Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

From the quantum random walk to classical mesoscopic diffusion in crystalline solids

Salvador Godoy, L. S. Garcı́a-Colı́n

Published Jun 1, 1996• 40 citations
Publisher

Abstract

We use a discrete approximation of quantum mechanics called the quantum random walk to study diffusion in one-dimensional crystalline nanostructures. There, intense fluctuations in the density are a consequence of quantum interference. As the size of the crystal increase, relative quantum fluctuations decrease and, if a coarse-grained averaging is taken, interference may be neglected. When this happens, we describe a far-from-equilibrium, classical, dynamic mesoscopic diffusion regime. In this regime, diffusion equations are given by a persistent-random-walk process. Density becomes a second-order Markov process, and in the continuum limit, the current satisfies the non-Fickian Maxwell-Cattaneo relationship. The generalized diffusion equation in the telegraphist's equation.

DOI
10.1103/physreve.53.5779

Related papers

Quantum Diffusion-Limited Aggregation

David Johnson, Ordóñez, Gonzalo

Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.

arXiv:1111.0626•Nov 2, 2011•1 citation

Quantum mesoscopic physics: Coherent backscattering of light by cold atoms

T. Chanelière, G. Labeyrie, Christian Miniatura +2 more

Propagation in a random medium is often described by a random walk and a diffusion coefficient. However in the mesoscopic regime, where wave properties cannot be neglected, interferences alter the diffusion process leading in extreme situations to strong (Anderson) localisation. The recent observation of coherent backscattering of light by laser cooled atoms has opened the way to new studies in previously not accessible regimes. In this paper we present some of the questions addressed by these experiments, allowing for a more complete understanding of the physical processes involved in this quantum mesoscopic regime

Journal de Physique IV (Proceedings)•Nov 1, 2004•1 citation

Exact diffusion coefficients for quantum--classical hybrid walks

Cook Hyun Kim, Jaesub Park, Namshik Han

One-dimensional quantum--classical hybrid walks are introduced by combining quantum and classical steps under periodic and random protocols. Exact diffusion coefficients are derived for both protocols throughout the diffusive regime. The exact results show how diffusion depends on the average frequency and temporal arrangement of classical steps. As the average frequency approaches zero, both diffusion coefficients diverge inversely with the frequency. At matched average frequencies, the ratio of the diffusion coefficients for the random and periodic protocols approaches two. A finite-mode approximation is also developed and tested against the exact diffusion coefficient for the random protocol. The exact results for the one-dimensional model provide a foundation for future studies of more general quantum--classical hybrid walks.

quant-pharXiv:2609.01951•Sep 3, 2026•0 citations

1 Random Time-Dependent Quantum Walks

Alain Joye

We consider the discrete time unitary dynamics given by a quantum walk on the lattice $\Z^d$ performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in $\Z^d$ when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided. An example of i.i.d. random updates for which the analysis of the distribution can be performed without averaging is worked out. The distribution also displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix whose law we compute. A large deviation principle is shown to hold for this example. We finally show that, in general, the expectation of the random diffusion matrix equals the diffusion matrix of the averaged distribution.

•Aug 16, 2016•19 citations

Quantum walks on a random environment

Y. Yin, D. E. Katsanos, S. N. Evangelou

Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength $W$, where a quantum to classical crossover at time ${t}_{c}\ensuremath{\propto}{W}^{\ensuremath{-}2}$ transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution, we relate our results to previously studied models of decoherence for quantum walks, and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.

arXiv:0708.1137•Physical Review A•Feb 5, 2008•119 citations

Quantum walks on random lattices: Diffusion, localization, and the absence of parametric quantum speedup

Rostislav Duda, Moein N. Ivaki, Isac Sahlberg +2 more

Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms, and quantum simulation of condensed-matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speedup in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct prediffusive and diffusive behaviors at different timescales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speedup, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speedup implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.

arXiv:2210.05310•Physical Review Research•Jun 5, 2023•13 citations
We use a discrete approximation of quantum mechanics called the quantum random walk to study diffusion in one-dimensional crystalline nanostructures. There, intense fluctuations in the density are a consequence of quantum interference. As the size of the crystal increase, relative quantum fluctuations decrease and, if a coarse-grained averaging is taken, interference may be neglected. When this happens, we describe a far-from-equilibrium, classical, dynamic mesoscopic diffusion regime. In this regime, diffusion equations are given by a persistent-random-walk process. Density becomes a second-order Markov process, and in the continuum limit, the current satisfies the non-Fickian Maxwell-Cattaneo relationship. The generalized diffusion equation in the telegraphist's equation.
Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.
Propagation in a random medium is often described by a random walk and a diffusion coefficient. However in the mesoscopic regime, where wave properties cannot be neglected, interferences alter the diffusion process leading in extreme situations to strong (Anderson) localisation. The recent observation of coherent backscattering of light by laser cooled atoms has opened the way to new studies in previously not accessible regimes. In this paper we present some of the questions addressed by these experiments, allowing for a more complete understanding of the physical processes involved in this quantum mesoscopic regime
One-dimensional quantum--classical hybrid walks are introduced by combining quantum and classical steps under periodic and random protocols. Exact diffusion coefficients are derived for both protocols throughout the diffusive regime. The exact results show how diffusion depends on the average frequency and temporal arrangement of classical steps. As the average frequency approaches zero, both diffusion coefficients diverge inversely with the frequency. At matched average frequencies, the ratio of the diffusion coefficients for the random and periodic protocols approaches two. A finite-mode approximation is also developed and tested against the exact diffusion coefficient for the random protocol. The exact results for the one-dimensional model provide a foundation for future studies of more general quantum--classical hybrid walks.
We consider the discrete time unitary dynamics given by a quantum walk on the lattice Zd\Z^dZd performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in Zd\Z^dZd when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided. An example of i.i.d. random updates for which the analysis of the distribution can be performed without averaging is worked out. The distribution also displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix whose law we compute. A large deviation principle is shown to hold for this example. We finally show that, in general, the expectation of the random diffusion matrix equals the diffusion matrix of the averaged distribution.
Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength WWW, where a quantum to classical crossover at time ${t}_{c}\ensuremath{\propto}{W}^{\ensuremath{-}2}$ transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution, we relate our results to previously studied models of decoherence for quantum walks, and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.
Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms, and quantum simulation of condensed-matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speedup in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct prediffusive and diffusive behaviors at different timescales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speedup, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speedup implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.