From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

Abstract

The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk, the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical systems.

How to Realize One-dimensional Discrete-time Quantum Walk by Dirac Particle

“…This originates from the discrete structure of space and time in the DTQW. Therefore, DTQWs can simulate several quantum and classical dynamics to unify the concept as a quantum dynamical simulator [6]. However, there remains the question of whether or not the relativistic particle can experimentally implement several DTQWs.…”

“…Recent progress on quantum complexity has been based on the quantum walk [1,2], and other applications and insights have been reported in recent review papers [3][4][5][6] and books [7,8]. It is remarked that the brief history of the DTQW was proposed by Feynman as the Feynman checkerboard [9] and thereafter independently formulated in quantum probability [10], quantum random walks [11], and quantum cellular automaton [12].…”

A walk on max-plus algebra

“…One of the reason for the studies on quantum walks is not only the efficiency on quantum search algorithm but also application as a quantum simulator of quantum phenomena (on a quantum device) as envisioned by Feynman [5] because of its universarity of the quantum computation [4]. For example, as a simulator of the Dirac equation e.g., [17,23] and its reference therein, while quantum graphs [6,18] which is a system of stationary Schrödinger equations on metric graphs, and topological insulator [8]. The time evolution of a discrete-time quantum walk is described by the iteration of a unitary operator on some Hilbert space generated by a discrete set.…”

Moments of coinless quantum walks on lattices

“…Moments and the mean square displacement (or variance) of the coined quantum walk on one-dimensional lattices were analyzed in Refs. [5][6][7][8][9][10] and on two-dimensional lattices in Refs. [7,[11][12][13].…”