Mapping quantum random walks onto a Markov chain by mapping a unitary transformation to a higher dimension of an irreducible matrix

TL;DR

This paper introduces a novel two-dimensional process that combines classical and quantum random walk behaviors, mapping a unitary transformation onto a higher-dimensional Markov chain to analyze quantum walk dynamics.

Contribution

It presents a new model that maps quantum random walks onto Markov chains by transforming unitary operations into higher-dimensional irreducible matrices.

Findings

The model reproduces quantum walk probability distributions.

Numerical simulations on infinite and finite lines validate the approach.

Abstract

Here, a new two-dimensional process, discrete in time and space, that yields the results of both a random walk and a quantum random walk, is introduced. This model describes the population distribution of four coin states |1>,-|1>, |0> -|0> in space without interference, instead of two coin states |1>, |0> .For the case of no boundary conditions, the model is similar to a Markov chain with a stochastic matrix, i.e., it conserves the population distribution of the four coin states, and by using a proper transformation, yield probability distributions of the two quantum states |1>, |0> in space, similar to a unitary operator. Numerical results for a quantum random walk on infinite and finite lines are introduced.