Quantum and random walks as universal generators of probability distributions

TL;DR

This paper demonstrates how quantum and classical random walks can be engineered to produce any desired probability distribution by adjusting their parameters, unifying their behavior as universal probability generators.

Contribution

It introduces a method to impose arbitrary stochastic behaviors on quantum and random walks through tailored time- and site-dependent coins, enabling control over their spreading characteristics.

Findings

Quantum walks can be made to exhibit diffusive spreading while maintaining coherence.

Random walks can be engineered to show rapid propagation similar to quantum particles.

Any probability distribution compatible with the continuity equation can be generated by these systems.

Abstract

Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without loosing coherence, as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.