On Random Quantum Random Walks

TL;DR

This paper studies quantum random walks, showing how averaging over coin choices results in spline-like amplitude patterns and establishing constraints on their localization and speed.

Contribution

It introduces a new analysis of averaged quantum walks, revealing spline patterns and providing bounds on their localization and velocity.

Findings

Averaging over coins produces spline-shaped amplitude patterns.

Constraints on the localization of quantum random walks.

Establishment of bounds on achievable speeds.

Abstract

Quantum random walks, - coined, lattice ones, - exhibit ballistic behavior with fascinating asymptotic patterns of the amplitudes. We show that averaging over the coins (using the Haar measure), these patterns blend into a spline. Also, we discuss the localizations of such quantum random walks, and establish some strong constraints on the achievable speeds.