1 e4at

4 if S ij = S 22 , S 33

S ij = e2at sinh 2[at ] if S ij = S 44

S ij = 0 otherwise

(3.33)

1111 =


S ij = 1 e4at4 if S ij = S 22 , S 33S ij = e2at cosh2[at ] if S ij = S 44

S ij = 0 otherwise

(3.34)

0101 =

S ij = 1 e4at

4 if S ij = S 11 , S 44

S ij = 1 + e4at

4 cos[Jt ]

2e2at if S ij = S 22; S 33

S ij = i sin[Jt ]

2e2at if S ij = S 23; S 32

S ij = 0 otherwise

(3.35)

1010 =

S ij = 1 e4at

4 if S ij = S 11 ; S 44

S ij = 1 + e4at

4 cos[Jt ]

2e2at if S ij = S 22; S 33

S ij =i sin[Jt ]

2e2at if S ij = S 23; S 32

S ij = 0 otherwise

(3.36)

0001 =

S ij = e2at

42cosh2[at ] (eiJt + cos[t]

iJ

sin[t]) if S ij = S 12; S 13

S ij = 18

(1 e4at ) 4ae2at

sin[t] if S ij = S 21; S 24

S ij = 18

(1 e4at ) + 4ae2at


S ij = e2at

4 2sinh 2[at ] (eiJ t cos[t] +

iJ

sin[t]) if S ij = S 42, S 43

S ij = 0 otherwise

(3.37)

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0010 =

S ij = e2at

42cosh2[at ] (eiJt + cos[t]

iJ

sin[t]) if S ij = S 12; S 13

S ij = 1

8(1

e4at ) + 4ae

2at


S ij = 18

1 e4at 4ae2at


S ij = e2at

4 2sinh 2[at ] (eiJ t cos[t] +

iJ

sin[t]) if S ij = S 42; S 43

S ij = 0 otherwise

(3.38)

0011 =


S ij

= e4at 1

4 if S

ij = S

23, S

32


S ij = 0 otherwise

(3.39)

0110 =

S ij = e4at 1

4 if S ij = S 14 , S 41

S ij = i sin[Jt ]

2e2at if S ij = S 22; S 33

S ij = 1 + e4at

4

cos[Jt ]

2e2at if S ij = S 23; S 32

S ij = 0 otherwise

(3.40)

0111 =

S ij = 18 (1 e4

at ) 4ae2at


S ij = 18 (1 e4

at ) + 4ae2at


S ij = e2at

4 2sinh 2[at ] (eiJ t cos[t]

iJ

sin[t]) if S ij = S 21; S 31

S ij = e2at

42cosh2[at ]

(eiJ t + cos[t] +

iJ

sin[t]) if S ij = S 24; S 34

S ij = 0 otherwise

(3.41)

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1011 =

S ij = 18 (1 e4

at ) + 4ae2at


S ij = 1

8 (1

e4at )

4ae2at


S ij = e2at

4 2sinh 2[at ] (eiJ t cos[t]

iJ

sin[t]) if S ij = S 21; S 31

S ij = e2at

42cosh2[at ] (eiJ t + cos[t] +

iJ

sin[t]) if S ij = S 24; S 34

S ij = 0 otherwise

(3.42)

3.3.3 Phase Flip Noise ( z )

Under HR x inuence

00 = 12

0

eat sin[t]2 + eat cos[t] + a

sin[t] 3 (3.43)

01 = 12

e2at 1 + ieat cos[t] a

sin[t] 2 + ieat

sin[t]3 (3.44)

11 = 12

0 +

eat sin[t]2 eat cos[t] + a

sin[t] 3 (3.45)

Under HR z inuence

00 = 120 + 3 (3.46)

01 = 12

e2at it 1 + i2 (3.47)

11 = 12

0 3 (3.48)

Under H USWAP inuence

0000 = |00 00| (3.49)1111 = |11 11| (3.50)

0101 =

S ij = 12

+ cos[t] + 2a sin[t]

2e2at if S ij = S 22

S ij = iJ sin[t]


S ij = iJ sin[t]2e2at if S ij = S 32S ij =

12

cos[t] + 2a sin[t]2e2at

if S ij = S 33

S ij = 0 otherwise

(3.51)

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1010 =

S ij = 12

cos[t] + 2a sin[t]2e2at

if S ij = S 22

S ij = iJ sin[t]2e

2at if S ij = S 23

S ij = iJ sin[t]


S ij = 12

+ cos[t] + 2a sin[t]


S ij = 0 otherwise

(3.52)

0001 =

S ij = 1 + eiJt


S ij = 1 eiJt


S ij = 0 otherwise

(3.53)

0010 =

S ij = 1 eiJt


S ij = 1 + eiJt


S ij = 0 otherwise

(3.54)

0011 =S ij = e4at if S ij = S 14

S ij = 0 otherwise(3.55)

0110 =

S ij = iJ sin[t]


S ij = 12e4at

+ cos[t]2a sin[t]


S ij = 12e4at

cos[t]2a sin[t]2e2at

if S ij = S 32

S ij = iJ sin[t]2e2at if S ij = S 33

S ij = 0 otherwise

(3.56)

0111 =

S ij = 1 + eiJt


S ij = 1 eiJt


S ij = 0 otherwise

(3.57)

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Extending To A Larger Number of Qubits GROVERS ALGORITHM IN NOISE

1011 =

S ij = 1 eiJt


S ij = 1 + eiJt


S ij = 0 otherwise

(3.58)

3.4 Extending To A Larger Number of Qubits

As the circuit model requires two-qubit rotation gates, and the cluster state model requires gates

that act on six qubits, we will need to extend the formulation such that it would be able to act on a

larger number of qubits. Extending the one-qubit noisy basis evolution under the inuence of HR intotwo-qubit basis evolutions is simple. Each basis evolution shall take the following form for,

1. Gates that rotate only the rst qubit

ijkl = ik | j l| (3.59)

2. Gates that rotate only the second qubit

ijkl = |i k| jl (3.60)

3. Gates that rotate both qubits

ijkl = ik jl (3.61)

Extending this to make a six-qubit rotation gate utilizes a similar concept. Noisy basis evolution under

H USWAP inuence can be extended to six-qubit basis evolutions as such:-

1. For a USWAP 12 effectijklmnpqrsuv = ijpq | klmn rsuv | (3.62)

2. For a USWAP 23 effectijklmnpqrsuv = |i p| jkqr | lmn suv | (3.63)

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GROVERS ALGORITHM IN NOISE Extending To A Larger Number of Qubits

3. For a USWAP 34 effectijklmnpqrsuv =

|ij pq

| klrs

|mn uv

| (3.64)

4. For a USWAP 41 effect, swap qubits 2 and 4, apply USWAP 12 , and swap it back. This is a purelymathematical constraint, necessary only when doing mathematical computation. Physically and

experimentally, there ought to be no complications when it comes to applying the USWAP gate toqubits 1 and 4, for they are or can indeed be made to be, side by side in order for the appropriate

H to be applied to both. Mathematically though, we cannot incorporate this as simplistically,because the generation of the noisy evolutions require a two-qubit H that acts on n and (n + 1),rather than n and (n + m) as can be seen from Eq. (3.3).

5. For a USWAP 25 effect, swap qubits 3 and 5 apply USWAP 23 , before swapping it back.6. For a USWAP 36 effect, swap qubits 4 and 6 apply USWAP 34 , before swapping it back.Each SWAP operation can be achieved by

SWAP 14 =1

i,j,k,l,m,n,p,q,r,s,u,v =0|ljkimn ijklmn | (3.65)

SWAP 25 =1

i,j,k,l,m,n,p,q,r,s,u,v =0|imkljn ijklmn | (3.66)

SWAP 36 =1

i,j,k,l,m,n,p,q,r,s,u,v =0|ijnlmk ijklmn | (3.67)

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Chapter 4

Analysis

4.1 Introduction

In our analysis, we shall focus on two particular quantities: delity and negativity. Fidelity is a

means of determining how close the output state mirrors the desired state. As the trace of a number is

the number itself, by the linearity of trace, we can reexpress delity as

P = |out |= Tr[ |out | ]= Tr[ out | |] (4.1)

where we have used P to denote delity. out is the outcome of our computation while | is ideal statethat the search ought to return. Please note that unless stated, delity and P shall always refer

to the delity of the output of the search, vout .

Negativity on the other hand, provides us with a means of studying entanglement for two-qubits

by distilling out entanglement. Using only local operations and classical communication (LOCC), we

transform our N copies of an arbitrarily entangled state to some number of purely entangled Bell state.

To obtain negativity we rst obtain the partial transpose of our density matrix, which is also a criterion

called partial positive tranpose, PPT to check for separability of density matrices [14, 15]. We obtain

the partial transpose by rst noting that a density matrix living in the Hilbert space H a H b can be

expressed as

=ijkl

pijkl |ij kl|=

ijkl

pijkl|i k

| | j l

| (4.2)


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ANALYSIS Introduction

where pijkl are the matrix elements of the |ij kl| basis. A partial transpose then consists of us transposingonly a part of the state, doing an identity map on system A and a transpose map on system B, for instance.

T B =ijkl

pijkl |i k| | l j | (4.3)

where T B is the partial transpose of system B. If the system is separable, such a partial transpose ought

to still return us a density matrix, and hence, positive eigenvalues. That is the basis of the PPT test, and

hence, a system whose PPT has negative eigenvalues is guaranteed to be entangled. Upon obtaining the

partial transpose, negativity consists in taking the absolute value of the sum of its negative eigenvalues,

N =i

Re[ei ] Re[ei ] (4.4)

When a system is maximally entangled, N = 1.

As our nal aim is to provide practical advice on ways to improve Grovers Search, we would rstly

need a list of the parameters that are within our control. The parameters within our control are:.

1. The ratio of the speed of the gates to the decay rate, a : J and a : .

2. The fraction or amount of gate completed: for the rotation gates and for the USWAP gates(refer Eq. (3.5) and Eq. (3.6)). Recall that controls the angle by which a state is rotated by,

whilst controls the extent to which the SWAP operation is applied onto the state.

3. The computational basis in which the search is carried out. The default basis this study is con-

ducted in is the {|0 , |1 } basis eigenbasis of the Pauli z matrix.

As a is the decay rate, the higher the a the larger the effect of the environment on the system. Recall

that = t and = J t , and thus can be rearranged to t = / and t = /J . From Eq. (3.13) through

to Eq. (3.58), we notice that the noisy basis maps depend on the the parameter at . Substituting t , and

keeping and constant, we nd that they depend on the ratio of a : and a : J such that the smaller

this ratio, the better the delity. The lower the decay rate, the quicker the gate, the better the results

of the search.

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Introduction ANALYSIS

In the rst parameter, the total and by which the unitary operation takes the state through

is kept at a constant. The second parameter then relaxes this constraint. Given the speed of a gate

(determined by setting the rst parameter), we can vary the time parameter to obtain the and such

that the search yields maximum delity possible. The and that yields maximum delity shall be

labelled max and max , whilst the full swing ones shall be full and full . In our discussions relating to

this issue, we shall speak in terms of the fraction of max / full and max / full . Theoretically, it is possible

for this ratio to exceed one. This may seem counter-intuitive for it is natural to assume the smaller the

angle of rotation, or the lesser the system is exposed to the SWAP operation, the less severe the effect

of the environment. And indeed this is most often than not, true. However, it is theoretically possible,

for the system becomes complicated when noise is present and the simple intuitive relationships may no

longer necessarily hold.

In this project, we will be studying and comparing the effects of noise for three different noise models.

In reality, we have no control over the environment, and hence the type of noise the environment imposes

on our system. However, what is really relevant to the performance of our search is the effect of the

environment relative to the basis chosen for computation. And this is within our control. Starting off

the search by encoding information in the {|+ , | } or the {|y , |y } basis changes the effect thatthe environment leaves on our search. This is better visualized through an example: assume that the

environment is characterized by a bit-ip noise. Since this project tells us how all three models of noise

affect a search whose computational basis is in {|0 , |1 }, we could identify the model that has the leasteffect on our search (phase ip, say) and choose our computational basis such that the bit ip noisy

environment that we are in has a phase ip effect on the search. Thus, we choose {|+ , | } as ourcomputational basis.

For the purposes of this project, we remind the reader that a delity of 0.5 shall be the minimum

requirement of a successful search. We shall present our analysis in the following manner: Firstly, we

discuss the effect of noise on just one gate. Thus, in that section, all other gates are idealized and

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ANALYSIS One Gate Noisy

assumed to run to full swing, i.e. = full and = full . As having only one gate noisy is an implausible

physical situation, this section is meant merely to deepen our understanding as to how the three noise

models may affect the two types of gates (rotation and USWAP ) as they act on the different states.It aims to deepen our appreciation and understanding of why the steps proposed in later sections do

indeed improve the search. Secondly, we study the more realistic situation where all the interactions are

infused with noise, but where all the gates are still running to full swing. Only in the last section do we

relax the constraint on and in attempts to maximize the delity of the search algorithm.

4.2 One Gate Noisy

In this section we shall rstly present a table that summarizes the effects of noise on individual

gates, before looking into some very interesting features of Grovers Search in noise. We then ask the

following question: what happens if entanglement fails to be generated in the course of the search?

As entanglement is an apparent benet that quantum mechanics has over classical mechanics, it would

be intriguing indeed to learn what happens to the quantum algorithm when no entanglement is generated.

Several clarications are in order before one can possibly make sense of the table presented in the

following page. The blue graphs plot for delity and negativity under x noise, the red, y and the

purple under z noise. Recall that indexing anything with ij species the i th stage of the search and the

j th operation of that stage (refer to section 2.3.1). The delity graphs in the table plots P (recall that

P is always the delity of the nal output state of the search) against tij , where i and j are indices of

the noisy gate. The negativity graphs on the other hand, plots for N ij against t ij .

The last three columns with the (s) and (s) is merely a summary the denotes that the delityof the search under that particular noise model is higher than the one with the . This evaluation isdone only at the point when the gate is completed, i.e. at full and full . The input states listed in this

table (second column) are for the GS11 search and are included merely for reference. Do refer to Table

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One Gate Noisy ANALYSIS

2.2 for the input states of the other searches. The delity and negativity plots look exactly the same for

all 4 searches, with the exception that for |01 and |10 , the last rotation gate is unnecessary. For thesegraphs, we have set a : = a : J = 1 : 10. In most plots, we see only two distinct lines rather than

three because two of the lines overlap. The and might assist in determining which are the linesthat overlap.

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The graphs plotted in the table, and for the rest of this section, has time tij as the x-axis as it

should for we are exploring the time-evolution and dynamics of the search. However, the maximum

value of t ij , as well as the meaning of the explicit values of t ij varies according to the different operations

applied (for and varies depending on the operation under scrutiny) and the speed of the operations,

as tij = / or /J . The y-axis represents either delity P , or negativity N as specied.

Our rst observation is that there is a direct correlation between the generation of entanglement and

of P . The following graphs plot for delity and negativity of the rst USWAP gate, G12 .

4Fid.pdf

(a) Fidelity

4Neg.pdf

(b) Negativity

Figure 4.1: Time evolution of P and N 12 against t12 a : J = 1 : 10

Comparing the delity plot (a) and negativity plot (b) of x (blue), or those of y and z (red and

purple), we observe that at the point where negativity is at its highest, delity reaches its maximum.

Included are graphs where delity and negativity are plotted on the same axis, with a certain constant

value added to the negativity plot to allow the two plots maximum value to coincide.

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4a.pdf

(a) L = x

4b.pdf

(b) L = y and z

Figure 4.2: Comparing delity and negativity

These graphs allows us to clearly see that the maximum point for delity and negativity occurs at the

same time 1.

This correlation with delity however, holds only for the generation of entanglement and not for

entanglement in general. Even when all other gates are idealized, this correlation does not hold for any

of the other three gates involving entangled states not even for the USWAP gate that disentangles thetwo qubits. Whilst it is simplistically true that the more entanglement generated, the better the delity

of the algorithm (other gates idealized); once we have generated a nite amount of entanglement, it is not

necessarily as clear anymore that further loss of entanglement would have as direct a correlation with -

delity. This is best seen in the G13 and G21 gates rotation gates whose application should not affect the

negativity of the states evolving through them. In the presence of noise, the negativity of any entangled

state evolving through a rotation gate will necessarily drop. Yet, it would be ridiculous to suggest that it

would then be better to skip over these gates to obtain a better delity. To understand why this is true,

consider a 1 2 (|01 + |10 ) state bound to enter a noisy R 2x []. The output of this operation would be1 2 (|00 + |11 ). If this gate were to be skipped over, we would nd that the delity of the state with re-

spect to 1 2 (|00 + |11 ) would be 0. If it undergoes noisy evolution, even if entanglement is lost, delityof the nal output state with respect to 1 2 (|00 + |11 ) would at least have a chance to be higher than 0.

1 No label was utilized to distinguish the negativity plot from the delity plot in the graphs below because the distinction

is of no consequence. This graph is but a tool to show that the maximum point of both coincide.

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Next, we direct out attention to the two USWAP gates. Notice that if the input state is in theeigenbasis of the the Pauli matrix that characterizes the noise in the environment, the effect of the

environment is signicantly lesser.

Gate Input Fidelity Graph Negativity Graph L = x L = y L = z

USWAP |

+

rootSWAP1fid.pdf rootSWAP1neg.pdf

USWAP |01 i |10rootSWAP2fid.pdf rootSWAP2neg.pdf

From the rst row, when the input state is an eigenstate of x , the x Lindblad operator (blue

plot) has less of a detrimental effect on the delity and negativity of the search, as compared to the

y (red plot) and z (purple plot) Lindblad operators. For the second column, where the input state

is a maximally entangled Bell state as expressed in the eigenstates of z , the z Lindblad operator has

less of a harmful effect on it. (Do refer to Table 2.2 to observe the forms the input states that the

other searches take. This connection holds true for all the four searches). Thus, if the input state is the

eigenbasis of the Lindblad operator, that state, as it evolves through a USWAP gate, will be more robust.

Furthermore, as v 12 evolves through G12 in an environment characterized by the x Lindblad oper-

ator, entanglement will always be generated regardless of the ratio a : J . For v22 , if the environment

is characterized by a z noise, the entanglement, when destroyed will always be immediately generated

again. The graphs below show this 2:-

2 The reason for this particular choice of a : J shall be apparent later

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3aNeg.pdf

(a) N 12 against t 12

3bNeg.pdf

(b) N 22 against t 22

Figure 4.3: Time evolution of

N ij against t ij a : J = 51 : 100

This places a natural constraint on the question raised what happens if G12 fails to generate entan-

glement? In our attempts to answer this question, we are limited to a discussion of only the bit-phase

ip and phase ip noise models. The condition that must be fullled to prevent entanglement from being

generated under these two noise models is a : J > 1 : 2. When a = 2J , J 2 4a2 as introduced inthe noisy maps of Section 3.3 reduces to 0.

Analysing the behaviour of delity when this ratio exceeds 12 , we shall nd that with no entanglement

generated, it is impossible for the delity of the search to reach 0.5. Recall that in this section, the rest

of the gates are idealized. Hence, even in idealized situations, the search will fail when no entanglement

is generated. Fidelity can, however, exceed 0.25, and thus, this search still returns us results that exceed

what we would have been able to obtain via classical means. The P and N 12 curve against t12 underL = z inuence where a : J 51 : 100 is plotted below (one will obtain the exact same curve withL = y ).

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1aFid.pdf

(a) Fidelity

1aNeg.pdf

(b) Negativity

Figure 4.4: Time evolution of P and

N 12 against t

12 a : J = 51 : 100

Comparing it with the delity plot of L = x inuence, under the same a : J ratio, we notice that the

state under the bit ip noise is much more robust than those under phase ip noise.

3aFid.pdf

Figure 4.5: Time evolution of P against t12 a : J = 51 : 100

Even more interesting is the following graphs:

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3cFid.pdf

(a) Fidelity

3cNeg.pdf

(b) Negativity

Figure 4.6: Time evolution of P and

N 12 against t

12 a : J = 100 : 1

These graphs show that when there is even a tiny amount of entanglement generated, given that the

rest of the gates are idealized, it is still possible to obtain a successful Grovers Search. This is a rather

amazing nd, considering the large ratio of a : J . This shows how fundamental a role entanglement

plays in quantum computing.

Thus, from this section, we learn the following:-

1. Entanglement is central to Grovers Search for the following reasons:

Firstly, there is a direct connection between the amount of entanglement one is able to generate and

the delity of the search such that if the USWAP operation is stopped at the point where maximumentanglement is generated, the search would have yielded an output with the highest delity

attainable under the conditions it was subjected to. Secondly, if entanglement is not generated,Grovers Search can never succeed even when it is idealized in every other consideration. If, on

the other hand, entanglement is generated ever so slightly, in a situation idealized in every other

manner, the search will succeed.

2. A state evolving through a USWAP gate will be relatively robust if it happens to be an eigenket of the Lindblad operator characterizing the environment. Entanglement will denitely be generated

in such cases.

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All Gates Noisy ANALYSIS

3. In the absence of entanglement, a quantum algorithm could still have the potential of surpassing

a classical algorithm.

4.3 All Gates Noisy

In this section, we consider what happens to the search when all the gates involved are noisy. We

rst consider cases where the speed of both the rotation gates and the USWAP gates are the same, i.e.it takes the same amount of time to do a 2 rotation as it does to do a USWAP gate. Then we vary therelative speed of the two operations. However, the and throughout this subsection still go to full

and full .

4.3.1 Same Speed Gates

When the speed of the rotation gates and the USWAP gates are the same, we observe the following:-

1. Fidelity of the marked state is least affected by an environment characterized by L = x

2. For GS01 and GS10, conducting a search in an environment characterized by a bit ip noise has a

higher probability of success than in the other two noisy environments. In GS00 and GS11 however,

no such preferences can be discerned.

3. Upon reevaluating the correlation between entanglement generated and delity of the output of

the search (now that all the gates are noisy), we nd that for two out of the three noise models

researched, the correlation still holds a happy conclusion for quantum computing indeed.

The following table tabulates the delity results for two particular initial conditions, a : = a : J =

1 : 15 and a : = a : J = 1 : 20. These two conditions were chosen because they provide us with delity

values that are close to a successful search.

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ANALYSIS All Gates Noisy

a : = 1 : 20 a : = 1 : 15

L = x L = y L = z L = x L = y L = z

P M 0.748 0.697 0.663 0.686 0.628 0.594

P 0.517 0.516 0.517 0.440 0.438 0.440

Table 4.2: Fidelity of GS00 and GS11

a : = 1 : 20 a : = 1 : 15

L = x L = y L = z L = x L = y L = z

P M 0.748 0.697 0.663 0.686 0.628 0.594

P 0.578 0.546 0.548 0.500 0.468 0.471

Table 4.3: Fidelity of GS01 and GS10

As a : = a : J , only one of those conditions was listed in the table. In the tables presented above,

P M refers to the delity of the marked state, whilst P , as always, refers to the delity of the nal

output. The values for GS00 and GS11 are the same because the structure of both these searches are

similar. The same goes for GS01 and GS10. The rst two observations mentioned can be easily drawn

just by referring to the tables above. Understanding or explaining them however, requires us to refer

back to a simplied version of Table 4.1.

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Gate Input L = x L = y L = z

R1z [] |++ USWAP |+

R12z [2 , 2 ] |+ + i |+R12x [2 , 2 ] |y y + i |y y USWAP |01 i |10

R2x [] |10

Table 4.4: Effect of Noise on Individual Gates

Recall that the and refer to the relative performances of the search under the different Lindbladoperators assuming rstly that all other gates are ideal, and secondly, that all the gates are applied to

full and full . From Table 4.4, it is clear that the effect of the x Lindblad operator (two s) on the

gates prior to obtaining the marked state is less detrimental than that of y (one ) which is then less

detrimental than the z Lindblad operator. This explains the trend of P M our rst observation above.

The second can be explained as such: As GS01 and GS10 involve only the rst 5 gates, it is clear

that the effect of a x Lindblad operator on the search is less than that of y or z as the former has two

s and one while the latter two have one and two s each. Over the 6 gates in GS00 and GS11,there are two s and two s for all three noise models. Thus, it is easy to conclude that we should indeed expect that the different noise models to return us roughly the same delity. Note that both

these explanations presuppose that the difference between a and a in a USWAP gate is exactlythe same as the difference between a and in a rotation gate. Thus, in comparing scenario a) wherewe have a on the USWAP and a on a rotation gate and scenario b) where we have a on the USWAP gate a on the rotation gate, we ought to nd that it yields the same effect. Referring backto the third column of Table 4.1, it can be seen that this is indeed roughly true. (Do note that the range

of the axes are different for different graphs).

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ANALYSIS All Gates Noisy

To prove the third observation, we plot the negativity of v 12 ( N 12), and the nal delity, P , againstt12 where the conditions have been set thus a : = a : J = 1 : 20. Recall that v 12 is the state upon

which the rst USWAP gate G12 is acting upon. In the following plots, a certain constant has beenadded to the value of delity. This allows us to compare the delity and negativity curves with ease 3.

The blue curve plots for a bit ip noise, red for bit-phase ip and purple for phase ip.

6a.pdf

(a) L = x

6b.pdf

(b) L = y

6c.pdf

(c) L = z

Figure 4.7: Time Evolution of P and N 12 Against t12 All Gates Noisy

Thus, we note that for the bit ip and bit-phase ip noise, assuming that all gates run to full swing,

there is a direct correlation between the entanglement generated and delity. These graphs were ob-

tained rather late into the project, and thus, we lacked the time needed to explore why the case is3 Again, the only important point is to determine if the maximum point of both curves coincide. Hence, it matters not

which curve is the delity curve and which, the negativity.

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different for a phase ip noise.

Taking the rst and second observation together, we seem to reach a rather disturbing conclusion:

that there is actually no simplistic correlation between the delity of the marked state and the delity

of the nal output, at least for GS00 and GS11. Comparing across the three noise models, it seems clear

that a higher P M does not necessarily entail a higher P . Would a search really be a search if there is no

correlation between how well I can mark my state and how well I can nd my state? Fortunately, this

is not necessarily the conclusion to be drawn from this observation. All that this study shows is that

there is no simplistic correlation between the delity of the marked state and the delity of the nal

output for the searches done across different environments. It is still possible then likely even that

for a particular search in one specic environment, characterized by one Lindblad op

Grover's search algorithm

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