From classical to quantum walks with stochastic resetting on networks

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From classical to quantum walks with stochastic resetting on networks

Sascha Wald and Lucas Böttcher

Phys. Rev. E 103, 012122 – Published 19 January 2021

Abstract

Random walks are fundamental models of stochastic processes with applications in various fields, including physics, biology, and computer science. We study classical and quantum random walks under the influence of stochastic resetting on arbitrary networks. Based on the mathematical formalism of quantum stochastic walks, we provide a framework of classical and quantum walks whose evolution is determined by graph Laplacians. We study the influence of quantum effects on the stationary and long-time average probability distribution by interpolating between the classical and quantum regime. We compare our analytical results on stationary and long-time average probability distributions with numerical simulations on different networks, revealing differences in the way resets affect the sampling properties of classical and quantum walks.

Received 2 August 2020
Accepted 14 December 2020

DOI:https://doi.org/10.1103/PhysRevE.103.012122

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Continuous-time random walk Quantum walks Random walks

Statistical Physics & ThermodynamicsQuantum Information, Science & Technology

Authors & Affiliations

Sascha Wald ^1,* and Lucas Böttcher^2,3,4,†

¹Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, D-01187 Dresden, Germany
²Department of Computational Medicine, University of California, Los Angeles, California 90024, USA
³Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
⁴Center of Economic Research, ETH Zurich, 8092 Zurich, Switzerland

^*Present address: Statistical Physics Group, Centre for Fluid and Complex Systems, Coventry University, Coventry, England; sascha.wald@coventry.ac.uk
^†lucasb@ethz.ch

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Issue

Vol. 103, Iss. 1 — January 2021

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Images

Figure 1
Degree distributions of different networks. We show the degree distributions of the different networks that we use throughout this work. Black disks represent the degree distribution of a particular network realization, and the gray solid line is the corresponding analytic degree distribution. (a) Erdös-Rényi network with a Gaussian degree distribution and $N = 1600$ nodes. (b) Barabási-Albert network with a power-law degree distribution. The corresponding exponent is $- 3$ [61]. Each new node is connected to $m = 2$ existing nodes (i.e., the degree of each node is at least 2), and the depicted realization has $N = 1600$ nodes. (c) Peer-to-peer network (p2p-Gnutella08) [62] with $N = 1600$ nodes and mean degree $\bar{k} \approx 2.5$ .
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Figure 2
Overview of different classical and quantum reset walks. Different types of random walks as a function of the classicality parameter $ε$ [see Eq. (26)] and reset rate $r$ [see Eq. (28)]. The classicality parameter $0 \leq ε \leq 1$ allows us to smoothly interpolate between a QW ( $ε = 0$ ) and a CRW ( $ε = 1$ ). In the time interval $[t, t + d t]$ , a reset to the initial state occurs with probability $r d t$ .
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Figure 3
Schematic of a quantum walk. A quantum walker starts from a certain initial state in panel (a). Here the initial state is fully localized on a certain node for illustration purposes. The dynamics of the quantum walker is governed by the Schrödinger equation. After some time, the wave function of the walker will be spread over the network as depicted in panel (b) and there will be quantum superpositions. Knowing the initial state, both the Hamiltonian that induces the dynamics and the waiting time uniquely determine this state. Observing (or measuring) the quantum walker will result in a collapse of the wave function as shown in panel (c), and there is no way of knowing to which state the wave function collapses. This introduces stochasticity to QWs.
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Figure 4
Occupation probability for different networks and reset rates. For different networks (row) and reset rates $r$ (column), we show the probabilities $p^{'} (r)$ and $q^{'} (r)$ [see Eqs. (13) and (19)] that a node of degree $k$ is occupied by a classical and quantum random walker, respectively. As initial conditions and reset states, we use a uniform distribution over all nodes. Our numerical results are based on solutions of Eqs. (9) and (46). The networks we consider have $N = 1600$ nodes. The shown data points are averages over $2.5 \times 10^{5}$ samples. We use gray markers to indicate the solutions of the analytic results Eqs. (34) and (36). The black solid line is the occupation probability of a CRW for $r = 0$ as a reference.
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Figure 5
Difference between the classical and quantum occupation probability for different networks and reset rates. For different networks, we show the distance $d (r)$ [see Eq. (50)]. As an initial condition, we use a uniform distribution over all nodes. Our numerical results are based on solutions of Eqs. (9) and (46). The networks we consider have $N = 1600$ nodes. The shown data points are averages over $2.5 \times 10^{5}$ samples.
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Figure 6
Occupation probability for quantum-to-classical walks. We show the probability that a node of degree $k$ is occupied by a classical-to-quantum random walker for (a) $r = 0$ and (b) $r = 0.3$ . As initial conditions and reset states, we use a uniform distribution over all nodes. Our numerical results are based on solutions of Eq. (48). Simulations were performed on an Erdös-Rényi network with $N = 100$ nodes. The shown data points are averages over $2 \times 10^{5}$ samples. The black solid line is the occupation probability of a CRW for $r = 0$ , and gray markers in panel (b) correspond to analytical solutions of a CRW ( $ε = 1$ ) and a QW $(ε = 0$ ).
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