Group Velocity of Discrete-Time Quantum Walks

TL;DR

This paper models certain quantum walks as wave phenomena with calculable velocities, proposing a new hitting time definition that is independent of initial conditions or probability thresholds, demonstrated on Abelian group Cayley graphs.

Contribution

It introduces a wave-based model for quantum walks and a novel hitting time measure that simplifies analysis and applies to Abelian group Cayley graphs.

Findings

Wave velocities can be explicitly calculated for certain quantum walks.

The new hitting time does not depend on initial conditions or thresholds.

Applicable to quantum walks on lines, hypercubes, and Abelian groups.

Abstract

We show that certain types of quantum walks can be modeled as waves that propagate in a medium with phase and group velocities that are explicitly calculable. Since the group and phase velocities indicate how fast wave packets can propagate causally, we propose the use of these wave velocities in a new definition for the hitting time of quantum walks. The new definition of hitting time has the advantage that it requires neither the specification of a walker's initial condition nor of an arrival probability threshold. We give full details for the case of quantum walks on the Cayley graphs of Abelian groups. This includes the special cases of quantum walks on the line and on hypercubes.