Time-inhomogeneous Quantum Walks with Decoherence on Discrete Infinite Spaces

TL;DR

This paper extends classical time-inhomogeneous random walk models to quantum walks with decoherence on infinite spaces, providing a representation theorem and analyzing their limiting distributions through numerical and statistical methods.

Contribution

It introduces a quantum analogue of classical models, establishing a representation theorem and analyzing the convergence of their distributions.

Findings

Representation theorem for quantum walks on infinite spaces.

Numerical estimation of distribution convergence.

Statistical analysis of quantum analogues of classical laws.

Abstract

In quantum computation theory, quantum random walks have been utilized by many quantum search algorithms which provide improved performance over their classical counterparts. However, due to the importance of the quantum decoherence phenomenon, decoherent quantum walks and their applications have been studied on a wide variety of structures. Recently, a unified time-inhomogeneous coin-turning random walk with rescaled limiting distributions, Bernoulli, uniform, arcsine and semi-circle laws as parameter varies have been obtained. In this paper we study the quantum analogue of these models. We obtained a representation theorem for time-inhomogeneous quantum walk on discrete infinite state space. Additionally, the convergence of the distributions of the decoherent quantum walks are numerically estimated as an application of the representation theorem, and the convergence in distribution of the quantum analogues of Bernoulli, uniform, arcsine and semicircle laws are statistically analyzed.