Study of Quantum Walk over a Square Lattice

TL;DR

This paper investigates how different types of noise affect the mixing time of discrete quantum walks on a square lattice, revealing that phase damping noise leads to uniform distribution while bit-flip noise does not.

Contribution

It provides a comparative analysis of the effects of phase damping and bit-flip noise on quantum walk mixing times using numerical methods.

Findings

Phase damping noise results in a uniform distribution over time.

Bit-flip noise prevents the distribution from becoming uniform.

Numerical analysis confirms different impacts of noise types on mixing behavior.

Abstract

Quantum random walk finds application in efficient quantum algorithms as well as in quantum network theory. Here we study the mixing time of a discrete quantum walk over a square lattice in presence percolation and decoherence. We consider bit-flip and phase damping noise, and evaluate the instantaneous mixing time for both the cases. Using numerical analysis we show that in case of phase damping noise probability distribution of walker's position is sufficiently close to the uniform distribution after infinite time. However, during the action of bit-flip noise, even after infinite time the total variation distance between the two probability distributions is large enough.