A New Limit Theorem for Quantum Walk in Terms of Quantum Bernoulli Noises.

Caishi Wang,Yuling Tang,Suling Ren

doi:10.3390/e22040486

Abstract

In this paper, we consider limit probability distributions of the quantum walk recently introduced by Wang and Ye (C.S. Wang and X.J. Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Inf. Process. 15 (2016), no. 5, 1897–1908). We first establish several technical theorems, which themselves are also interesting. Then, by using these theorems, we prove that, for a wide range of choices of the initial state, the above-mentioned quantum walk has a limit probability distribution of standard Gauss type, which actually gives a new limit theorem for the walk.

Highlights

Quantum walks, known as quantum random walks [1], are quantum analogs of classical random walks, but usually behave quite differently from the classical ones [2]
In 2016, a model of discrete-time quantum walk on the one-dimensional integer lattice Z was introduced in terms of quantum Bernoulli noises [8], which we call the one-dimensional QBN (Quantum Bernoulli noises) walk below, where
By using Theorem 5, we come to the theorem, which shows that, for all ξ ∈ Z, the one-dimensional QBN walk with Φ0 (0) = ξ has a limit probability distribution of standard

Summary

Introduction

Known as quantum random walks [1], are quantum analogs of classical random walks, but usually behave quite differently from the classical ones [2]. Due to their wide application in quantum information and quantum computing, quantum walks have received much attention in the past two decades (see e.g., [3,4,5] and references therein). In 2016, a model of discrete-time quantum walk on the one-dimensional integer lattice Z was introduced in terms of quantum Bernoulli noises [8], which we call the one-dimensional QBN (Quantum Bernoulli noises) walk below, where

Objectives

Results

Conclusion