The uniform measure for quantum walk on hypercube: a quantum Bernoulli noises approach

TL;DR

This paper introduces a quantum Bernoulli noises approach to quantum walks on hypercubes, providing explicit formulas, limit theorems, and conditions under which the walk converges to a uniform distribution, highlighting a novel analytical framework.

Contribution

It develops a new quantum Bernoulli noises method for analyzing quantum walks on hypercubes, deriving explicit probability formulas and convergence results.

Findings

The probability distribution can be explicitly calculated at any time.

The averaged distribution converges to the uniform distribution.

The walk's stationary measure is the uniform measure under mild initial conditions.

Abstract

In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube and then, based on the alternative description, we find that the operators $\partial_k^* + \partial_k$ behave actually as the shift operators, where $\partial_k$ and $\partial_k^*$ are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the above operators as the shift operators on the position space, we introduce a discrete-time quantum walk model on a general hypercube and obtain an explicit formula for calculating its probability distribution at any time. We also establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution. Finally, we show that the walk produces the uniform measure as its stationary measure on the hypercube provided its initial state satisfies some mild conditions. Some other results are also proven.