Discrete Randomness in Discrete Time Quantum Walk: Study via Stochastic Averaging

TL;DR

This paper investigates how classical noise affects discrete-time quantum walks, revealing a transition from quantum to classical diffusion and showing that averaged randomized quantum walks behave like a new type of classical walk.

Contribution

It introduces a CPTP-based analysis of classical noise in quantum walks and demonstrates the equivalence of averaged randomized quantum walks to a novel classical walk model.

Findings

Transition from quantum ballistic to classical diffusive regime

Averaged randomized quantum walk behaves as a new classical walk

Classical randomness can be effectively modeled via CPTP maps

Abstract

The role of classical noise in quantum walks (QW) on integers is investigated in the form of discrete dichotomic random variable affecting its reshuffling matrix parametrized as a SU2)/U(1) coset element. Analysis in terms of quantum statistical moments and generating functions, derived by the completely positive trace preserving (CPTP) map governing evolution, reveals a pronounced eventual transition in walk's diffusion mode, from a quantum ballistic regime with rate O(t) to a classical diffusive regime with rate O(\surdt), when condition (strength of noise parameter)^{2}\times(number of steps)=1, is satisfied. The role of classical randomness is studied showing that the randomized QW, when treated on the stochastic average level by means of an appropriate CPTP averaging map, turns out to be equivalent to a novel quantized classical walk without randomness. This result emphasizes the dual role of quantization/randomization in the context of classical random walk.