Bond percolation on the square lattice - Quantum walk on square lattices

4.3 Quantum walk on square lattices

4.3.2 Bond percolation on the square lattice

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10^-1 10⁰ 10¹ 10²

t

10^-8 10^-6 10^-4 10^-2 10⁰

P

(t)

quantum classical

Figure 4.9: The return probabilities, P_r(t) and P_r^cl(t), for the quantum and classi-cal random walks on a 2D lattice of size 33². The blue dashed line, proportional to 1/t², indicates the decay behavior of the quantum probability in the intermediate range, where the classical probability decays as P_r^cl(t)⇠ 1/t before it converges to P_r^cl(1) = 1/N

contrast, the probability P_r(t) for the quantum case does not decay to a constant value at t ! 1, but oscillates over time. In the intermediate range (between t ⇡ 0.5 and t ⇡ 100) the classical probability decays algebraicly as Pr^cl(t)⇠ 1/t, while the quantum probability decays faster (Pr(t) ⇠ 1/t²) before it oscillates around the long time average.

4.3.2 Bond percolation on the square lattice

Percolation models, introduced by Broadbent and Hammersley [32], are math-ematical models of random media. The analysis of di↵usion (or transport) in higher dimensional disordered media by means of random walk on a percolation system was suggested by de Gennes [33], for which he coined the term ”ant in the labyrinth”.

Consider the square latticeZ²(infinitely large); each edge between two nearest-neighbor vertices is present with probability p and absent with probability 1 p

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4.3 Quantum walk on square lattices

(a) bond percolation (b) site percolation

Figure 4.10: Illustrations of (a) bond percolation and (b) site percolation on two-dimensional square lattices with open boundary conditions. There are two clusters, denoted by red and blue, on each lattice.

[see Fig. 4.10(a)]. This model is called a bond percolation model (In analogy with the bond percolation model, one defines the site percolation model in which vertices (sites) are randomly removed with probability 1 p [see Fig. 4.10(b)]).

A ”cluster” on this graph is a set of connected vertices. As the occupation prob-ability, p, is increased from zero, the average or typical clusters become larger, both in terms of the number of vertices (mass) and geometric size. The graph is said to be ”percolate” if there is an infinite cluster containing the origin; if the graph is translation-invariant there is no di↵erence between the origin and any other vertex. There exists a critical value pc (called the percolation threshold ) for the occupation probability p such that all clusters in the graph are finite when p < pc, but there exists an infinite cluster when p > pc. The change in behavior while crossing the percolation threshold is an example of a phase transition. For the square bond percolation the percolation threshold is exactly known, pc = 1/2 [34]. On a finite lattice, ”infinite clusters” defined above correspond to spanning clusters that touch opposite boundaries. This assumption is valid for large lattice sizes and preferably when periodic boundary conditions are imposed.

The dilute lattice provides a constrained arena for random walk. The ran-dom walker (the ”ant”) can only move within a cluster, but not move between disjoint clusters. There are two interesting aspects for studying random walk in percolation. First, percolation graphs are quenched disordered media for random

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(a) p = 0.49 (b) p = pc= 0.50 (c) p = 0.51

Figure 4.11: Phase transition in bond percolation on a two-dimensional square lattice of size 512⇥ 512 with periodic boundary conditions. Only the largest clus-ter at each occupation probability p is shown. A clusclus-ter that spans to opposite boundaries (a percolating cluster) appears only at p pc.

walks. Second, percolation clusters at p pc are fractal objects¹; they are irreg-ular geometric objects with an infinite nesting of structure at all scales and with noninteger dimensions; for example, at pc, the mass M of the largest cluster scales with the linear lattice size (L) as M ⇠ L^d^f, with df = 91/48 for two dimensions [34]. Both randomness and fractal contribute anomalous di↵usion in percolation;

the time dependence of the mean square displacement becomes

[hr²i]^av ⇠ t^2/d^w (4.39)

with dw > 2 [36, 37], where [· ]av denotes an average over di↵erent disorder real-izations. We note that the slowdown of the random walk in a higher dimensional random medium is in general less pronounced than the slowdown in 1D, where a logarithmically slow di↵usion is observed (see Sec. 2.1.2). A heuristic explanation for such a di↵erence with the 1D case is that due to a less restricted topology of space in higher dimensions, it is much harder to force the random walk to visit traps.

Unlike a classical random walk which is essentially a di↵usion process, a quan-tum walk manifests quanquan-tum coherences, which can lead to faster spreading, as

1A fractal is a mathematical set that has a fractal dimension that usually exceeds its topo-logical dimension. A fractal is not smooth at every point. Mandelbrot [35] coined the term

”fractal” to describe the property of being fractured at every point.

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4.3 Quantum walk on square lattices

10^-2 10^-1 10⁰ 10¹ 10²

Figure 4.12: The mean square displacements [hr²i]av averaged over 1280 samples both for the quantum walk (black line) and the classical random walk (red line) on a bond-diluted lattice of size 33⇥ 33 at the percolation threshold pc = 0.5. The grey dashed line indicates [hr²i]av ⇠ t² for the quantum case in early times; the blue dashed line corresponds to the normal di↵usion behavior [hr²i]av ⇠ t for the classical case in early times, and the orange dashed line indicates an anomalous di↵usion, given by [hr²i]av⇠ t^0.67. The inset shows the data up to t = 10⁵.

we have seen in the preceding sections. On the other hand, the interplay between quantum e↵ects and disorder e↵ects may lead to localization of motion; the most prominent example is Anderson localization [38] which describes a remarkable transformation from metal to insulator due to the localization of electronic wave-functions by the presence of disorder in system.

Here we use the 2D square bond percolation with periodic boundary conditions as a random environment to study disorder e↵ects on both classical and quantum random walks in 2D. We use the same notation for the lattice vertices as defined in the previous section, and set the transition rate = 1 for all present edges in the bond-diluted lattice. For a given disorder realization, the transition rate matrix for the classical random walk and the Hamiltonian for the quantum walk are given by the same matrix. By diagonalizing the matrix, we obtained the probability, Pn!m(t), that the walker starting at t = 0 at vertex vn arrives at

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vertex vm at time t; the expression for Pn!m(t) in terms of the eigenvalues and the eigenvectors is given in Eq. (4.9) for the quantum case, and in Eq. (4.10) for the classical case. We first calculated the mean square displacement hr²i at the percolation threshold pc = 1/2. For a given percolation realization, the mean square displacement for a walk starting at vertex (nx, ny) is calculated as

hr²i(t) =X

Pn!m(t)



(mx nx)²+ (my ny)² . (4.40)

The disorder average [hr²i]av was then obtained by averaging over 1280 samples.

Our numerical results [Fig. 4.12] show there are three time regimes for both the classical and quantum cases: (i) In early times (t. 1), disorder is irrelevant; the mean square displacement [hr²i]avscales as [hr²i]av/ t² for the quantum case, and [hr²i]av / t for the classical case, corresponding to the results for homogeneous systems (i.e. lattices without dilution). (ii) At t ! 1, [hr²i]^av converges to a finite constant value for our finite system; for the classical case, this value corresponds to the average cluster radius [34]; for the quantum case, the value of [hr²i]^av appears to be smaller than the classical value, which may indicate that the quantum walker is ”trapped” in one region of the cluster where the starting point is located. (iii) In intermediate times (1 . t . 100), we have found that [hr²i]av / t^0.67 for the classical case; this implies that the ”anomalous di↵usion exponent” defined in Eq. (4.39) is dw ⇡ 0.335 ¹, in agreement with the theoretical prediction¹ [34]. For the quantum case, the time dependence of [hr²i]av appears to be slower than a power law.

We have also calculated the average return probabilities Pr(t)⌘ 1

as well as the probability (denoted by PL(t)) of spreading from the origin (0,0) to the boundary (see. Fig. 4.13) at the percolation threshold. It is observed in

1Another way to obtain the anomalous di↵usion exponent is to consider random walk in the infinite cluster only. The exponent, denoted by d⁰_w for this case, is di↵erent than dwfor walks averaged over all clusters. d⁰_w ⇡ 0.352 ¹ was found analytically and numerically in previous studies [36, 37].

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4.3 Quantum walk on square lattices

3 2 1 0 1 2 3

0 1 2 3

1 2 3

Figure 4.13: P_L(t) is the accumulation of probabilities of spreading from the origin (the blue vertex) to one of the vertices on the boundary (red vertices).

10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵

10^-1 10⁰

P r(t)

quantum

classical 0.01

0.02 0.03 0.04

P L(t)

Figure 4.14: The return probabilities P_r(t) and the probabilities P_L(t) of going from the origin to the boundary vertices on a percolation lattice of size 65⇥ 65 at p = p_c.

Fig. 4.14 that the return probability for the quantum walk decays faster than the probability for the classical walk (except for the short times regime t < 1);

its average value for t ! 1 is also higher than the value of the classical case.

At the same time, the probability P_L(t) for t 1 for the quantum cases is smaller than PL(t) for the classical case. All of these provide further evidence of

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slow spreading of quantum walks on percolation graphs, compared with classical anomalous di↵usion.

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Part II

Quantum Adiabatic Optimization

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5 Part II – Introduction

A discrete optimization problem consists of finding minimum (or maximum) val-ues of a function (called the cost function) of a large number of independent variables. Many such problems, including the traveling salesman problem [39]

and circuit design, can be reduced to that of finding the ground state of a sys-tem of interacting spins; however, finding such a ground state remains difficult because of the vast number of nearly degenerate solutions. Research in this area aims at developing efficient techniques without using brute-force search.

A physically motivated optimization problem is the complex free-energy struc-tures of glassy systems at low temperastruc-tures and the difficulty in reaching their ground states. Some magnetic alloys with frustrated interactions, called spin glasses, are examples for such glassy materials (for a review, see e.g. [41]). Theo-retical models for spin glasses with Ising variables are well described by a Hamil-tonian of the form

H({si}) = X

Jijsisj (5.1)

on a d (> 1) dimensional lattice, where Jij is the coupling between the spins on site i and j represented by the variables si =±1 and s^j =±1. A positive value of J_ij denotes a ferromagnetic interaction, and a negative value corresponds to an antiferromagnetic interaction. For a spin glass model, the couplings Jij are quenched random variables and have either sign. The model defined in (5.1) exhibits frustration, i.e. no spin configuration can simultaneously satisfy all cou-plings (see Fig. 5.1). Since the spin orientations are ”in conflict” with each other,

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Figure 5.1: Examples of Ising spin systems with (a) ferromagnetic couplings (b) antiferromagnetic couplings, and (c) frustrated couplings. Frustration in panel (c) occurs since" and # spin states are ”in conflict” with each other.

no long-range order of ferromagnetic or antiferromagnetic type can be established.

Below certain temperature spin glasses undergo a phase transition to a glassy phase, where the whole free-energy landscape becomes rugged and is divided into many valleys (local minima) separated by high free-energy barriers. Thus the system, once trapped in a local minimum, remains there for a long time; this reflects extremely slow dynamics at low temperatures. While theories for mean-field spin glasses are well-developed, their applicability to finite-dimensional spin glasses with short-range couplings is still in question. Progress in verifying those theories using simulations have been hampered by the complexity of the task.

Just finding the ground state is an NP-hard problem, which belongs to the class of the most difficult computational problems.

LiHoxY1 xF4 is a dilute, insulating, dipolar-coupled Ising system [42]. In this insulator, the Li⁺ and F ions have equal number of up and down spins and are non-magnetic. The magnetic moments contributed from Ho³⁺ ions have only two possible magnetic states along a preferred crystalline axis, thus represent perfect physical realization of Ising variables. The material is ferromagnetic with a Curie temperature T_c = 1.53K. In the dilute system, the magnetic Ho³⁺ ions are randomly substituted with nonmagnetic Y³⁺ with probability x. The nature of the ground state can be tuned by x and a field transverse to the Ising axis; it has been an especially useful experimental realization of the Ising spin glass and the quantum Ising model. The corresponding Hamiltonian for the quantum Ising

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model is given in terms of Pauli matrices by

H = X

J_ij ^z_i _j^z X

i x

i, (5.2)

where is the on-site transverse field (controlled by an external laboratory field Ht applied perpendicular to the Ising axis in the experiments). With 6= 0 the commutator [H, ^z] is finite and the -term can cause ”spin flips” even at zero temperature. Indeed, quantum fluctuations controllable by can drive the classical order-disorder transition point to zero temperature, resulting in a quan-tum critical point [43]. In Ref. [44], this material was used to probe two di↵erent routes, via thermal fluctuations and quantum fluctuations, for convergence to the ground state of the Ising spin glass.

In this part of the thesis, we will compare the low-temperature energies of the two dimensional Ising spin glass achieved by two optimization methods, one is classical annealing based on statistical mechanics, and the other is quantum Monte Carlo annealing.

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6 Optimization by Simulated Annealing

Simulated annealing (SA) is a probabilistic method proposed by Kirkpatrick et al. in 1983 [39] for finding the global minimum of a cost function by using the concepts of statistical mechanics. In statistical mechanics and solid state physics, annealing is known as a thermal process for obtaining low-energy states of a system in a heat bath. The introduction of a variable ”temperature” for optimization permits the system to become unstuck from a local minimum of the cost function (the free energy), and as the temperature is slowly reduced to zero the system settles into a minimum that should be comparable to the global minimum (the ground state). The name of the algorithm comes from the technique of annealing metals, i.e. heating to a high temperature and then cooling slowly in order to have perfect crystals formed.

At the heart of the SA method is a simple stochastic algorithm introduced by Metropolis et al. [40] for simulating the evolution of a thermodynamic system in a heat bath to thermal equilibrium. Consider for example the Ising model (defined in Eq. 5.1) with N spins. There are 2^N microstates (spin configurations) {c}, and each has a corresponding energy Ec ⌘ H({c}). For a given temperature T , each configuration of the system is weighted by its Boltzmann probability factor, exp( Ec/kBT ). If we wish to calculate the expectation value of some quantity,

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we could randomly assign the spin values, weight the contribution of that con-figuration by exp( Ec/kBT ) and repeat this process until the expectation value has converged. However, this method will not work in practice since there are so many microstates, and all of them appear with equal probability, include those whose weight is so small that they in e↵ect do not contribute to the average.

The idea of the Metropolis algorithm is, instead, to generate a sequence of states according to a Markov process, i.e. a random walk in the configuration space, which asymptotically converges to the stationary probability

P_c^⇤ = 1

Ze ^E^c^/k^B^T. (6.2)

The process is defined as Markovian, on which each new state c⁰ is generated from the current state c with a transition probability w_c,c0 that does not depend on the previous history of the system. The time evolution of the probability of having configuration c at time step t is then described by the master equation

Pc(t + 1) = Pc(t) +X

c⁰

wc⁰,cPc⁰(t) wc,c⁰Pc(t) . (6.3)

To construct the algorithm to have a stationary distribution lim_t=1P (t) = ~~ P^⇤, one requires extra conditions: (i) the algorithm is ergodic (i.e. can reach ev-erywhere in the configuration space and aperiodic); (ii) the algorithm satisfies detailed balance given by

wc,c⁰P_c^⇤ = wc⁰,cP_c^⇤0, (6.4) for each state c and c⁰. This (sufficient but not necessary) detailed balance con-dition makes the terms of the summation in Eq. (6.3) disappear separately at equilibrium. There are in principle infinitely many choices for the transition probabilities that satisfy the detailed balance condition

wc,c⁰

wc⁰,c

= P_c^⇤0

P_c^⇤ = e ^(E^c0 ^E^c^)/k^B^T. (6.5)

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The choice in Metropolis algorithm for the transition probability of moving from state c to c⁰ is

wc,c⁰ = 8<

1 for Ec⁰  Ec (P_c^⇤0 P_c^⇤),

e ^(E^c0 ^E^c^)/k^B^T for Ec⁰ > Ec (P_c^⇤0 < P_c^⇤). (6.6) or in a more compact form:

wc,c⁰ = min



1, e ^(E^c0 ^E^c^)/k^B^T (6.7)

By generating such a Markov chain of configurations c⁽¹⁾, c⁽²⁾, c⁽³⁾,· · · c^{(M )} (with M ! 1), which is distributed like e ^E^c^/k^B^T, one can compute the expectation values hOi by

hOi ⇡ 1 M

XM m=1

O(c^(m)) (6.8)

We now return to simulated annealing for the Ising spin glass. The algorithm based on the Metropolis method is implemented as follows:

(i) Choose an initial temperature T , and an initial spin configuration c.

(ii) Propose a transition (a trial move) c ! c⁰ by flipping a spin at random or sequentially.

(iii) Accept or reject the transition using the criterion (6.6); the spin configura-tion c is replaced with c⁰ if the transition is accepted.

(iv) Execute (ii) and (iii) N times (N : number of spins on the lattice), i.e. one Monte Carlo sweep.

(v) Update the temperature T according to a slowly decreasing ”cooling sched-ule”.

(vi) Repeat steps (ii) to (v) until the temperature is reduced to zero, or when it reaches any other stopping criterion.

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Figure 6.1: Residual energy per spin as a function of annealing time T (MC sweeps) for the±J spin glass of size 64 ⇥ 64. The initial temperature is T0= 3.5.

Before annealing, N_mc(0) Monte Carlo sweeps were performed at the temperature T0. We take averages over 64 samples for coupling randomness. There is no sig-nificant di↵erence between outcomes using di↵erent values of N_mc(0). The red line indicates a power law with decay exponent 0.2. Inset: the same data (with Nmc(0) = 10000) as in the main figure, replotted by assuming the Huse-Fisher logarithmic scaling (Eq. (6.11)); the red solid line is a fit with annealing exponent

⇣ = 1.93

In our simulation, we used a linear annealing schedule

T (t) = T0 to reduce the temperature T ; the annealing time T is the number of MC sweeps (1 MC sweep is equal to N spin-flip attempts) used in the simulation, i.e. 1 MC sweep was done at each temperature step T (t). To achieve the equilibrium state at each T (t), one needs to run a sufficiently large number of MC sweeps at fixed T before the temperature is reduced to the next lower value. We did not do so in our simulation, but rather followed a ”quasi-equilibrium” route towards the optimal ground state.

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Figure 6.2: Residual energy per spin as a function of annealing time T (MC steps) for the Gaussian spin glass of size 64⇥64. The initial temperature is T0 = 3.5. The data are averaged over 64 samples of coupling randomness. The red line indicates a power law with decay exponent 0.2. Inset: the same data with Nmc(0) = 10000 as in the main figure, replotted to assume the Huse-Fisher logarithmic decay (Eq. (6.11)); the red solid line is a fit with annealing exponent ⇣ = 2.14, which is larger than the upper bound ⇣^⇤ = 2 (indicted by the blue dashed line) in Huse-Fisher conjecture.

We focused only on 2D spin glasses with nearest-neighbor interactions {Jij};

two types of {Jij}-distributions were investigated: (i) the binary distribution, in which the couplings take two values +1 and 1 with equal probability, (the so-called ±J spin glass); (ii) the Gaussian distribution around zero with standard deviation one. For a given set of couplings, the ground state energy per spin E0 can be calculated using the Spin Glass Ground State Server at University of Cologne [45]. The residual energy (per spin) "res(T), the di↵erence between the final annealed energy Efinal(T) and the ground state energy E0, as a function of annealing time T is depicted in Fig. 6.1 for the±J model and in Fig. 6.2 for the Gaussian spin glass. For both cases, our data seem compatible with a power-law decay

"res(T)/ T ^↵, (6.10)

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with ↵ ⇡ 0.2, albeit a small deviation from a pure power law is seen at large T.

Theoretical arguments by Huse and Fisher [46] predict a slow logarithmic decay of the residual energy

"res(T)/ log ^⇣(T), (6.11) with ⇣  2. Comparison with this logarithmic scaling is shown in the inset of Fig. 6.1 and Fig. 6.2. For the±J model, the data at large T are in good agreement with the Huse-Fisher logarithmic scaling with ⇣ ⇡ 1.93, while for the Gaussian spin glass, we cannot conclude from our data whether ⇣  2.

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7 Optimization by Quantum Annealing

In simulated annealing the solution space for a given optimization problem is ex-plored by Metropolis random walks in which thermal fluctuations are used to get the system out from local minima. For some optimization problems, the barriers separating the local minima in the solution space diverge with the system size, thermal dynamics may become inefficient and break ergodicity when it is difficult

在文檔中量子模擬: 量子隨機行走法則與量子退火式最佳化演算 - 政大學術集成 (頁 58-103)