quantum walks in quantum games and quantum graphs

QUANTUM WALKS IN QUANTUM GAMES AND QUANTUM GRAPHS

A thesis Submitted in Partial Fulfilment of the Requirements for the Degree of

MASTER OF SCIENCE

by

JISHNU RAJENDRAN

to the

School of Physical Sciences National Institute of Science Education and Research Bhubaneswar June 28, 2017

Dedicated to my inspiring parents, brother and friends, for being the pillows, role models, cheer-leading squad and sounding boards I have needed.

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DECLARATION I hereby declare that I am the sole author of this thesis in partial fulfillment of the requirements for a postgraduate degree from National Institute of Science Education and Research (NISER). I authorize NISER to lend this thesis to other institutions or individuals for the purpose of scholarly research.

Signature of the Student Date:

The thesis work reported in the thesis entitled Quantum walks in quantum games and quantum graphs was carried out under my supervision, in the School of Physical Sciences at NISER, Bhubaneswar, India.

Signature of the thesis supervisor School: Date:

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ACKNOWLEDGEMENTS Foremost, I would like to express my sincere gratitude to my advisor Dr. Colin Benjamin for the continuous support of my project and research, for his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my MSc project.Also, I would like to thank my fellow batch mates for their feedback, cooperation and friendship. In particularly Blessy, Param, Chandan, Rajat, Biswajeet, Rohith and Sayoni for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of research and writing this thesis. This accomplishment would not have been possible without them. Last but not least, I would like to thank my family: my parents and to my brother for supporting me throughout my journey.

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List of Publications

• How to implement a genuine Parrondo’s paradox with quantum walks?, Jishnu Rajendran and Colin Benjamin, arXiv:1702.05927

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ABSTRACT Quantum algorithms are of a great importance in the field of quantum information and computation. As a universal computational primitive quantum walks are at the heart of quantum computation. Exploring quantum walks in the regime of quantum games has important role in the field of quantum ratchets which explains many interesting physical phenomena. In this thesis, we have formally introduced a genuine implementation of Parrondo’s paradox with quantum walk, further exploring its connection with quantum ratchets along with the extension of quantum walks into finite graphs and implementation of it into graphene .

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Contents 1 Introduction to Quantum Walks 1.1 Introduction . . . . . . . . . . . . . . 1.2 Classical Random walk . . . . . . . . 1.3 Quantum walk . . . . . . . . . . . . 1.3.1 Discrete-Time quantum walk

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1 1 3 4 5

2 Parrondo’s paradox and quantum ratchets 2.1 Quantum ratchets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 16

3 Implementing Parrondo’s paradox with quantum walks 3.1 Entangled Quantum Walk . . . . . . . . . . . . . . . . . . 3.2 Implementation using quantum walks . . . . . . . . . . . . 3.3 Parrondo’s paradox using two coin initial state . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18 18 20 22 27

4 Quantum walks on simple graphs 4.1 Quantum walks on finite graphs . . . . . 4.1.1 Continuous-time quantum walk . 4.1.2 Discrete-time quantum walk . . . 4.1.3 Perfect State Transfer Conditions 4.2 Quantum walks on simple graphs . . . . 4.3 Quantum walks on graphene . . . . . . .

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References

39

Appendix A

42

vi

List of Figures 1.1 1.2 1.3 1.4 1.5 2.1 3.1 3.2 3.3

3.4

3.5

3.6

4.1 4.2 4.3 4.4 4.5

Probability of the particle being in the position n at time t, assuming it starts the random walk at the origin. The probability is zero in empty cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . Probability of finding the quantum particle in position n at time t, assuming that the walk starts at the origin with the quantum coin in ”tails” state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . Probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . .

8 11 13

Parrondo’s games a). Definition of two games b). When he games are played individually and alternatively . . . . . . . . . . . . . . . . . .

15

3 4

Coin initial state is |Φ+ i = √12 (|00i + |11i) and the number of steps is 200. Coin operators are given by Eq. (3.3). . . . . . . . . . . . . . . . 20 Pictorial illustration of the conditions for win or loss for QWs on a line. 22 a) PR − PL of the walker after t steps, with initial state | Ψ0 i = √12 |

0i ⊗ (| 0i + i | 1i), and coin operator A = U S (−51, 45, 0) (red line) or B = U S (0, 88, −16) (green line). b) PR − PL of the walker with games played in sequence ABBBABBB . . . (i.e., q=4), A = U S (−51, 45, 0), B = U S (0, 88, −16) (1600 steps), herein initially you win(steps < 100) but in asymptotic limits you lose. . . . . . a) Parrondo walk is evident in the asymptotic limit for partially entangled coin states when ABAB . . . is played on first coin & BABA . . . on second coin. b) However, when AAAA . . . is played on first & BBBB . . . on second coin, one gets a losing outcome. . . . . . . . . . . . . . . . . . . . . . .

In c) we show similar to a partially entangled state a non-entangled state also gives a Parrondo’s paradox in the asymptotic limit when ABAB . . . is played on first coin & BABA . . . on second coin and finally in d) we show that PR − PL is negative in the asymptotic limits when AAA . . . and BBB . . . are played on the two coins. . . . . . . . Plot of Concurrence(Green), PR − PL (red, solid) for ABAB.. on first coin and BABA...on second coin, and finally PR − PL (red, dashed) for AAAA.. on first coin and BBBB...on second coin. Note that Parrondo’s paradox is observed for 0 < θ < π/2 and 3π/2 < θ < 2π with the definition as in Fig. 1. In the region π/2 < θ < 3π/2 there is a role reversal and thus our definition for Parrondo’s paradox as used in Fig. 1 is also reversed. . . . . . . . . . . . . . . . . . . . . . . . . . . Labeled graphs K3 and P4 and their corresponding adjacency matrix Complete graphs: K4 and K3 . . . . . . . . . . . . . . . . . . . . . . Path graphs: P4 and P3 . . . . . . . . . . . . . . . . . . . . . . . . . Cycle graphs: C4 and C3 . . . . . . . . . . . . . . . . . . . . . . . . . Structure of Graphene: a) Graphene nit cell b) Honeycomb lattice of graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

23

25

25

26 29 29 30 30 35

LIST OF FIGURES 4.6

Probability distribution after 15 steps using H ⊗ H as the coin operator. 37

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Chapter 1 Introduction to Quantum Walks 1.1

Introduction

The discovery of quantum mechanics has changed our view of reality. Over time Quantum mechanics is established as a fundamental theory of physics in such a way that every single theory has to be built in its framework. Ever since then the world of physics has come to face more and more counter-intuitive characteristics of nature. Entanglement, Interference, Superposition are some of such counter-intuitive aspects of quantum mechanics, and they show profound effects in physical systems. In 1982 Richard P Feynman proposed his ideas about a Quantum computer [1]. He showed that a classical Turing machine would experience an exponential slowdown when simulating some quantum phenomena and hence the need of a quantum version. In past years one of the main focus of quantum information theory was implementing its laws to develop devices of huge computational power. The essential idea of such a quantum computer was suggested by Feynman[1],Manin[2] and Benioff [3]. Later this notion of a quantum computer was taken to next level by Deutsh [4] as a universal computing machine. One of the first application of such a quantum computer was investigated in quantum cryptography [5]. Earliest success in this field was Shor’s Factorizing algorithm [6]. Shor was able to show that Quantum version of Factorizing algorithm is much faster that its classical counterpart. Due to its importance in applicability there was a huge development in these and related areas. The main focus of quantum computing is to develop faster and efficient algorithms

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1 Introduction to Quantum Walks for computational problems. From a physics perspective, it mainly concerns with the physical properties of devices used for performing computational or informational processing tasks and the ways to implement such a task efficiently in a given device. Quantum computation can be defined as the interdisciplinary scientific field mainly concerned with quantum computer and quantum information processing systems. Research in this area concentrates primarily on building and running algorithms which exploits the quantum mechanical nature of a quantum computer or a quantum information processing device.

The counter-intuitive nature of quantum mechanics makes designing a quantum algorithm a challenging task. The basic requirement for a good quantum algorithm is that it should perform the task more efficient that any classical algorithms.

Quantum walks, the quantum mechanical version of the classical random walks is an important tool in designing quantum algorithms. Recent studies by Andrew M. Childs [7] showed that quantum walk can be regarded as a universal computational primitive. There are two types of quantum walks: discrete and continuous time quantum walk [23]. The main difference between these two sets is the timing used to apply corresponding evolution operators. As the name suggests, in discrete time quantum walk, corresponding evolution operator is applied only in discrete time steps, whereas in continuous quantum walk, the evolution operator can be applied at any time.

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1 Introduction to Quantum Walks

1.2

Classical Random walk

The classical walks are a crucial concept in field of stochastic used in many different fields like economics, biology. Considering a simple example, random walk describes a trajectory in which the walker makes consecutive left or right moves. In case of 1-D case, the resulting distribution is of Gaussian form. Depending upon the model in consideration this walk can be in a discrete as well as in continuous fashion.

In case of one dimension, a walker starting from the origin moves according to a coin toss. If the coin gives a heads he moves towards right otherwise left. So at anytime the position of walker is described by a probability distribution which gives us the probability at each point at anytime. Table 1.1 shows the probability of walker to be found in corresponding locations after time T when an unbiased coin is used. i @@ -5 -4 -3 -2 T 0 1 2 1/4 3 1/8 4 1/16 1/4 5 1/32 5/32

-1

0 1

1/2

1

2

3

4

5

1/2 1/2

3/8

1/4 3/8

3/8 5/16

1/8 1/4

5/16

1/16 5/32

1/32

Figure 1.1: Probability of the particle being in the position n at time t, assuming it starts the random walk at the origin. The probability is zero in empty cells.

A general term in this table is given by p(t, n) = where ( ab ) =

1 t t+n 2t 2

a! . (a−b)!b!

3

(1.1)


Figure 1.2: Probability distribution of the random walk in a classical 1D lattice for t = 10,t = 40 and t = 100.

In Figure 1.2 probability distribution of the random waler in a 1-D lattice is plotted for t=10,40 and 100 steps.

1.3

Quantum walk

Quantum walks are the quantum mechanical analog of random walks. One of the main difference between classical and quantum walks is that in case of quantum walks the walker is defined by a quantum state, i.e. at anytime there is a quantum state associated with the walker which holds the complete information regarding the walker. In the usual context, a quantum walker has a position state and a coin state associated with it. The evolution of the walker is determined by a coin operator and a shift operator. Similar to classical random walks, two models of quantum walks are

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1 Introduction to Quantum Walks proposed: 1. Discrete-Time quantum walk. 2. Continuous-Time quantum walk.

1.3.1

Discrete-Time quantum walk

Discrete-time quantum walk(DTQW) consists of two quantum mechanical systems, a walker and a coin. An evolution operator governs the evolution of both these systems over discrete time. Mathematically the evolution is determined by a unitary operator.(i.e. |ψit2 = Uˆ |ψit1 ) For the discrete model, the walker’s position n is defined as a vector in a Hilbert space HP of infinite dimension, whose computational basis is {|ni : n ∈ Z}. The quantum ”coin” determines the evolution of the walk. If one obtains ”heads” after tossing the ”coin” and the walker is described by vector |ni,then in the next step it will be described by |n + 1i. If it is ”tails”, it will be described by |n − 1i. This scheme can be realized in physical terms too. If we consider electron as the ”random” walker on a one-dimensional lattice. The state of the electron is described not only by its position in the lattice but also by its spin, which can be in up spin or down spin. Thus, the direction of motion is determined by its spin value. If the electron is in position |ni with its spin up, it should go to |n+1i with spin value unchanged. Similarly, when its spin is down, it should go |n − 1i. The Hilbert space of the system should be H = HC ⊗ HP , where HC is the two-dimensional Hilbert space associated with the ”coin”, the computational basis of which is {|0i, |1i}. The coin toss can be defined as any unitary matrix C with dimension 2, which acts on vectors in Hilbert space HP . It is called coin operator. The shift from |ni to |n + 1i or |n − 1i must be described by a unitary operator,

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1 Introduction to Quantum Walks called the shift operator S. It should operate as follows: S|0i|ni = |0i|n + 1i.

(1.2)

S|1i|ni = |1i|n − 1i.

(1.3)

If we know the action of S on the computational basis of H, we have a complete description of this linear operator. Therefore, we can conclude that S = |0ih0|⊗

∞ X

n=−∞

|n + 1ihn|+|1ih1|⊗

∞ X

|n − 1ihn|

(1.4)

n=−∞

We can re-obtain 1.2 and 1.3 by applying S to the computational basis.

The coin operator C is applied to the initial state in the starting of quantum walk. This is analogous to tossing a coin in the classical case. C operator produces a rotation of the given coin state. If the coin is initially described by one of the states of the computational basis, the result may be a superposition of states. Each term in this superposition will generate a shift in one direction. In order to generate a symmetrical walk around we choose a fair coin. Considering an initial state where the walker is located at origin |n = 0i and the coin state with spin up |0i . |ψ(0)i = |0i|n = 0i

(1.5)

where |ψ(0)i denotes the state at the initial time and |ψ(0)i denotes the state of the quantum walk at time. The coin used for most one-dimensional quantum walks is the Hadamard operator 1 H=√ 2

1 1 1 −1

.

(1.6)

One step consists of applying H in the state of the coin, i.e. applying H ⊗ I , where I identity operator of the Hilbert space HP , followed by the application of the shift 6

1 Introduction to Quantum Walks operator S: H⊗I

|0i ⊗ |0i −−→

|0i + |1i √ ⊗ |0i 2

1 − → √ (|0i ⊗ |1i + |1i ⊗ |−1i) 2

(1.7)

S

The result is a superposition of the particle both in position n = 1 and in position n = −1. The superposition of positions is a result of the superposition generated by the coin operator. We can see that the coin H is non-biased when applied to |0i, since the amplitude of the right part is equal to the amplitude of the left part. If we apply H to |1i, there is a sign difference between the amplitudes of the right and left parts. When we calculate the probability of finding the particle at position n, the sign plays no role. So we can call H a non-biased coin. In the quantum case, we need to measure the quantum system in the state (1.7) to know what the position of the particle is. If we measure it using the computational basis of HP , we will have a 50% chance of finding the particle at position n = 1 and a 50% chance of finding it at the position n = −1. This result is the same, compared to the first step of the classical random walk. If we repeat the same procedure successively, i.e. (1) we apply the coin operator, (2) we apply the shift operator, and (3) we measure using the computational basis, we will re-obtain the classical random walk. Our goal is to use quantum features to obtain new results, which cannot be obtained in the classical context. When we measure the particle position after the first step, we destroy the correlations between different positions, which are typical of quantum systems. If we do not measure and apply the coin operator followed by the shift operator successively, the quantum correlations between different positions can have constructive or destructive interference, effectively generating a behavior different from the classical context, which is a characteristic of quantum walks. We will see that the probability distribution does not go to the normal distribution and 7

1 Introduction to Quantum Walks the standard deviation is not

√

t.

@i −5 −4 −3 T @ 0 1 2 1 3 8 1 4 16 1 17 5 32 32

−2 −1 1 4 5 8

1 2 5 8 1 8

0 1 1 2 1 8

1 1 2 1 8 1 8

2

1 4 1 8

3

1 8 5 32

4

1 16

5

1 32

Figure 1.3: Probability of finding the quantum particle in position n at time t, assuming that the walk starts at the origin with the quantum coin in ”tails” state

The quantum walk consists in applying the unitary operator U = S(H ⊗ I)

(1.8)

a number of times without intermediate measurements. One step consists in applying U one time, which is equivalent to applying the coin operator followed by the shift operator. In the next step, we apply U again without intermediate measurements. A time t, the state of the quantum walk is given by |ψ(t)i = U t |ψ(0)i

(1.9)

Let us calculate the initial steps explicitly to compare with the classical random walk. We will take (1.5) as initial condition. The first step will be equal to (??). The second step can be calculated using the formula |ψ(2)i = U |ψ(1)i and so on. 1 |ψ(1)i = √ (|1i|−1i + |0i|1i) 2 1 |ψ(2)i = (−|1i|−2i + (|0i + |1i)|0i + |0i|2i) 2 1 |ψ(3)i = √ (|1i|−3i − |0i|−1i + (2|0i + |1i)|1i + |0i|3i) 2 2 8

1 Introduction to Quantum Walks These few initial steps have already revealed that the quantum walk differs from the classical random walk in several aspects. We use a non-biased coin, but the state |ψ(3)i is not symmetric with respect to the origin. The table in Figure 1.3 shows the probability distribution up to the fifth step, without intermediate measurements. Besides being asymmetric, the probability distribution is not concentrated in the central points. A comparison with the table in Figure. 1.1 clearly illustrates this fact.

To find the probability distribution for a number of steps much larger (say 100) we use the help of computers.For this purpose I have used two main methods

First way is to calculate matrix U explicitly. We have to calculate the tensor product H ⊗ I. The tensor product is also required to obtain a matrix representation of the shift operator as defined in (1.4). These operators act on vectors in an infinite vector space. However, the number of nonzero entries is finite. Therefore, these arrays must have dimensions slightly larger than 200 × 200. After calculating U , we calculate U 100 , and the product of U 100 with the initial condition kψ(0)i written as a column vector with a compatible number of entries. The result is |ψ(100). Finally, we can calculate the probability distribution. This method is suitable to be implemented in computer algebra systems, such as Mathematica, Maple, or Sage, and is inefficient in general. Second one and more efficient one uses a recursive formula obtained as follows: the generic state of the quantum walk can be written as a linear combination of the computational basis as |ψ(t)i =

∞ X

(An (t)|0i + Bn (t)|1i)|ni

n=−∞

9

(1.10)

1 Introduction to Quantum Walks where the coefficients satisfy the constraint ∞ X

n=−∞

|An (t)|2 +|Bn (t)|2 = 1

(1.11)

ensuring that |ψ(t)i has norm equal to 1 in all steps. When applying H ⊗ I followed by the shift operator in expression (1.10), we can obtain recursive formulas involving the coefficients A and B, which are given by An (t) + Bn (t) √ 2 An (t) − Bn (t) √ An−1 (t + 1) = 2 An+1 (t + 1) =

Using the initial condition ( 1 if, n == 0; An (0) = 0 otherwise and Bn (0) = 0, we can calculate the probability distribution using the formula p(t, n) = |An (t)|2 +|Bn (t)|2

10

(1.12)


Figure 1.4: Probability distribution after 100 steps of a quantum walk with the Hadamard coin starting from the initial condition |ψ(0)i = |0i|n = 0i. The points where the probability is zero were excluded (n odd)

By employing any of the aforementioned methods the graph in 1.4 for the probability distribution after 100 steps will be obtained. Analogous to the classical random walk, we will ignore the null values of the probability. At t = 100, the probability is zero for all odd values of n. The asymmetry of the probability distribution is evident. The probability of finding the particle on the right side of the origin is larger than √ on the left. In particular, for n around 100/ 2, the probability is much higher than at the origin. This fact is not exclusive to the value t = 100. It is valid for any value of t. This suggests a ballistic behavior of the quantum walk. The particle can be found away from the origin as if it were in a uniform motion rightward. It is natural to ask whether this pattern would be held if the distribution were symmetric around the origin. In order to obtain a symmetrical distribution, one must understand why the pre11

1 Introduction to Quantum Walks vious example has a tendency to go rightward. The Hadamard coin introduces a negative sign when applied to state |0i. This means there are more cancellations of terms with coin state equals |0i than of terms with coin state equals |1i. Since the coin state |0i induces movement rightward and |1i leftward, the final effect is the asymmetry with large probabilities on the left. We can confirm this analysis by calculating the resulting probability distribution when the initial condition is |ψ(0)i = |0i|n = 0i

In this case, the number of negative terms will be greater than positive terms and there will be more cancellations of terms with the coin state in |0i. The final result will be the mirror distribution in Figure 1.4 around the vertical axis. To obtain a symmetrical distribution, one must superpose the quantum walks resulting from these two initial conditions. This superposition should not cancel terms before the calculation of the probability distribution. The trick is to multiply the imaginary complex number i to the second initial condition and add to the first initial condition, as follows: |ψ(0)i =

|0i + i|1i √ |n = 0i 2

12

(1.13)


Figure 1.5: Probability distribution after 100 steps of a quantum walk with the Hadamard coin starting from the initial condition |ψ(0)i = √12 (|0i + i|1i|n = 0i. The points where the probability is zero were excluded (n odd)

The entries of the Hadamard coin are real numbers. When we apply the evolution operator, terms with the imaginary unit are not converted into terms without the imaginary unit and vice versa. There will be no cancellations of terms of the walk that goes rightward with the terms of the walk that goes leftward. At the end, the probability distributions are added. In fact, the result is the graph in Figure 1.5.

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Chapter 2 Parrondo’s paradox and quantum ratchets In many games, a sacrificial moves is needed in order to win the overall game.one of the best example is chess. Similarly, two unstable systems, if combined in the right way, can paradoxically become stable. In the same line of thought two losing gambling games be set up such that, when they are played one after the other, they becoming winning. This striking result in game theory is known as Parrondos paradox, discovered by Juan Parrondo citeParrondo. Parrondo’s game as originally introduced in Refs.[9, 10] as a gambling game. A player plays against a bank with a choice of two games A and B, whose outcomes are determined by the toss of biased coins. Each of these games is losing when played in isolation but when played alternately or in some other deterministic or random sequence (such as ABB . . . , ABAB . . ., etc.) can become a winning game. Owing to this counter-intuitive nature, Parrondo’s games are also referred to as Parrondo’s paradox. The apparent paradox that two losing games A and B can produce a winning outcome when played in an alternating sequence was originally devised by Juan M. R. Parrondo as a pedagogical illustration of the Brownian ratchet[11]. Parrondo’s games have important applications in many physical and biological systems, e.g., in control theory-the random/deterministic combination of two unstable systems can produce a overall stable system[14].

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2 Parrondo’s paradox and quantum ratchets

Figure 2.1: Parrondo’s games a). Definition of two games b). When he games are played individually and alternatively

In Figure 2.1 Game rules and simulation. a , An example of two games, consisting of only three biased coins, which demonstrate Parrondos paradox, where p1 , p2 and p3 are the probabilities of winning for the individual coins.For a game A,if = 0.005 and p1 = 1/2 − , then it is a losing game.For game B,if p2 = 1/10 − , p3 = 3/4 − and M = 3 then we end up with coin 3 more often that coin 2 .But coin 2 has a poor probability of winning, so B is a losing game. The paradox is that playing games A and B in any sequence leads to a win.. b, The progress of playing games A and B individually and when switching between them. The simulation was performed by playing game A twice and game B twice, and so on, until 100 games were played; this is indicated by the line labelled Periodic. Randomly switched games result in the line 15

2 Parrondo’s paradox and quantum ratchets labelled Random. The results were averaged from 50000 trials with = 0.005. There are actually many ways to construct such gambling scenarios, the simplest of which uses three biased coins.A model of such a game is presented in the Figure (2.1(a)).Game A consists of tossing a biased coin (coin 1) that has a probability (p1 )of winning of less than half, so it is a losing game. Let p1 = 1/2 − where , the bias, can be any small number, say 0.005.Game B (2.1(a)) consists of playing with two biased coins. The rule is that we play coin 2 if our capital is a multiple of an integer M and play coin 3 if it is not. The value of M is not important, but for simplicity let us say that M = 3. This means that, on average, coin 3 would be played a little more often than coin 2. If we assign a poor probability of winning to coin 2, such as p2 = 1/10 − , then this would outweigh the better coin 3 with p3 = 3/4 − , making game B a losing game overall. Thus both A and B are losing games, as can be seen in Figure 2.1b, where the two lower lines indicate declining capital. If we play two games of A followed by two of B and so on, this periodic switching results in the upper line in Figure. 2.1b, showing a rapid increase in capital this is Parrondos paradox. What is even more remarkable is that when games A and B are played randomly, with no order in the sequence, this still produces a winning expectation (Figure.2.1b).

2.1

Quantum ratchets

Brownian particles under spatially asymmetric potential has been shown to produce nonzero net current under specific conditions [12, 13]. As these particles are subjected to thermal noise this kind of systems are known as thermal ratchets. Thermal ratchets have already shown its application in different fields. In biology, it provides a possible mechanism for biological motors. In physics, it is closely related o noise induced

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2 Parrondo’s paradox and quantum ratchets transportation. In context to parrondo’s paradox, ratchets can be visualized as physical realization of parrondo’s paradox. One of the simplest ratchet model is a Brownian particle in a saw-tooth potential, this potential is flashed on and off to affect the motion of the particle. Without any other biasness, just with the randomness and asymmetry, the particle can move towards one of the sides.Parrondo’s games were designed as an illustration of Brownian ratchets. It is a system where Brownian noise is converted into drifting motion in specific direction using the ratchet potential. Parrondos games helps in understanding the Brownian ratchet through the analogy obtained from both of them.

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Chapter 3 Implementing Parrondo’s paradox with quantum walks 3.1

Entangled Quantum Walk

As in the Chapter 1, the elements of entangled quantum walk are the walker, coins, evolution operators for both the coins, walker and a set of observables. The walker is a quantum system with its position denoted as |positioni residing in a Hilbert space of infinite but countable dimension HP . The basis states |iiP which span HP , P P and any superposition of the form i αi |iip which are subject to i |αi |2 = 1, are valid states for the walker [23]. The walker is usually initialized at the ‘origin’, i.e., |positioni0 = |0iP . The entangled initial state is a quantum system in a 4-D Hilbert space HEC . We denote the two coin initial state as |coini0 , which can be one for the following bell states 1 |Φ+ i = √ (|00i + |11i) 2 1 |Φ− i = √ (|00i − |11i) 2 1 |Ψ+ i = √ (|01i + |10i) 2

(3.1a) (3.1b) (3.1c)

above bell states are maximally entangled pure bipartite states with concurrence equal to unity. The initial state of the quantum walker resides in the Hilbert space HT = HP ⊗ HEC and has the form: |ψi0 = |positioni0 ⊗ |coini0 18

(3.2)

3 Implementing Parrondo’s paradox with quantum walks Evolution operators used are unitarity as before and since the coin is a bipartite system, the coin is defined as the tensor product of two single-qubit coin operators: H CEC = Uα ⊗ Uβ . For example,, we can define the operator CÊC as the tensor product

ˆ ⊗2 . H 1 H = (|00ih00|+|01ih00|+|10ih00|+|11ih00| CÊC 2 + |00ih01|−|01ih01|+|10ih01|−|11ih01|

(3.3)

+ |00ih10|+|01ih10|−|10ih10|−|11ih10| + |00ih11|−|01ih11|−|10ih11|+|11ih11|). The evolution operator is fully separable, thus any entanglement in the coins is due to the initial states used. The conditional shift operator SEC allows the walker to move either forward or backward, depending on the state of the coin. The operator X SEC = |i + 1ipp hi|⊗|00icc h00|+ i

X |iipp hi|⊗|01icc h01|+ i

X |iipp hi|⊗|10icc h10|+

(3.4)

i

X |i − 1ipp hi|⊗|11icc h11| i

incorporates the stochastic behavior of the random walk with entangled state. It is only when the coin is in the |00i or |11i state that the walker moves either forward or backward else the walker does not move.Figure 3.1 shows the probability distribution obtained for the entangled quantum walk with coin initial state: |Φ+ i = H ˆ ⊗2 . |11i) and coin operator CÊC == H

19

√1 (|00i 2

+

3 Implementing Parrondo’s paradox with quantum walks

Figure 3.1: Coin initial state is |Φ+ i = √12 (|00i + |11i) and the number of steps is 200. Coin operators are given by Eq. (3.3).

3.2

Implementation using quantum walks

Our main goal is to implement a genuine Parrondo’s paradox via quantum walk. We show that while previous attempts at implementing Parrondo’s paradox with quantum walks failed in asymptotic limits[16]-[17] our method using two coin initial states gives a genuine Parrondo’s paradox even in the asymptotic limits. The 1D discrete time quantum walk implementation of Parrondo’s paradox is as follows: Consider two games A and B alternately played according to time. Game A and B are represented by different quantum operators U (αA , βA , γA ) and U (αB , βB , γB ),

U (α, β, γ) =

eiα cos β −e−iγ sin β eiγ sin β e−iα cos β

The initial state of the quantum walker is | Ψ0 i =

√1 2

,

(3.5)

| 0i ⊗ (| 0i + i | 1i), where first

ket refers to the position space and second ket refers to the single coin space which

20

3 Implementing Parrondo’s paradox with quantum walks is initially in a superposition of heads and tails. The shift in the position space, say from |ni to |n − 1i or |n + 1i, is defined by a unitary operator called shift operator(S) defined as, S=

∞ X

n=−∞

|n + 1ihn|⊗|0ih0|+

∞ X

|n − 1ihn|⊗|1ih1|

(3.6)

n=−∞

Games A and B are played alternately in different time steps, i.e., game A is played on time steps t = nq and game B is played on time steps t 6= nq, where q is the period and n is an integer. The evolution operator can be written as: S.U (αA , βA , γA ) if t = nq, n ∈ Z U= S.U (αB , βB , γB ) if t 6= nq, n ∈ Z

(3.7)

and the final state after N steps is given by | ΨN i = U N | Ψ0 i. For q = 3, it means we play games with the time sequence ABBABB . . . As denoted in Fig. 3.2, after N steps, if the probability PR of the walker to be found to the right of the origin, is greater than the probability PL to be found to the left of the origin, i.e., PR − PL > 0, we consider the player to win. Similarly, if PR − PL < 0, the player losses. If PR − PL = 0, it means the player neither loses nor wins, it’s a draw. By making use of the above scheme, Parrondo’s games using 1-D DTQW are formulated. We use PR −PL to indicate the player win or lose and not expectation value of position because as already shown in Refs.[17] expectation value may be positive but PR − PL may ybe negative this would be an absurd result. The game is constructed with two losing games A and B having two different biased coin operators UA (αA , βA , γA ) and UB (αB , βB , γB ), if we set αA = −51, βA = 45, γA = 0, αB = 0, βB = 88, γB = 0, UAS = U S (−51, 45, 0), UBS = U S (0, 88, −16) as in Fig.3.3(a). We form a game with sequences ABBB . . .. This results in winning at the beginning but in the asymptotic limit the player will lose as in Fig.3.3(b), one can check for different sequences like ABAB . . . ABBABB . . . etc. and in all cases in the asymptotic limits we lose. Hence Parrondo’s paradox does not exist in case of 1-D DTQW. This fact was noted in 21


Figure 3.2: Pictorial illustration of the conditions for win or loss for QWs on a line. Refs. [16, 17] also. In particular, Ref. [16] shows with many different sequences like ABAB.., AABAAB.., etc, in asymptotic limits there is no Parrondo’s paradox. Hence our motivation to find circumstances for the existence of a genuine Parrondo’s paradox in quantum walks.

3.3

Parrondo’s paradox using two coin initial state

As in the previous section, the elements of our two coin quantum walk are the walker, coins, evolution operators for both the coins, walker and a set of observables. The walker is a quantum system with its position denoted as |positioni residing in a Hilbert space of infinite but countable dimension HP . The basis states |iiP which span HP , P P and any superposition of the form i αi |iip which are subject to i |αi |2 = 1, are valid states for the walker [23]. The walker is usually initialized at the ‘origin’, i.e., |positioni0 = |0iP . The two coin initial state is a quantum system in a 4-D Hilbert space HEC . We

22


(a)

(b)

Figure 3.3: a) PR − PL of the walker after t steps, with initial state | Ψ0 i =

√1 2

| 0i ⊗ (|

0i + i | 1i), and coin operator A = U S (−51, 45, 0) (red line) or B = U S (0, 88, −16) (green line). b) PR − PL of the walker with games played in sequence ABBBABBB . . . (i.e., q=4), A = U S (−51, 45, 0), B = U S (0, 88, −16) (1600 steps), herein initially you win(steps < 100) but in asymptotic limits you lose.

denote the two coin initial state as |coini0 , which may or may not be entangled θ θ |coini0 = cos |10i + i sin |01i 2 2

(3.8)

The initial state of the quantum walker resides in the Hilbert space HT = HP ⊗ HEC and has the form: |ψi0 = |positioni0 ⊗ |coini0 which using Eq.3.8, gives |ψi0 = |0i ⊗ cos

θ 2

|10i + i sin

(3.9) θ 2

|01i Evolution operators

used are unitary as before and since the coin is a bipartite system, the coin is defined as the tensor product of two single-qubit coin operators: CEC = Uαk ,βk ,γk ⊗ Uαl ,βl ,γl , where k,l can be any of the Game A and B. The evolution operator is fully separable, thus any entanglement in the coins is due to the initial states used. The conditional shift operator SEC allows the walker to move either forward or backward, depending

23

3 Implementing Parrondo’s paradox with quantum walks on the state of the coin. The operator X SEC = |i + 1ipp hi|⊗|00icc h00| i

+ +

+

X i

X i

|iipp hi|⊗|01icc h01|

i

|iipp hi|⊗|10icc h10|

X

(3.10)

|i − 1ipp hi|⊗|11icc h11|

incorporates the stochastic behavior of the random walk with a two coin initial state. It is only when the coin is in the |00i or |11i state that the walker moves either forward or backward else the walker does not move. The full evolution operator has the structure UT = SEC .(Ip ⊗ CEC ) and one can mathematically represent a two coin quantum walk after N steps as |ψiN = (UT )N |ψi0 , where |ψi0 denotes the initial state of the walker and the coins. As defined before, winning and losing in context of Parrondo’s game, after N time steps if the probability PR of the walker to be found to the right of the origin is greater than the probability to be found left of the origin, i.e., PR − PL > 0 we consider the player to win. However if, PR − PL < 0 then the player loses and if PR − PL = 0 it implies a draw. In order to obtain a genuine Parrondo’s paradox the two games A and B are now played on the two coin space as follows: UA ⊗ UB is operated on the two coins and in the next step UB ⊗ UA is played on the two coins. Thus, for the first coin we have the series ABAB . . . while on the second coin we have BABA . . .. The coin operators can as before be defined asX = A ⊗ B = CEC = U (−51, 45, 0) ⊗ U (0, 88, −16) 0 Y = B ⊗ A = CEC = U (0, 88, −16) ⊗ U (−51, 45, 0)

and are played alternately in time, i.e, in sequence XY XY . . . and the plot for PR −PL as shown in Fig.3.5(a) is obtained. It is evident that the sequence XY XY . . . 24

3 Implementing Parrondo’s paradox with quantum walks provides a winning outcome for two losing games even in asymptotic limits. The fact that individually the sequence AAA . . . on first coin and BBB . . . on second coin give a losing outcome can be seen from PR − PL plot in Fig.3.5(b).

(a)

(b)

Figure 3.4: a) Parrondo walk is evident in the asymptotic limit for partially entangled coin states when ABAB . . . is played on first coin & BABA . . . on second coin. b) However, when AAAA . . . is played on first & BBBB . . . on second coin, one gets a losing outcome.

(a)

(b)

Figure 3.5: In c) we show similar to a partially entangled state a non-entangled state also gives a Parrondo’s paradox in the asymptotic limit when ABAB . . . is played on first coin & BABA . . . on second coin and finally in d) we show that PR − PL is negative in the asymptotic limits when AAA . . . and BBB . . . are played on the two coins. From Fig.3.5 we can conclude that one can obtain a genuine Parrondo’s paradox with a non-entangled or a partially entangled two coin state using quantum walks. 25

3 Implementing Parrondo’s paradox with quantum walks When a single coin was considered from Fig.3.3 the outcome of Parrondo’s games did not give rise to the paradox in the asymptotic limit. In order to obtain a genuine Parrondo’s paradox, what is needed is a two-coin state. Maximally entangled coins lead to a draw as the probability distribution is perfectly symmetric as noted before in Ref.[23], on the other hand non-entangled or partially entangled coins lead to a Parrondo’s paradox. Further, in Fig. 3.6, we plot the amount of entanglement present in a quantum system, i.e., the concurrence[24]. The concurrence is zero for a separable state and one for a maximally entangled state. Fig.3.6 shows the concurrence for our arbitrary two coin state as a function of θ. One sees that Parrondo’s paradox is observed for 0 < θ < π/2 and 3π/2 < θ < 2π with the definition as in Fig. 3.2. In the region π/2 < θ < 3π/2 there is a role reversal and thus our definition for Parrondo’s paradox as used in Fig. 3.2 is also reversed.

Figure 3.6: Plot of Concurrence(Green), PR − PL (red, solid) for ABAB.. on first coin and BABA...on second coin, and finally PR − PL (red, dashed) for AAAA.. on first coin and BBBB...on second coin. Note that Parrondo’s paradox is observed for 0 < θ < π/2 and 3π/2 < θ < 2π with the definition as in Fig. 1. In the region π/2 < θ < 3π/2 there is a role reversal and thus our definition for Parrondo’s paradox as used in Fig. 1 is also reversed.

26


3.4

Conclusion

Our goal in this work was to show evidence of a genuine Parrondo’s paradox using quantum walks and we show this using a two coin state. We also considered entanglement between the two coins and showed that maximally entangled states do not show any paradox while non-entangled as well as partially entangled states do show the paradox. Our work can be considered as a demonstration of a quantum ratchet too, implying particle transport against an applied bias in presence of noise or perturbations. In our case the noise parameter can be considered to reduce entanglement, thus looking at Fig. 3.6, from zero asymmetry in probability distribution, i.e., non-directed transport, when there is maximal entanglement to finite asymmetry in probability distribution, i.e., directed transport when there is no entanglement, is a clear marker of quantum ratchet like behavior in our system. The quantum ratchet analogies in Parrondo’s paradox with quantum walks were also noticed in Ref. [19], however without any entanglement.

27

Chapter 4 Quantum walks on simple graphs Quantum walks on graphs are proven to be the universal computational primitive for any quantum computation i.e,. any desired quantum computation can be encoded entirely in some underlying graph. In light of this, Now we discuss quantum walks on graphs. In order to fully appreciate Quantum walks on graphs, we start with a basic introduction to graph theory. Definition 4.0.1. Graph: A graph is an ordered pair G = {E, V } where • V is the vertex set whose elements are the vertices of the graph. This set is often denoted V or V (G). • E is the edge set whose elements are the edges between vertices, of the graph. This set is often denoted E(G) or E. If the graph is undirected, individual edges are unordered pairs {u, v} and if the graph is directed, edges are ordered pairs (u, v), where u and v are vertices in V . Definition 4.0.2. Adjacency matrix: An Adjacency matrix is a square matrix which represents a finite graph. Elements in the matrix denotes whether a pair of vertices are directly connected or not(adjacent or not).(See Fig. 4.1)

28

4 Quantum walks on simple graphs

3 1



 0 1 1 AG =  1 0 1  1 1 0

2 2

3

1



0  1 AG =   0 0

4

1 0 1 0

0 1 0 1

 0 0   1  0

Figure 4.1: Labeled graphs K3 and P4 and their corresponding adjacency matrix Definition 4.0.3. Complete graph: A Complete graph (Kn ) is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.

2

3

1

4



0 1  1 0 AG =   1 1 1 1

1 1 0 1

3



1 1   1  0

1



 0 1 1 AG =  1 0 1  1 1 0

2

Figure 4.2: Complete graphs: K4 and K3 Definition 4.0.4. Path graph: A Path graph or a linear graph (Pn ) is a graph with two of the vertex with degree 1, and the other n-2 nodes of vertex degree 2.

29


1



2

0 1  1 0 AG =   0 1 0 0

3 0 1 0 1



4

1

2

3



 0 1 0 AG =  1 0 1  0 1 0

0 0   1  0

Figure 4.3: Path graphs: P4 and P3 Definition 4.0.5. Cycle graph: A Cycle graph or a circular graph (Cn ) is a graph with a single cycle, or in other words, all the vertices are connected in a closed chain. Each vertices have a degree 2.

1

4

2

3



0 1  1 0 AG =   0 1 1 0

0 1 0 1

1



 0 1 1 AG =  1 0 1  1 1 0 

1 0   1  0

2

3

Figure 4.4: Cycle graphs: C4 and C3 The complement of a graph G, denoted by G, shares the same vertices as G but its set of edges is E = {(i, j) | (i, j) ∈ / E}. The join of graphs G and H is denoted G + H and its adjacency matrix is given by:

AG+H =

AG J J AH

,

where J is the all ones matrix of the relevant dimensions.

30

(4.1)


4.1

Quantum walks on finite graphs

Similar to the classical random walks, Quantum walks for both discrete and continuous time can also be defined for finite graphs.

4.1.1

Continuous-time quantum walk

Continuous time quantum walk evolves over graph G = {E, V } according to the Schrdinger equation. The adjacency matrix AG gives rise to the Hamiltonian, which is taken to be H = AG . As there is no coin space, the walk evolves entirely in the position basis.

|φ(t)i = e−iAG t |φ(0)i.

(4.2)

The amplitude at vertex v at time t is hv|φ(t)i.

4.1.2

Discrete-time quantum walk

Discrete-time quantum walk in a finite graph is very similar to that of DTQW in one dimension. A discrete time quantum walk over a graph structure G = {E, V } is controlled by a quantum ‘coin.’ The coin is a unitary operator applied at each vertex v ∈ V , and its dimension is d × d where d is the degree of the vertex. The state of the quantum walk must describe both the position of the walker, and the configuration of coin states. A general state is written

ψ(T ) =

X

αv,c (T )|v, ci

v,c

31

(4.3)

4 Quantum walks on simple graphs where αv,c ∈ C and |v, ci denotes a basis state on vertex v with coin state c. After the coin toss, a shift operation is applied. This is simply a permutation between the relevant coin states at different vertices, and is hence a unitary transformation. The shift operator acts like S|v, ci = |w, di, so moves amplitude from the cth coin state of v to the dth coin state of w [?]. The probability of the walker being measured at position v after T steps is the summation over coin states at v, P p(v, t) = i |v, ci |2 .

4.1.3

Perfect State Transfer Conditions

For the continuous time walk, condition forperfect state transfer between vertices v and w at time t is:

hw|e−iAG t |vi = 1

(4.4)

where |vi and |wi are unit vectors at vertices v and w respectively and t > 0. If the perfect state transfer condition holds for some v, w ∈ V then we say the graph G has perfect state transfer. A special case of perfect state transfer occurs when v = w, in which case the graph G is periodic. In the discrete time quantum walk, perfect state transfer occurs between vertices v and w after T steps if

X c,d

hw, d|(SC)T |v, ci = 1.

(4.5)

where S and C are shift operator and coin operator respectively and d, c is some coin state. 32


4.2

Quantum walks on simple graphs

Considering K2 , whose adjacency matrix is: A=

0 1 1 0

(4.6)

the time-dependent amplitude vector in continuous-time quantum walk is of the form −itA

|ψ(t)i = e

|0i =

cos(t) −i sin(t)

(4.7)

where p0 (t) = cos2 (t) and p1 (t) = (−i sin(t))2 = sin2 (t) which implies P (t) = T cos2 (t) sin2 (t) . This exhibits perfect state transfer t = nπ for n ∈ Z Considering P3 , with adjacency matrix   0 1 0 A= 1 0 1  0 1 0

(4.8)

the time-dependent amplitude vector in continuous-time quantum walk is of the form √   1 + 1 cos 2t 2 √   i sin( 2t) |ψ(t)i = e−itA |0i =  (4.9)  − √2 √ 1 cos 2t − 1 2

∴ On calculating individual probability we will obtain, t 4 p0 (t) = cos √ 2 √ 1 2 p1 (t) = − sin 2t 2 t 4 p2 (t) = sin √ 2 √ Considering p0 (t), p0 (t) = 1 for t = 2πn and p2 (t) = 1 for t =

(4.10)

2πn+π √ , 2

which

means there exist different t0 s such that probability at two different vertices reaches √ unity. Thus P3 has perfect state transfer from vertex 1 to 2 because at t = 2πn the probability is unity at vertex 1 and zero at vertex 3 and later for t = 33

2πn+π √ 2

vertex 1

4 Quantum walks on simple graphs becomes zero and vertex 3 reaches unity. Now if we consider K3 , whose adjacency matrix is   0 1 1 A= 1 0 1  1 1 0  1  (2 cos(t) + cos(2t) + i(2 sin(t) − sin(2t))) 3 |ψ(t)i = e−itA |0i =  13 (− cos(t) + cos(2t) + i(− sin(t) − sin(2t)))  1 (− cos(t) + cos(2t) + i(− sin(t) − sin(2t))) 3

(4.11)

(4.12)

now the probability at each vertices are

1 p0 = (i(2 sin(t) − sin(2t)) + 2 cos(t) + cos(2t))2 9 1 (4.13) p1 = (i(− sin(t) − sin(2t)) − cos(t) + cos(2t))2 9 1 p2 = (i(− sin(t) − sin(2t)) − cos(t) + cos(2t))2 9 there exist not real value for t such that any of the above probability reaches unity. Thus we can conclude that there is no perfect state transfer for K3

4.3

Quantum walks on graphene

As a physical realization of quantum walks, here we implement quantum walks on graphene. Quantum walk on graphene needs slight modification in its walker state as the graphene lattice has a honeycomb structure. The basic definitions will be discussed below.

Definition 4.3.1. Position: Position of the walker, denoted by |x, yi, is a vector in Hilbert Space HP of infinite dimension,the computational basis of which is {|x, yi : x, y ∈ Z} Definition 4.3.2. Coin:

A Four Dimensional Hilbert space is associated with

"coin",HC ,has computation basis of which is {|00i, |01i, |10i, |11i},denotes the quantum equivalent for "Up","Down","Left" and "Right". 34

4 Quantum walks on simple graphs A general coin in HC will be of the form 

 a0  a1   |ci =  (4.14)  a2  ; a3 When we consider graphene lattice, the above four amplitudes are associated with the four directions of motion of the quantum walker on the honeycomb lattice(See Fig 4.5 a)).

(a)

(b)

Figure 4.5: Structure of Graphene: a) Graphene nit cell b) Honeycomb lattice of graphene Definition 4.3.3. Total system: The basis state of the quantum walk are the tensor product of position and coin states. |QW i = |i, ji ⊗ |ci which evolves in the Hilbert space H, given by H = Hc ⊗ Hp Definition 4.3.4. Coin operator: Coin operator(C) is a unitary operator acting on the Hilbert space HC , with dimension 4 Definition 4.3.5. Shift operator: Shift operator is defined for both red and blue atoms separately :For blue atom(starting with (i, j)) shift operator is 35






  a0   a1      S |x, y + 1i ⊗  a2  = |x, y + 1i ⊗  a3

  a0  0   + |x − 1, yi ⊗    0 0    0  0    + |x, yi ⊗  +|x + 1, yi ⊗   a2   0

For red atom located at (i,j+1), shift operator is given by       a0 a0   a1       = |x, yi ⊗  0  + |x + 1, y + 1i ⊗  S |x, yi ⊗   a2   0   a3 0    0  0     +|x − 1, y + 1i ⊗   a2  + |x, y + 1i ⊗  0

 0 a1   0  0  0 0   0  a3

 0 a1   0  0  0 0   0  a3

(4.15)

(4.16)

By running the simulation using the Mathematica code given in the Appendix, a

3D probability distribution is obtained with the coin operator C = H ⊗ H. Using the Mathematica code provided in Appendix, using different quantum gates as the coin operator we can implement quantum walk on graphene. From the results obtained one can verify that each probability distribution is unique and from this observation we can conclude that one can use quantum walks on graphene to physically realize quantum gates and circuits with the aid of quantum walk.

36


Figure 4.6: Probability distribution after 15 steps using H ⊗ H as the coin operator.

37

Conclusion In conclusion, through quantum walks we were able to explorer many areas of quantum computation, particularly quantum game theory and its relation with quantum ratchets. This study enabled us to find some new interesting results regarding Parrondo’s paradox and how to implement it with quantum walks. From a physics perspective, application of quantum walks on a real physical system can help us in developing a quantum computer. Quantum walks on graphene using quantum gates helped to realize the physical significance in real systems. Finally Quantum walks on a general graph is an important step towards formulating any quantum algorithm and we investigated quantum walks on some simple graphs and this can shed light into not only quantum algorithms but also into quantum transportation.

38

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REFERENCES [9] G. P. Harmer, D. Abbott, P. G. Taylor and J. M. R. Parrondo, in Proc. 2nd int. Conf. on Unsolved Problems of Noise and Fluctuations (UPoN ’99), Adelaide, Australia, 511, 189 (1999). [10] J. M. R. Parrondo, G. P. Harmer, and D. Abbott, Phys. Rev. Lett. 85, 5226–5229 (2000). [11] Harmer, Gregory P. AU - Abbott, Derek,Game theory: Losing strategies can win by Parrondo’s paradox,Nature,1999/12/23/print,0028-0836 [12] A. Ajdari, and J. Prost: C. R. Acad. Sci. Paris, 315 (1992) 1635. [13] M. O. Magnasco: Phys. Rev. Lett. 71 (1993) 1477. [14] A. Allison and D. Abbott, ”Control systems with stochastic feedback,” Chaos, Vol. 11, No. 3, pp. 715-724, 2001. [15] J.M.R. Parrondo and L. Dinis, ”Brownian motion and gambling: from ratchets to paradoxical games,” Contemporary Physics, Vol 45, pp. 147-157, 2004 [16] A. P. Flitney,Quantum Parrondo’s games using quantum walks, arXiv:quantph/1209.2252 (2012). [17] Min Li, Yong-Sheng Zhang, Guang-Can Guo, Qunatum Parrondo’s games constructed by quantum random walk, arXiv:quant-ph/1303.6831 (2013). [18] C. M. Chandrashekar, S. Banerjee, Physics Lett. A 375 (2011) 1553 [19] David A. Meyer, Noisy quantum Parrondo games,Fluctuations and Noise in Photonics and Quantum Optics, Proceedings of SPIE 5111 (2003) 344-350. [20] Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A 48, 1687 (1993).

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REFERENCES [21] V. Kendon, A random walk approach to quantum algorithms, Phil. Trans. R. Soc. A (2006) 364, 3407-3422. [22] Sagnik Chakraborty, Arpan Das, Arindam Mallick, C. M. Chandrashekar, Quantum ratchet in disordered quantum walk, arXiv:1611.03323 [23] S.E. Venegas-Andraca, J.L. Ball, K. Burnett, S. Bose, Quantum Walks with Entangled Coins, New Journal of Physics 7 (2005) 221, 10.1088/1367-2630/7/1/221 [24] Roland Hildebrand, Concurrence revisited, Journal of Mathematical Physics, 10.1063/1.2795840 (2007) [25] José Luis Gómez-Mu˜ noz and Francisco Delgado, Mathematica add-on for Dirac Bra-Ket Notation,

Quantum Algebra and Quantum Computing,

http://homepage.cem.itesm.mx/lgomez/quantum/

41

Appendix A Successful completion of this Master’s thesis would have been impossible without the Mathematica add-on for Dirac Bra-Ket Notation, Quantum Algebra, Quantum Computing and the QHD approximation to the Heisenberg Equations of Motion by José Luis Gómez-Mu˜ noz and Francisco Delgado. Below are the codes used for simulating quantum walks for 1D,entangled states,parrondo’s paradox with two coin state and for graphene. Starting with 1-D quantum walk Python code 1 2 3

# ∗ coding : utf 8 ∗ ””” Created on Wed Dec 28 1 6 : 1 6 : 0 2 2016

4 5 6 7 8 9 10 11 12 13

@author : J i s h n u ””” import numpy import m a t p l o t l i b . p y p l o t from numpy import ∗ from m a t p l o t l i b . p y p l o t import ∗ N = 3 # number o f random s t e p s P = 2∗N+1 # number o f p o s i t i o n s

14 15 16

coin0 = array ( [ 1 , 0 ] ) coin1 = array ( [ 0 , 1 ] )

17 18 19 20 21 22 23

C00 C01 C10 C11

= = = =

outer ( coin0 outer ( coin0 outer ( coin1 outer ( coin1

, , , ,

# |0> # |1>

coin0 ) coin1 ) coin0 ) coin1 )

# # # #

|0 > # |01 >

43

A

16 17

c o i n 1 0 = np . a r r a y ( [ 0 , 0 , 1 , 0 ] ) c o i n 1 1 = np . a r r a y ( [ 0 , 0 , 0 , 1 ] )

18 19 20 21 22

# |10 > # |11 >

C00 C00 C00 C00

00 01 10 11

= = = =

np . o u t e r ( c o i n 0 0 np . o u t e r ( c o i n 0 0 np . o u t e r ( c o i n 0 0 np . o u t e r ( c o i n 0 0

, , , ,

coin00 ) coin01 ) coin10 ) coin11 )

# # # #

|00 >