Breakdown of an Electric-Field Driven System: a Mapping to a Quantum Walk

Oscillatory Localization of Quantum Walks Analyzed by Classical Electric Circuits

We examine an unexplored quantum phenomenon we call oscillatory localization, where a discrete-time quantum walk with Grover's diffusion coin jumps back and forth between two vertices. We then connect it to the power dissipation of a related electric network. Namely, we show that there are only two kinds of oscillating states, called uniform states and flip states, and that the projection of an arbitrary state onto a flip state is bounded by the power dissipation of an electric circuit. By applying this framework to states along a single edge of a graph, we show that low effective resistance i…

Electric circuit induced by quantum walk

We consider the Szegedy walk on graphs adding infinite length tails to a finite internal graph. We assume that on these tails, the dynamics is given by the free quantum walk. We set the $\ell^\infty$-category initial state so that the internal graph receives time independent input from the tails, say $\boldsymbolα_{in}$, at every time step. We show that the response of the Szegedy walk to the input, which is the output, say $\boldsymbolβ_{out}$, from the internal graph to the tails in the long time limit, is drastically changed depending on the reversibility of the underlying random walk. If t…

电量子行走的区间谱及相关的斜移CMV矩阵

Interval spectrum for electric quantum walk and related skew-shift CMV matrices

本文研究含电场量子行走模型的谱性质，证明对任何无理场其谱均为单位圆，该结果对斜移定义的相关CMV矩阵同样成立，并推广至高维环面上的斜移CMV矩阵。

Decoherence on a two-dimensional quantum walk using four- and two-state particle

We study the decoherence effects originating from state flipping and depolarization for two-dimensional discrete-time quantum walks using four-state and two-state particles. By quantifying the quantum correlations between the particle and position degree of freedom and between the two spatial ($x-y$) degrees of freedom using measurement induced disturbance (MID), we show that the two schemes using a two-state particle are more robust against decoherence than the Grover walk, which uses a four-state particle. We also show that the symmetries which hold for two-state quantum walks breakdown for …

Elfs、树与量子游走

Elfs, trees and quantum walks

本文提出基于电流采样的 elfs 马尔可夫过程，证明其到达分布等价于随机游走且树上电命中时间为对数级，并据此得到二次加速的量子游走采样算法。

Quantum Walks in artificial electric and gravitational Fields

The continuous limit of quantum walks (QWs) on the line is revisited through a recently developed method. In all cases but one, the limit coincides with the dynamics of a Dirac fermion coupled to an artificial electric and/or relativistic gravitational field. All results are carefully discussed and illustrated by numerical simulations.

Propagation and spectral properties of quantum walks in electric fields

We study one-dimensional quantum walks in a homogeneous electric field. The field is given by a phase which depends linearly on position and is applied after each step. The long time propagation properties of this system, such as revivals, ballistic expansion and Anderson localization, depend very sensitively on the value of the electric field $Φ$, e.g., on whether $Φ/(2π)$ is rational or irrational. We relate these properties to the continued fraction expansion of the field. When the field is given only with finite accuracy, the beginning of the expansion allows analogous conclusions about th…

电场量子行走中输运的指数抑制

Exponential Suppression of Transport in Electric Quantum Walks

本文建立了周期性电场中平移不变量子行走最大群速度的精确标度关系，证明其随场周期指数衰减，证实并扩展了相关猜想，还给出复兴关系及绝对连续谱刻画，结果适用于广义CMV矩阵。

Anderson localization for electric quantum walks and skew-shift CMV matrices

We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the L…

Revivals in Quantum Walks with quasi-periodically time-dependent coin

We provide an explanation of recent experimental results of Xue et al., where full revivals in a time-dependent quantum walk model with a periodically changing coin are found. Using methods originally developed for "electric" walks with a space-dependent, rather than a time-dependent coin, we provide a full explanation of the observations of Xue et al. We extend the analysis from periodic time-dependence to quasi-periodic behaviour with periods incommensurate to the step size. Spectral analysis, one of the principal tools for the study of electric walks, fails for time-dependent systems, but w…