Mapping a finite and an infinite Hadamard quantum walk onto a unique case of a random walk process

TL;DR

This paper presents a novel mapping of a Hadamard quantum walk onto a classical random walk model using a Markov chain, revealing the underlying stochastic process and extending to finite chains with reflecting boundaries.

Contribution

It introduces a method to represent a Hadamard quantum walk as a Markov chain with positive transition rates, bridging quantum and classical random walk models.

Findings

Quantum walk mapped to a Markov chain with positive transition rates

Probability distributions of quantum states are obtained through a transformation

Model extended to finite chains with reflecting boundaries

Abstract

A new model that maps a quantum random walk described by a Hadamard operator to a particular case of a random walk is presented. The model is represented by a Markov chain with a stochastic matrix, i.e., all the transition rates are positive, although the Hadamard operator contains negative entries. Using a proper transformation that is applied to the random walk distribution after n steps, the probability distributions in space of the two quantum states |1>, |0> are revealed. These show that a quantum walk can be entirely mapped to a particular case of a higher dimension of a random walk model. The random walk model and its equivalence to a Hadamard walk can be extended for other cases, such as a finite chain with two reflecting points