Discrete-time quantum walks in two-dimensional amorphous topological matter

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DISCRETE-TIME QUANTUM WALKS IN TWO-DIMENSIONAL AMORPHOUS TOPOLOGICAL MATTER

Master of Science Thesis Faculty of Engineering and Natural Sciences Examiners: Prof. Teemu Ojanen, Prof. Esa Räsänen December 2021

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ABSTRACT

Rostislav Duda: Discrete-time quantum walks in two-dimensional amorphous topological matter

Master of Science Thesis Tampere University

Master’s Programme in Science and Engineering December 2021

Discrete-time quantum walks have been steadily growing in relevance as a research topic due to their potential applications as building blocks for universal quantum computation and tools for designing fast quantum search algorithms. Recently, it has been discovered that they can be used as a platform for the study of topological phases of matter. In this thesis, we construct a model of amorphous topological matter through a two-dimensional discrete-time quantum walk protocol propagating on a diluted lattice. We perform a computational study of transport properties of this model and discover that the ballistic spread of the quantum walk dissipates into the diffusive regime immediately when any amount of site disorder is introduced into the lattice. We report that the non-trivial topological phase reduces the degree of Anderson localization in the quantum walk and decreases its localization threshold to p≈0.67, which is 17 to 32 percent lower compared to quantum walks in the trivial topological phase. We also observe the onset of anomalous subdiffusion in the vicinity of the localization threshold with diffusion exponent α ≈0.9.

Our results are reinforced by simulations of the two-dimensional discrete-time quantum walk protocol up to 10000 time steps, surpassing formerly conducted numerical studies by an order of magnitude.

Keywords: discrete-time quantum walks, topological quantum matter, topological insulators, amorphous topological matter, percolation theory

The originality of this thesis has been checked using the Turnitin OriginalityCheck service.

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PREFACE

I want to offer my sincerest gratitude to Teemu Ojanen for giving me an opportunity to work on a fascinating and challenging thesis topic that has elevated my research skills to new heights and pushed me to the limits in the best way possible. I would also like to thank Kim Pöyhönen and Moein Najafi Ivaki for insightful discussions, help with debugging, and well-timed advice. I have had the privilege of working alongside some truly outstanding scientists during my time in the Computational Physics Unit at Tampere University (and Tampere University of Technology as it was known back in the day). To that end, I thank Esa Räsänen for initially giving me the chance to become a part of the research department and endorsing my work throughout the years. Finally, I would like to thank my friends and family for providing the necessary mental and emotional support in these turbulent times. As this chapter of my formal education comes to a close, I am thrilled to continue the never-ending journey of scientific discovery in my doctoral studies.

Tampere, 1st December 2021 Rostislav Duda

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LIST OF FIGURES

2.1 Six separate instances of (a) unbiased one-dimensional discrete-time random walk and (b) unbiased two-dimensional discrete-time random walk propagated to 10000 time steps. . . 4 2.2 Probability density of a 1D DTQW after 10, 100 and 1000 time steps.

In these figures the probability densities at the odd-numbered positions are all equivalently zero and as such have been omitted from the plots to make them easier to read. . . 11 2.3 Comparison of RMS displacement as a function of time step between a

discrete-time quantum walk and a discrete-time random walk iterated over the first 100 steps. Statistical averaging of the random walk has been performed over 10000 instances. . . 12 2.4 Probability density of a 2D DTQW after 16, 64 and 256 time steps

with rotation parameters θ₁ =θ₂ =π/2. . . 15 2.5 Effects of site percolation with decreasing site occupancy density p

on the structure of 50-by-50 square lattice Z². . . 19 2.6 Comparison of spreading speeds in different diffusive regimes. . . 22 2.7 Examples of randomly diluted 2D DTQWs for decreasing values of p

propagated up to 256 steps. In these figures, grey spots represent the missing sites, and the scale of the probability density is logarithmic in order to see the diffusive spread better. Rotation parameters are set to θ₁ =θ₂ =π/2. . . 24 2.8 Vector d(k) as a mapping from reciprocal space to the Bloch sphere. 29 2.9 Topological phase diagram of the Haldane model in terms of param-

eters ϕ and ∆/t₂ . . . 30 2.10 Topological phase diagram of the 2D DTQW protocol. . . 32 5.1 Energy gap diagram of the 2D DTQW. Red and blue points corre-

spond to the 2D DTQW processes in trivial and non-trivial topological phases respectively that were chosen for the comparative study of transport properties. White spots correspond to NaN values obtained during the computation. . . 48

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5.2 Comparison between generalized diffusion coefficients and exponents of trivial and non-trivial 2D DTQWs propagated to 1000 time steps with ∆p = 0.1 and p ∈ [0.4,1.0]. Configuration averaging has been performed over 1000 walk iterations for each p. Line widths correspond to the 99% confidence intervals of the estimates. . . 49 5.3 Comparison between generalized diffusion coefficients and exponents

of trivial and non-trivial 2D DTQWs propagated to 10000 time steps with ∆p = 0.01 and p ∈ [0.91,0.99]. Configuration averaging has been performed over 100 walk iterations for each p. Line widths correspond to the 99% confidence intervals of the estimates. . . 50 5.4 Comparison between generalized diffusion coefficients and exponents

of trivial and non-trivial 2D DTQWs propagated to 10000 time steps with ∆p = 0.01 and p ∈ [0.51,0.60]. Configuration averaging has been performed over 1000 walk iterations for each p. Line widths correspond to the 99% confidence intervals of the estimates. . . 54 5.8 Conductance phase diagrams of the 2D DTQW processes with varying

lattice density p. For p ̸= 1.00, the diagrams have been averaged over 50 different lattice configurations. Conductance σxx has been evaluated in KWANT on 100-by-100 diluted square lattices with two infinite leads connected from left and right. White spots correspond to NaN values obtained during the computation. . . 57

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C.1 The power law fit (dashed lines) to the mean squared displacement of the two-dimensional quantum walk in a trivial topological phase, θ= (0, π/2)after(a) 1000 time steps with∆p= 0.1and (b-f )10000 time steps with ∆p= 0.01. . . 78 C.2 The power law fit (dashed lines) to the mean squared displacement

of the two-dimensional quantum walk in a trivial topological phase, θ = (π/8, π/2) after (a) 1000 time steps with ∆p = 0.1 and (b-f ) 10000 time steps with ∆p= 0.01. . . 79 C.3 The power law fit (dashed lines) to the mean squared displacement of

the two-dimensional quantum walk in a non-trivial topological phase, θ = (π/2, π/2) after (a) 1000 time steps with ∆p = 0.1 and (b-f ) 10000 time steps with ∆p= 0.01. . . 79 C.4 The power law fit (dashed lines) to the mean squared displacement of

the two-dimensional quantum walk in a non-trivial topological phase, θ = (π/2,2π/5) after (a) 1000 time steps with ∆p = 0.1 and (b-f ) 10000 time steps with ∆p= 0.01. . . 80

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LIST OF SYMBOLS AND ABBREVIATIONS

D_α generalized diffusion coefficient E, E(k) quasi-energy spectrum

Xt random walk time series

∆p lattice density spacing

α diffusion exponent

δ random walk step length

γ_n(t) Berry phase of the nth instantaneous eigenstate Cˆ

H Hadamard coin flip operator

Hˆ Hamiltonian operator (generic)

Hˆ_2D,per, Hˆ_2D,per(θ1, θ2) topologically equivalent Hamiltonian of a two-dimensional discrete-time quantum walk on a diluted lattice

Hˆ

2D, Hˆ

2D(θ₁, θ₂) topologically equivalent Hamiltonian of a two-dimensional discrete-time quantum walk on a regular lattice

Hˆ

Hal(k) Haldane Hamiltonian operator Hˆ

eff effective Hamiltonian operator

Iˆ identity operator

Mˆ measurement operator

Rˆ_y(θ) y-axis spin rotation by angle θ coin flip operator

Tˆ shift operator (generic)

Tˆ_1,per, Tˆ_2,per, Tˆ_3,per shift operators of a two-dimensional discrete-time quantum walk on a diluted lattice

Tˆ₁(k), Tˆ₂(k), Tˆ₃(k) shift operators of a two-dimensional discrete-time quantum walk on a regular lattice in quasi-momentum space

Tˆ₁, Tˆ₂, Tˆ₃ shift operators of a two-dimensional discrete-time quantum walk on a regular lattice

Uˆ propagator (generic)

Uˆ_1D propagator of a one-dimensional discrete-time quantum walk

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Uˆ_2D,per, Uˆ_2D,per(θ1, θ2) propagator of a two-dimensional discrete-time quantum walk on a diluted lattice

Uˆ

2D(k) propagator of a two-dimensional discrete-time quantum walk on a regular lattice in quasi-momentum space

Uˆ

2D, Uˆ

2D(θ₁, θ₂) propagator of a two-dimensional discrete-time quantum walk on a regular lattice

σˆ vector of Pauli matrices (σˆ_x, σˆ_y, σˆ_z) σˆ_x, σˆ_y, σˆ_z Pauli matrices

|↓⟩ spin up state vector

|ψ0⟩ initial quantum state vector

|ψ⟩ quantum state vector (generic)

|↑⟩ spin up state vector

A Landau gauge

d(k) Haldane wrapping vector

e_n nth standard basis vector

k quasi-momentum

n,n(k) wrapping vector

p momentum

x position

C Chern number

H Hilbert space

H_c coin space of the discrete-time quantum walk H_p position space of the discrete-time quantum walk

L set of allowed lattice sites

O average asymptotic performance

P set of missing lattice sites

S state space

T index space

i imaginary unit

∇ gradient operator

∂kx, ∂kx quasi-momentum partial derivatives

∂_t time derivative

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σxx longitudinal conductance

σ_xy transverse conductance

p lattice density

p_c percolation threshold

t time

1D one-dimensional

2D two-dimensional

AL Anderson localization

AQHE anomalous quantum Hall effect

ATM amorphous topological matter

CTQW continuous-time quantum walk

DTQW discrete-time quantum walk

FBZ first Brillouin zone

FQHE fractional quantum Hall effect

GCC giant connected component

H.c. Hermitian conjugate

IQHE integer quantum Hall effect

MSD mean squared displacement

PL percolation-induced localization

RMS root mean squared

TDSE time-dependent Schrödinger equation TKNN Thouless-Kohmoto-Nightingale-den Nijs

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1. INTRODUCTION

If one were to imagine the history of natural sciences as a star cluster of humanity’s greatest intellectual achievements and trace its constellations from past to present, the quantum theory would certainly shine as one of the brightest supernovae. In- deed, more than a century since its inception, quantum mechanics still currently stands as the most accurate model of reality on the smallest of scales that allows us to make experimentally verifiable predictions. Typically, a quantum system maps to some classical system in the correspondence limit, allowing us to build a connection between macroscopic objects and the nanoscale. As the semiconductor industry is steadily approaching the size threshold for the manufacture of devices that can be approximately described with classical theories, novel developments in the field of quantum computing are gaining more relevance with each passing day. In recent times, it has been discovered that discrete-time quantum walks, quantum coun- terparts of discrete-time random walks, can be used to realize universal quantum computation, implement efficient quantum search algorithms, and even develop se- cure quantum communication protocols. Due to such an abundance of potentially groundbreaking practical applications, we have decided to set discrete-time quantum walks on two-dimensional square lattices as the focal point of this thesis project.

Perhaps one of the most surprising features of discrete-time quantum walks is their connection to condensed matter physics. In particular, it turns out that it is possible to use suitably defined discrete-time quantum walk protocols as models of different topological phases of matter. The field of topological quantum matter originated in the 1980s from the discovery of integer quantum Hall state with exact quantization of conductance in two-dimensional electron gas subject to a strong magnetic field.

Since then, more exotic topological states of matter have been discovered, and three Nobel prizes in physics have been awarded directly in relation to these findings in the past 35 years. Unlike other phases of matter that are characterized by their local symmetry properties, topological phases of matter are quantified by a presence of global invariance that is robust to different forms of local perturbation. This natural tolerance to impurities and disorder makes materials that exhibit topological properties particularly appealing in the manufacturing context. Nowadays, such materials are used to design superconductors and insulators. Topological properties

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of matter also persist in the absence of spatial order, so the study of amorphous topological matter provides an exciting research platform. The discrete-time quantum walk protocol we consider in this work serves as a model of the Chern insulator, which we subsequently transform into a model of amorphous topological matter by random removal of lattice sites in the process of dilution.

In this thesis, we conduct a computational study of two-dimensional discrete-time quantum walks to determine whether the non-trivial topological phase protects their transport properties when site disorder is introduced into the lattice. To that end, we build a theoretical model for the propagation of discrete-time quantum walk on a diluted lattice. Quantum walks on regular lattices are characterized by ballistic spreading that is quadratically faster than the diffusive spreading of random walks.

By simulating discrete-time quantum walks up to 10000 time steps for different values of lattice density p, we discover that even the most minor of lattice pertur- bations lead to the immediate transition of quantum walk from ballistic to diffusive transport regime regardless of topological phase. We also learn that the non-trivial topological phase prevents the onset of Anderson localization in the quantum walk to an extent and lowers the localization threshold of the process to about p≈0.67, while quantum walks in trivial topological phase may localize as early as p≈ 0.99. Finally, we observe that quantum walks exhibit anomalous subdiffusive behavior in the vicinity of the localization threshold. Our results confirm the positive impact of the non-trivial topological phase on the robustness of spreading of the quantum walk, opening new possibilities for improving noise tolerance in systems that incor- porate quantum walks as parts of their pipeline. The simulations we have conducted set a new record for the number of numerically propagated two-dimensional discrete- time quantum walk steps, surpassing current state-of-the-art research by an order of magnitude.

This thesis is structured as follows. In Section 2 we outline the relevant theoretical background, showing how the seemingly separate notions of quantum walks, percolation theory, and topological quantum matter can be merged into a two-dimensional discrete-time quantum walk model of amorphous topological matter. In Section 3 we motivate our research by providing an overview of practical applications of discrete-time quantum walks in the fields of quantum computing, computer science, and cryptography. In Section 4 we conduct a literature review of experimental approaches for the realization of discrete-time quantum walks in trivial and non-trivial topological phases. In Section 5 we show the results of our numerical simulations and perform their analysis. We finish by providing a summary of our key findings and concluding remarks regarding the future research directions in Section 6.

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2. THEORETICAL BACKGROUND

The subject of this thesis is a synthesis of concepts that come from many diverse subfields of physics. In particular, our study builds an interface that unifies various ideas from statistical physics, complex systems theory, condensed matter physics and quantum theory. In this brief theoretical exposition, we shall trace a coherent path of reasoning from one relevant construct to another. Specifically, we first demonstrate how the notion of a discrete-time quantum walk in quantum theory emerges as a counterpart of the archetypal random walk omnipresent in statistical physics. Then, we employ percolation theory as a tool to investigate the diffusive properties of a discrete-time quantum walk on a lattice with site defects. Finally, we integrate the above with the study of the topological matter in condensed matter physics by observing the similarities between the tight-binding Hamiltonians of the two-dimensional discrete-time quantum walk and the Chern insulator model.

2.1 Discrete-time random walks

In order to understand discrete-time quantum walks and what makes them so in- teresting, we must first look at their classical physics analog, the random walk.

In this introduction to random walk, we borrow information from Chapters 1 and 2 of [1], as well as Chapters 3 and 4 of [2]. The random walk is a quintessential example of a stochastic process, which is a process defined as a set of random variables X that share the same state space S and are ordered in index space T, X = {X_t|X_t∈ S, t∈ T }. In the context of natural sciences, stochastic processes are typically used as models of various kinds of dynamics, so the index space of the process corresponds to the time dimension. Herein we make the distinction between discrete-time andcontinuous-time stochastic process. The index space of a discrete- time stochastic process is finite or countably infinite (e.g., T = N). In contrast, for continuous-time stochastic process, it is necessarily uncountably infinite (e.g., T =R). In this work, we restrict ourselves to the study of discrete-time dynamics, so it is important to keep this distinction in mind. The state space of a stochastic process can also be discrete (e.g., a graph) or continuous (e.g., a manifold).

The most basic example of a random walk, also known as the simple random walk

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(a) Unbiased 1D random walk (b) Unbiased 2D random walk

Figure 2.1. Six separate instances of (a) unbiased one-dimensional discrete-time random walk and (b) unbiased two-dimensional discrete-time random walk propagated to 10000 time steps.

is constructed as follows: consider a set ofT −1independent identically distributed random variables Z = {Z_t}^T_t=1⁻¹ which all share the state space S = {−1,1}. The values of elements of Z are assigned by uniform sampling from S with random selection probability p(Z_t = −1) = p(Z_t = 1) = _|S|¹ = ¹₂. With this sequence in place, the simple random walk of T steps is defined as a sequence of cumulative sums of Z starting from the origin, i.e. X = {X_t}^T_t=0⁻¹ such that X₀ = 0 and X_t = ∑︁t

i=1Z_i. The resulting random walk process is one-dimensional, with the corresponding state space spanning the integers Z as N → ∞. Similarly, one can define a two-dimensional random walk on Z² by constructing the setZ via uniform sampling of the state space S = {(−1,0),(0,−1),(1,0),(0,1)} and then following the same cumulative sum procedure to create the random walk X.

Generalizing further, the discrete-time random walk procedure can be just as easily extrapolated to a d-dimensional integer lattice Z^d by uniformly sampling elements for the set Z from the state space S = {e1,−e1,e2,−e2, . . . ,ed,−ed}, where en is the nth standard basis vector. Random walks constructed in such a manner are known as symmetric random walks since at every step direction at any axis can be randomly chosen. One can alternatively construct random walks by switching the step axis sequentially after each time step, with such walks known as alternating random walks. In the case when the sampling procedure follows some non-uniform selection probability distribution, we call the resulting random walk biased. Fig- ure 2.1 illustrates a few instances of unbiased 1D and 2D random walks. With basic definitions and examples of random walks in place, we outline some of their essential properties. It is easily shown that the expected location of an unbiased random walker after T time steps is the origin for an integer lattice of any dimensionality.

Indeed, since members of the set Z are all independent random variables, it follows

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that:

⟨Xt⟩=

⟨︄ _t

∑︂

i=1

Zi

⟩︄

=

t

∑︂

i=1

⟨Zi⟩. (2.1)

Now, from the uniform sampling ofZ_n, we may expand the summation further and get the final result as:

⟨X_t⟩=

t

∑︂

i=1

⟨Z_i⟩=

t

∑︂

i=1 d

∑︂

j=1

1

|S|(e_j+ (−e_j)) =

t

∑︂

i=1 d

∑︂

j=1

1

|S|0=0, (2.2) where 0 is the null vector. Discrete-time random walks follow the binomial probability distribution. In an asymptotic limit, probability distribution of the random walk approaches Gaussian distribution, which is a result one may obtain through straightforward application of the Central Limit Theorem.

The speed with which the walker moves away from the origin is another natural topic of investigation when studying the random walk process. In essence, it comes down to understanding the functional relationship between the number of time steps that the random walk has been iterated for and the displacement of the walker at a given time step. In particular, it can be shown that the root mean squared (RMS) displacement of the walker from the origin is proportional to the square root of the number of steps taken, i.e., √︁

⟨∥X_t∥²⟩ ∝ √

t. This property can be derived by looking at the mean squared displacement (MSD) and applying the independence of random variables property as before:

⟨∥X_t∥²⟩=⟨X_t^TX_t⟩=

⟨︄(︄ _t

∑︂

i=1

Z_i^T )︄ (︄ _t

∑︂

j=1

Z_j )︄⟩︄

=

⟨︄ _t

∑︂

i=1

Z_i^TZ_i

⟩︄

+ 2

⟨︄

∑︂

1≤i<j≤t

Z_i^TZ_j

⟩︄

=

t

∑︂

i=1

⟨Z_i^TZ_i⟩+ 2 ∑︂

1≤i<j≤t

⟨Z_i^TZ_j⟩

=

t

∑︂

i=1

⟨Z_i^TZ_i⟩+ 2 ∑︂

1≤i<j≤t

⟨Z_i^T⟩⟨Z_j⟩.

(2.3)

Noting that ⟨Z_i^TZi⟩ = 1 and ⟨Zi⟩ = 0 for all i, the second summation term in the last expression is reduced and we arrive to the desired conclusion:

⟨∥X_t∥²⟩=

t

∑︂

i=1

1 =t. (2.4)

Consequently, for an unbiased random walk with a step of unit length we get the equality√︁

⟨∥Xt∥²⟩=√

t. In case the walker makes steps of lengthδ, the step length

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is introduced into the relation as

√︁⟨∥Xt∥²⟩ ∝δ√

t. (2.5)

The linear relationship between MSD and time is essential to our understanding of the random walk process, as it places a fundamental limit on the traversal capabil- ities of the walker. Stochastic processes that spread out according to this proportionality law are said to bediffusive. To summarise, unbiased discrete-time random walks evolve in a diffusive fashion and form a Gaussian distribution centered at the origin in the asymptotic limit. As it turns out, these properties are very different for discrete-time quantum walks, and that difference serves as one of the primary motivators for this study.

2.2 Discrete-time quantum walks

With background information on discrete-time random walks in place, we are ready to set the stage for the main target of this study, the quantum walk. The emergence of quantum theory in the early 20th century initiated one of the most drastic paradigm shifts in physics. Due to the fundamental "fuzziness" of reality as precip- itated by Heisenberg’s uncertainty principle, we have come to understand physical objects as probabilistic entities that only take definite shape when measured. This thesis follows the Hartee atomic unit convention, so the reduced Planck’s constant ℏ, elementary chargee, Bohr radius a₀ and electron rest mass m_e are omitted from the equations to follow.

2.2.1 Basics of quantum theory

Before diving into quantum walks, let us give a brief overview of the main postulates of the quantum theory. This overview is based on Chapters 1 and 2 of [3]. The state of any quantum system is described by its corresponding wavefunction |ψ⟩ (using the bra-ket notation). Wavefunctions are mathematical objects that reside in a construct known as the Hilbert space H, a vector space endowed with the inner product operation. The time evolution of the wavefunction is governed by thetime- dependent Schrödinger equation (TDSE):

Hˆ |ψ⟩= i∂_t|ψ⟩, (2.6)

where Hˆ is the Hamiltonian, an operator that represents the total energy of the system. Given an initial state|ψ₀⟩, solution to this partial differential equation can be expressed as

|ψ⟩=e^−iH^ˆ^t|ψ₀⟩=Uˆ |ψ₀⟩, (2.7)

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where the unitary time evolution operator Uˆ = e^−iH^ˆ^t is known as the propagator. The framework of quantum theory that utilizes the propagator to evolve the time- dependent wavefunction state is known as the Schrödinger picture. For a system in a closed environment, the total probability ⟨ψ|ψ⟩ is conserved and is equal to 1.

Degrees of freedom of the wavefunction are dictated by the nature of the system.

For our purposes, it will be sufficient to consider the spin-¹₂ systems with position (or momentum) degrees of freedomx(or p) and a spin degree of freedoms. Spin-¹₂ degree of freedom is spanned by the spin up and spin down basis vectors, |↑⟩ and

|↓⟩, commonly expressed in C² as

|↑⟩=

⎡

⎣ 1 0

⎤

⎦ and |↓⟩=

⎡

⎣ 0 1

⎤

⎦. (2.8)

Using tensor product notation, one may rewrite the wavefunction with degrees of freedom separated, e.g. for a spin-¹₂ system we can write |ψ⟩=|x⟩ ⊗ |s⟩.

Naturally, given the state of a system, we would like to make measurable predictions by operating on it in some fashion. In quantum theory, parameters of the system that a physical device can measure are calledobservables. Mathematically, they are represented by Hermitian operators that act on the wavefunction and transform it in some way. Given an observable represented by an operator Oˆ, one may measure its expectation value (the average outcome of the measurement of an observable) as:

⟨Oˆ⟩=⟨ψ|Oˆ |ψ⟩. (2.9)

From any observable, one may solve the eigenstates {|λ_i⟩} and the corresponding eigenvalues {λi} using the eigenvalue equation:

Oˆ|λi⟩=λi|λi⟩. (2.10) Specifically, eigenvalues of the Hamiltonian Hˆ correspond to the allowed energies of the system, and are thus called eigenenergies. Some well-known examples of observables include positionxˆ_i, momentumpˆ_i, and spin components σˆ_i, represented byPauli matrices:

σˆ1 =

⎡

⎣ 0 1 1 0

⎤

⎦, σˆ2 =

⎡

⎣ 0 −i i 0

⎤

⎦, and σˆ3 =

⎡

⎣ 1 0 0 −1

⎤

⎦. (2.11) While the mathematical representation of observables depends on the wavefunction basis, the operators mentioned above must always obey the following commutation

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relations:

[xˆ_i, pˆ_j] = iε_ijIˆ and [σˆ_i, σˆ_j] = 2iε_ijkσˆ_k, (2.12) whereIˆis the identity operator,ε_ij, ε_ijkare the Levi–Civita symbols. In Schrödinger picture, observables are time-independent, but one may also formulate quantum theory by looking at state evolution as an action of time-dependent operators acting on a time-independent state wavefunction. This approach is known as theHeisenberg picture.

Since quantum theory is formulated on vector spaces, choice of basis plays a crucial role in any given problem. State wavefunctions are typically represented in the basis spanned by the eigenstates of the observables. In particular, if one has a maximal set of mutually commuting observables, the set of their simultaneous eigenstates {λ_i} spans the Hilbert space of the system. This fact is expressed by the completeness relation:

∑︂

i

|λ_i⟩ ⟨λ_i|=Iˆ. (2.13)

The term|λ_i⟩ ⟨λ_i| is also known as theprojection operator, as it determines the |λ_i⟩ component of any state |ψ⟩. Applying the completeness relation, we can re-express the state |ψ⟩ as

|ψ⟩=Iˆ|ψ⟩= (︄

∑︂

i

|λ_i⟩ ⟨λ_i| )︄

|ψ⟩=∑︂

i

|λ_i⟩ ⟨λ_i|ψ⟩=∑︂

i

⟨λ_i|ψ⟩ |λ_i⟩. (2.14) Denoting α_i = ⟨λ_i|ψ⟩, we conclude that one may express the wavefunction as a linear combination of eigenvalues of the observables:

|ψ⟩=∑︂

i

α_i|λ_i⟩. (2.15)

Coefficient set{αi}then defines the components of the state|ψ⟩in the basis spanned by{λ_i}. Conveniently, set of eigenstates of the Hamiltonian operatoralways forms the basis for the Hilbert space of a given system. The procedure of finding the basis set spanned by the operator is also known as operatordiagonalization, and in case of the Hamiltonian it provides powerful insights into the nature of a quantum system. In quantum theory, one most often deals with computing expectation values, solving eigenstates, performing diagonalizations and projecting wavefunctions from one state representation to another. This exposé serves as a summary of these aspects of the field.

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2.2.2 Quantum walk protocols for 1D and 2D

Finally, it is time to learn about quantum walks. The term "quantum random walk", or more briefly "quantum walk", was introduced in 1993 in the work of Aharonov et al [4]. Like random walks, quantum walks can be broken down into two main categories: continuous-time (CTQW) and discrete-time quantum walks (DTQW).

We will concern ourselves with the latter, and unless otherwise stated, any mention of term "quantum walk" herein refers specifically to DTQWs. The basis states of a discrete-time quantum walk are described by the position space H_p of the walkers, and the space Hs corresponding to the internal "coin" degree of freedom, so the complete Hilbert space of the quantum walker may be written as H = H_p ⊗ H_s. Typically one considers DTQWs on graph structures, and in this thesis we will focus on square integer lattice, H_p = Z². The coin degree of freedom in DTQW enables the conditional translation of the walker. Given that basis states {|↑⟩,|↓⟩} of a spin-¹₂ system can be naturally interpreted as the heads and tails states of a coin flip, we shall use them as basis for H_s.

In essence, the step of a DTQW protocol for any choice of position and coin space follows a sequential application of two main operations:

1. Coin flip;

2. Coin-dependent walker translation (also referred to as shift).

Here, the coin flip is represented by an operator that acts on the coin space Hs. In order for the total spin of the quantum walk process to be conserved, this operator is chosen to be unitary. We will be utilizing the Hadamard coin Cˆ_H and the coin Rˆ_y(θ)that corresponds to the spin rotation iny-direction by an angleθin this study, defined as:

Cˆ_H = 1

√2

⎡

⎣ 1 1 1 −1

⎤

⎦ and Rˆ_y(θ) =e⁻¹²^iθσ^ˆ^y =

⎡

⎣

cos(θ/2) sin(θ/2)

−sin(θ/2) cos(θ/2)

⎤

⎦. (2.16) The shift operatorTˆ depends on the graph structure on which the DTQW is propagated. The purpose of this operator is to "split" the walker, so that the probability density of one of its spin components moves in one direction, while the probability density of the other spin component moves in the opposite direction. The continuous-time counterpart of DTQWs is formed by utilizing the infinitesimal gen- erator matrix of the continuous-time random walk process as the Hamiltonian of the corresponding continuous-time quantum walk process. As a result, formulation of CTQWs does not require the introduction of coin space or coin flip operators.

In this work we focus only on DTQWs mainly due to this drastic difference in the formalism.

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Perhaps the best way to learn something about the properties of a DTQW is by means of an example. The simplest possible DTQW procedure may be defined for a spin-¹₂ particle on a 1D integer lineZ, s.t. H =H_p⊗ H_s =Z⊗C² with the walker state given as |ψ⟩=|x⟩ ⊗ |s⟩. In the study of 1D DTQW we follow a review article authored by Kempe [5]. We shall use the Hadamard coin as the coin flip operation and we will define the shift operatorTˆ as

Tˆ = ∑︂

x∈Z

[︂|x+ 1⟩ ⟨x| ⊗ |↑⟩ ⟨↑|+|x−1⟩ ⟨x| ⊗ |↓⟩ ⟨↓|]︂

. (2.17)

This operator splits the walker such that its spin up component moves to the right, while its spin down component moves to the left. The propagator Uˆ_1D for this quantum walk process can then be written simply as

Uˆ

1D=Tˆ ·(︂

Iˆ⊗Cˆ

H

)︂

. (2.18)

The state of the system after t time steps |ψ_t⟩ would then be |ψ_t⟩ = Uˆ^t

1D|ψ₀⟩. In order to completely define a specific instance of a quantum walk we still need to specify the initial state |ψ₀⟩, which we will set to

|ψ₀⟩=|0⟩ ⊗ 1

√2(|↑⟩ −i|↓⟩). (2.19) Figure 2.2 shows the probability density function of such a 1D DTQW up to 1000 time steps. From this figure we can see that this particular choice of initial state results in a symmetrical probability density distribution for the 1D DTQW. Change in the initial state would result in a heavier tail in one direction of the probability distribution or another. However, the most important properties of the process would remain unaffected.

In addition, we may observe significant differences in the behavior of this quantum walk compared to the random walk. While random walks are centered at the origin, the DTQW follows a roughly uniform distribution in the vicinity of the walk origin, with the maxima of the probability density accumulated at the edges of the walk.

One may show (and confirm by looking at Fig. 2.2) that the distribution of the 1D DTQW aftertsteps is contained entirely in the region[︂

−^√^t₂,^√^t₂

]︂. Most importantly, evaluation of RMS displacement of 1D DTQW demonstrateslinear proportionality between the standard deviation of the walk and the number of time steps, i.e.

√︁⟨x²⟩ ∝t. (2.20)

This property indicates that a quantum walk propagates quadratically faster than a random walk! To illustrate this, Figure 2.3 provides a comparison between time

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(a) 1D DTQW after 10 time steps (b) 1D DTQW after 100 time steps

(c) 1D DTQW after 1000 time steps

Figure 2.2. Probability density of a 1D DTQW after 10, 100 and 1000 time steps. In these figures the probability densities at the odd-numbered positions are all equivalently zero and as such have been omitted from the plots to make them easier to read.

step dependent RMS displacements of the two processes. Consequently, processes that employ random walks as a part of their algorithmic pipeline may potentially receive a significant performance boost by replacing them with quantum walks. The linear relation encoded by Eq. (2.20) indicates that the transport mechanism of the 1D DTQW process is ballistic in nature, in contrast to the diffusive transport of a random walk.

Upon some observation, one may note that the DTQW protocol provided above is, in fact, deterministic (in the quantum sense of the word), i.e. evolution of the walker state at any point in time is governed by a well-defined propagator, and knowledge of the initial state is sufficient to infer the walker state at any other point in time. This is in stark contrast to the classical random walk process, which is stochastic by definition and consequently non-deterministic, meaning that at any time step, the walker’s location is independent of its location at the preceding time step. So how exactly do we recover a random walk process from a quantum walk?

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Figure 2.3. Comparison of RMS displacement as a function of time step between a discrete-time quantum walk and a discrete-time random walk iterated over the first 100 steps. Statistical averaging of the random walk has been performed over 10000 instances.

In quantum theory, stochastic element comes into play during the measurement, when an observable takes a definite value as a consequence of a process known as wavefunction collapse. Denoting the measurement operator as Mˆ, we may apply it after any step of the quantum walk process, leading to the observation of a spin-¹₂ particle in a definite spin up or down state at a definite location x on the integer line. The walker wavefunction immediately prior to application of Mˆ describes the probability distribution of these observable states.

With the introduction of measurement operatorMˆ, we can perform a little thought experiment to establish the connection between discrete-time quantum walks and random walks. We return to the 1D DTQW example studied above and note that the coin operatorCˆ

H isbalanced, i.e. it splits the amplitudes of spin up and down walker states evenly with weighting factors of identical magnitude, ^√¹₂. Now, let’s trace the evolution of initial state |ψ₀⟩ defined in Eq.(2.19) after sequential application of Cˆ

H →Tˆ →Mˆ:

|0⟩ ⊗ 1

√2(|↑⟩ −i|↓⟩)−−→ |0⟩ ⊗^C^ˆ^H 1 2 [︂

(1−i)|↑⟩+ (1 + i)|↓⟩]︂

Tˆ

−→ 1

2(1−i)|1⟩ ⊗ |↑⟩+1

2(1 + i)|−1⟩ ⊗ |↓⟩

Mˆ

−→either |1⟩ ⊗ |↑⟩ or |−1⟩ ⊗ |↓⟩,

(2.21)

where probability of measuring state |1⟩ ⊗ |↑⟩ is |¹⁻ⁱ₂ |² = ¹₂ and probability of mea-

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suring state|−1⟩ ⊗ |↑⟩is|¹⁺ⁱ₂ |² = ¹₂ as well. Consequently, we note that by making a measurement right after the first step of the 1D DTQW with the balanced coin flip operatorCˆ_H, we get a process identical to a single step of an unbiased discrete-time random walk. Generalizing further, one can show that by making a measurement after every step of a discrete-time quantum walk, one obtains a discrete-time random walk! Whether the recovered random walk process is biased or unbiased depends on whether the coin flip operator of the associated quantum walk is balanced or unbalanced. In any case, the act of measurement slows down the quantum walk process asymptotically and connects it to the random walk. Conversely, one might also say that the quantum interference generated as a result of keeping the quantum walker in an unobserved state is the most likely cause of ballistic propagation of the process. For further reading, the work by Andrade et al [6] formally establishes the equivalence conditions between random and quantum walks on arbitrary graphs.

Now let us make a dimensional leap and present a "clean" version of the two- dimensional DTQW process that we will be analyzing in greater detail throughout the course of this thesis. Herein we present a quantum walk protocol outlined by Kitagawa in [7]. We consider a spin-¹₂ quantum walker moving on a square integer lattice Z² denoted by a state |ψ⟩ =|x, y⟩ ⊗ |s⟩. This walker would then belong to a Hilbert space H =Z² ⊗C². Additionally, the procedure is parametrized by two spin rotation angles, θ₁ and θ₂. A single step of this walk consists of three different sequential applications of the spin filp + shift operations, making the 2D DTQW protocol a six step procedure, which we can formulate as follows:

1. Rotate the spin state of each walker iny-direction by angleθ₁ via coin operator Rˆ

y(θ₁).

2. Shift the |↑⟩ components of each walker one lattice site to the right and up and the |↓⟩ components one lattice site to the left and down via operator Tˆ₁ defined as

Tˆ

1 = ∑︂

(x,y)∈Z²

[︂|x+ 1, y+ 1⟩ ⟨x, y| ⊗ |↑⟩ ⟨↑|+|x−1, y−1⟩ ⟨x, y| ⊗ |↓⟩ ⟨↓|]︂

. (2.22) 3. Rotate the spin state of each walker iny-direction by angleθ₂ via coin operator

Rˆ_y(θ2).

4. Shift the|↑⟩components of each walker one lattice site up and the |↓⟩components one lattice site down via operator Tˆ₂ defined as

Tˆ₂ = ∑︂

(x,y)∈Z²

[︂

|x, y+ 1⟩ ⟨x, y| ⊗ |↑⟩ ⟨↑|+|x, y−1⟩ ⟨x, y| ⊗ |↓⟩ ⟨↓|]︂

. (2.23)

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5. Rotate the spin state of each walker iny-direction by angleθ1 via coin operator Rˆ

y(θ₁).

6. Shift the |↑⟩ components of each walker one lattice site to the right and the

|↓⟩ components one lattice site to the left via operatorTˆ

3 defined as Tˆ

3 = ∑︂

(x,y)∈Z²

[︂|x+ 1, y⟩ ⟨x, y| ⊗ |↑⟩ ⟨↑|+|x−1, y⟩ ⟨x, y| ⊗ |↓⟩ ⟨↓|]︂

. (2.24)

The propagator of this 2D DTQW process is then given by Uˆ_2D(θ₁, θ₂) =Tˆ₃·(︂

Iˆ⊗Rˆ_y(θ₁))︂

·Tˆ₂·(︂

Iˆ⊗Rˆ_y(θ₂))︂

·Tˆ₁·(︂

Iˆ⊗Rˆ_y(θ₁))︂

. (2.25) It is worthwhile to mention that a modified version of this protocol which excludes steps 1 and 2 (i.e. the diagonal transition) is another common choice when studying 2D DTQW typically referred to as a split-step quantum walk. Throughout the course of this thesis, we will learn how parametrization of the propagatorUˆ

2D with rotation angles θ₁ and θ₂ enables control of robustness and diffusive properties of this quantum walk. As before, state after t steps given the initial walker state|ψ₀⟩ can be evaluated via |ψ_t⟩ = Uˆ^t

2D|ψ₀⟩. We shall fix the initial walker state in the same way as the 1D case:

|ψ₀⟩=|0,0⟩ ⊗ 1

√2(|↑⟩ −i|↓⟩). (2.26) Figure 2.4 illustrates the iteration of this protocol up to 256 time steps. Similarly to the 1D case, we observe that a considerable amount of walker probability density is located at the edges of the density distribution, which are now spread out in an elliptical fashion. Additionally, the inner pattern of the distribution resembles as six-edged star which is aligned with and stretched along they=xline. The presence of this six-fold symmetry is a consequence of the construction of the propagatorUˆ_2D. Specifically, it represents the six directions in which the walker moves during the iteration of one step: Tˆ

1 moves the walker along the y =x diagonal, Tˆ

2 moves the walker along the y-axis and Tˆ₃ moves the walker along the x-axis. One may note that the 2D DTQW on a clean square lattice propagates linearly with the time step, making the transport process ballistic as in 1D case. We will validate and analyze this observation in further detail in Sec. 5.2.

As we conclude this introduction to quantum walks, it is worthwhile to point out that the quantum walk protocols provided above are not the only ways in which the DTQWs can be defined. Generally, one does not have to restrict the position space of the walk to just square lattices, as quantum walks may be studied on arbitrary graphs with an appropriate formulation of shift operators. To name just

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(a) 2D DTQW after 16 time steps

(b) 2D DTQW after 64 time steps

(c) 2D DTQW after 256 time steps

Figure 2.4. Probability density of a 2D DTQW after 16, 64 and 256 time steps with rotation parameters θ₁ =θ₂ =π/2.

a few, studies of quantum walks on general graphs [8], carbon nanotube structures [9], graphene structures [10] and triangular lattices [11, 12] have previously been conducted. Choice of the coin space may also be quite arbitrary, and while we have selected spin-¹₂ basis states to form the walk coin space due to its simplicity and direct mapping to the simple random walk process, systems of any spin could be used as quantum walkers. To that end, one does not have to use spin states to form coin space at all, any finite-dimensional quantity of a system would work just as well. Coin flip operators are also free to be any unitary mapping from the coin space to itself, soCˆ

H andRˆ

y(θ)are just a couple of examples of such operators from an infinitude of possibilities. It is also possible to define a DTQW process where the coin flip operator is systematically or randomly switched to some other one after every time step. Such quantum walks are known as inhomogeneous quantum walks and as a matter of fact, the 2D DTQW protocol we have described may be considered inhomogeneous when we set θ₁ ̸= θ₂. Finally, incorporation of periodic or aperiodic applications of the measurement operator in a quantum walk enables transition between random walk and quantum walk behaviors.

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2.2.3 Effects of decoherence and disorder

In the preceding section, we have outlined the fundamental reason for ballistic transport of a quantum walk process: the interference effects caused by the coherent spread of the quantum walk wavefunction. In a realistic physical scenario, where the quantum system is coupled to the external environment, the coherency of the quantum walk may be affected in a controlled or unwanted way. We have already encountered one of the approaches for introducing decoherence in the quantum walk:

the application of intermediate measurements in the walk protocol. By measuring the quantum walk after each step of the protocol, we revert it to the classical random walk, resulting in a shift from ballistic to diffusive transport. A study by Romanelli et al [13] has shown that for a Hadamard 1D DTQW on an integer line, any form of aperiodic measurement eventually makes the quantum walk diffusive. In addition to measurement-induced decoherence, quantum walks can be affected by different forms of disorder in the system. A method of adding disorder to the walk process through the application of random phase shifts to the coin flip operator has been employed by Kosik et al [14], resulting in diffusive behavior for Grover and Fourier coin DTQWs inZ^d. A similar approach has been taken by Asboth and Edge [15] and applied to a family of Hadamard coin-based 2D DTQWs, also resulting in diffusion.

Overall, the three most general coin operator-based approaches to introduction of disorder in the DTQW are: addition of temporal disorder by changing the coin flip operator every time step [16], addition of spatial disorder in the coin flip operator [17] by making it position-dependent and addition of spatio-temporal disorder through assignment of different coin flip operators for each position and time step [18].

So far, we have only mentioned that some quantum walks revert to diffusive transport akin to random walks in disordered environments. However, decoherence and disorder can also lead to a complete halt of the walk process, known aslocalization. More specifically, in a localized state, the RMS displacement of the quantum walk process becomes constant with respect to the time step, and the probability density function of the quantum walk decays exponentially away from the walk origin:

⟨ψ|ψ⟩ ∼e^{−∥x∥/ξ}, (2.27)

whereξis known as the localization length. The phenomenon of wavefunction localization due to the presence of uncorrelated onsite disorder in quantum systems was first discovered by Anderson in 1958 [19]. Nowadays, this particular type of localization mechanism is known asAnderson localization (AL). By varying the strength of the disorder, one may induce a transition in states of a quantum system from delo- calized to localized behavior, also known as Anderson transition. Examples of such

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transitions include metal-insulator transitions and topological quantum-Hall type transitions. A comprehensive review of Anderson transitions and their classification has been compiled by Evers and Mirlin [20]. A good example of AL in DTQW can be found in [21], where Vakulchyk et al studied a 1D DTQW with a generalized spin-¹₂ coin flip operator parametrized by anglesϕ, ϕ1, ϕ2 and θ:

Cˆ =e^iϕ

⎡

⎣

e^iϕ¹cosθ e^iϕ²sinθ

−e^−iϕ²sinθ e^−iϕ¹cosθ

⎤

⎦. (2.28)

They have discovered that introduction of disorder in angles θ, ϕ and ϕ2 leads to AL of the quantum walk eigenstates. AL in quantum walks has also been detected experimentally by Schreiber et al [17] who observed localization after introducing static disorder in a photonic implementation of 1D DTQW. An experimental setup for the study of Anderson transitions has been realized by Ghosh [22] through the implementation of a 1D DTQW using superconducting qubits on a lattice with controllable disorder introduction methods. A study by Chandrashekar [18] provides evidence that the presence of temporal and spatio-temporal disorder induces AL in a particular family of 1D and 2D DTQWs, but also counterintuitively enhances the spin-position entanglement. This discovery hints that in some contexts, one may benefit from state localization. Moving on to 2D DTQW, Asboth and Edge [15] have found that split-step quantum walks generically exhibit AL when phase disorder is introduced.

In some cases, quantum walks may localize without any disorder whatsoever. We have previously mentioned that the choice of the initial state of the walker affects the final probability density function, and occasionally this choice affects its asymptotic behavior as well. For instance, it has been shown by Lyu et al [23] that Grover walks on honeycomb networks have a considerable probability of localization when the initial walker state is chosen at random. Similar conclusions have also been drawn for Grover coin DTQW on an integer line [24] and a triangular lattice [25].

Conversely, for Fourier coin based quantum walks, it has been mathematically proven that localization does not occur for any choice of the initial state in Z^d [26]. While there are many ways to disturb a DTQW system, most studies of its effects uncovered by the author of this work are restricted to particular instances of DTQW protocol which were in turn confined to some specific graph structures and some specific coin operator families. Consequently, no truly general results or theorems regarding disorder-induced properties that apply to all quantum walks have been discovered as of now.

All of the results regarding the emergence of diffusion and localization in quantum walks presented above share a common trait: the disorder employed to perturb the

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walk process would always act on the coin flip operator of the walk. After noting this shared commonality in all the presented studies, a natural question emerges: what will happen to the quantum walk if, instead of distorting the coin flip operator, we distort the coin-dependent translation operator? The most straightforward way to answer this question is to consider a quantum walk process defined on an imperfect, or diluted lattice. The missing lattice sites play a role of disorder in position space rather than the coin space, so their impact on the overall properties of the walk can be directly studied by modifying the translation operators of the quantum walk to take forbidden transitions into account. In this thesis, our primary focus is to study the effects of lattice percolation on various aspects of the 2D DTQW. To that end, we require some understanding of the effects of percolation on graph structures, which is precisely the topic of the following section.

2.3 Percolation theory

Percolation theory is a field of research that originally emerged as a branch of statistical physics. It has recently blossomed into an increasingly relevant topic of inquiry due to its applications in the natural sciences and the more modern data-driven context of complex networks. The main subject of study of percolation theory is relatively easy to define. Consider a graph G= (V, E), where V is the set of nodes representing possible occupation sites, and E is the set of edges that connect these nodes. Then, introduce "disorder" into the graph by randomly marking the nodes as "occupied" with probability p and leaving them in an "unoccupied" state with probability 1−p. Percolation theory studies the statistical properties of clusters formed by the occupied nodes and how they change with occupancy probability p. In this context, when we say "statistical", we mean properties averaged over all possible lattice configurations for a given value of p, since percolation is an inherently random phenomenon. Figure 2.5 provides an illustrative example of effects different values of p have on the 50-by-50 lattice in Z². This description of the percolation problem is also known assite percolation. One can also formulate abond percolation problem, where occupancy is instead assigned to the edges of the graph, but we shall focus on the site percolation. In the study of complex networks, percolation theory is often used to assess the robustness of the network in response to the removal of nodes or edges. Percolation theory is essential in the study of spreading phenomena, such as epidemics in virology or diffusion in disordered media. Coming back to discrete-time quantum walks, we are most curious about how the transport properties of the walk change as we distort the regularZ² lattice further and further.

This section provides an overview of concepts and ideas in percolation theory that will be relevant to our study, mostly following Chapters 1, 2, 6 of [27] and Chapters 9, 12 of [28].

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(a) p= 1 (b) p= 0.75

(c) p= 0.5 (d) p= 0.25

Figure 2.5. Effects of site percolation with decreasing site occupancy density p on the structure of 50-by-50 square lattice Z².

2.3.1 Phase transitions and order parameters

Let us start this brief exposition by considering the effects of percolation on the square lattice Z² with edges connecting only the nearest neighboring sites. We then introduce site percolation into this lattice, such that the occupation probability of the sites isp. As a rudimentary analogy, we may imagine this lattice to correspond to a compound with impurities, where electrons may flow freely only from one occupied site to another along the lattice edges (and are not allowed to tunnel where no edge is present). Now suppose we can measure conductance across this compound to determine whether it exhibits a metallic (with non-zero conductance) or an insulator-

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like (with zero conductance) behavior. In this very simplistic picture, we infer that for the compound to have a non-zero conductance, its lattice must have a connected component that extends from one of its ends to another, as otherwise there is no path an electron can take to traverse the entire length of the compound at all. The connected component in the percolated lattice that has a size of the same order as the lattice itself is known as a giant connected component (GCC). Suppose we want to find how many impurities we can dope the compound with and still have non-zero conductance. This question is equivalent to finding the smallest value of pfor which GCC exists in the lattice populated with an increasing amount of missing sites.

Quite surprisingly, it turns out that for a particular choice of lattice geometry, this critical value of p, known as the percolation threshold p_c is universal for any system size and length scale! Specifically, for square lattice with nearest-neighbor edges, percolation threshold comes out to bep_c≈0.5927. Below this critical value, the lattice breaks down into fragmented and disjointed sets of smaller connected components, leading to a zero conductance state following our physical analogy.

This abrupt qualitative change in behavior induced by the effects of percolation is a quintessential example of aphase transition. Let us think about the effects this phase transition has on the transport inside the lattice. Belowp_c, GCC is not formed in the lattice, so regardless of where the random (or quantum) walker is initially, it will be trapped in one of the smaller clusters, leading to localization. Thus, the percolation threshold provides an upper bound to the amount of disorder that can be introduced in the lattice before the walker has to localize. While Anderson localization is an inherently quantum effect, this percolation-induced localization (PL) is classical in nature and, as such, serves as a different localization mechanism.

A large portion of modern statistical physics deals with the quantification and cat- egorization of phase transitions that occur in nature. In physics, the phase change is attributed to symmetry breaking. For instance, one may consider the difference between ice and water. Ice possesses a discrete rotational symmetry and a discrete translational symmetry due to the periodic nature of its lattice. On the other hand, water as a statistical ensemble of all of its possible configurations is self-similar un- der any form of translation or rotation, hence possessing complete rotational and translational symmetry. To that end, one needs to have a quantitative measure that could be used to detect a symmetry breaking and subsequent phase transition. This measure in statistical physics is known as the order parameter. The order parameter of a system forms a field that serves as a mapping from physical space to order parameter space. For instance, in the percolated lattice example considered above, the density of GCC (i.e., ratio of the number of sites in GCC to the total number of occupied sites) can be used as an order parameter, as in large lattices it is close to one for p > p_c and to zero for p < p_c. As another example, in magnets, one

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may use the direction of magnetization M(x) as the order parameter that defines an order parameter space mappingxto a location on a sphereS². Topological phase transitions in the quantum matter can also be quantified with the order parameter approach, and in Sec. 2.4 we will learn about that in greater detail.

2.3.2 Diffusive regimes and power law scaling

We may recall from previous sections that the MSD of random walks was shown to grow linearly with the time step on a clean lattice, while the MSD of quantum walks had a quadratic proportionality to the time step. Both of these behaviors can be generally classified using the power law:

⟨∥x(t)∥²⟩=D_αt^α, (2.29)

where D_α is called the generalized diffusion coefficient, and α is the diffusion exponent. Random walks and quantum walks on a regular lattice obey Eq. (2.29) with α = 1 and α = 2 respectively. Processes that follow the power law with α = 1 correspond to a standard linear relation characteristic of classical diffusion. On the other hand, processes that spread according to the power law with α ̸= 1 exhibit the phenomenon known as anomalous diffusion. Within the anomalous diffusion regime, one may further distinguish between subdiffusion (α < 1) and superdif- fusion (α > 1). Consequently, ballistic transport in the quantum walk is a good example of a superdiffusive process. Localization in the power law framework occurs when the exponent α converges to zero, as in that case, we get a constant MSD.

One striking feature of the processes that obey some form of a power law is the fact that they are scale invariant, i.e. they look similar to each other on all scales as long as one observes them from a distance where individual steps of the process are nearly indistinguishable. Figure 2.6 shows a simple comparison between the spreading phenomenon in classical and anomalous diffusive regimes. Of particular interest to us are the effects site percolation may have on the diffusion exponent of the associated walk process.

We shall consider the random walk process in this section. There are two main approaches in which the missing sites of the disordered lattice can be incorporated into the random walk protocol. First, the walker can be programmed to ignore them during the random step process, which is accomplished by setting the probability of jumping to the missing sites to zero and then readjusting the probabilities of going to the present neighboring sites so that they still sum up to unity. Second, one can keep the probabilities the same as in the case of the clean lattice but force the walker to stay in place whenever it is about to visit the missing site. In either case, for random walk on a disordered lattice, it can be shown that for p far above p_c one observes

Discrete-time quantum walks in two-dimensional amorphous topological matter