Quantum state revivals in quantum walks on cycles

TL;DR

This paper investigates the conditions under which quantum walks on cycles exhibit full quantum state revivals, extending previous incomplete descriptions to establish general criteria for such phenomena.

Contribution

It provides the first comprehensive set of conditions for full quantum state revival in quantum walks on cycles with constant, uniform coin operators.

Findings

Derived general conditions for quantum state revivals

Extended understanding of recurrence in quantum walks

Identified specific scenarios for full state revival

Abstract

Recurrence in the classical random walk is well known and described by the P\'olya number. For quantum walks, recurrence is similarly understood in terms of the probability of a localized quantum walker to return to its origin. Under certain circumstances the quantum walker may also return to an arbitrary initial quantum state in a finite number of steps. Quantum state revivals in quantum walks on circles using coin operators which are constant in time and uniform across the path have been described before but only incompletely. In this paper we find the general conditions for which full-quantum state revival will occur.