Subspace State Transfer in Coined Quantum Walks

TL;DR

This paper investigates subspace state transfer in coined quantum walks, providing characterizations, algorithms for detection, and examples of perfect transfer in complex subspaces.

Contribution

It introduces spectral characterizations for subspace transfer, a polynomial-time testing algorithm, and constructs infinite families of perfect transfer examples.

Findings

Spectral conditions characterize perfect and pretty good subspace transfer.

A polynomial-time algorithm tests for perfect transfer at integer steps.

Infinite families of examples with high-dimensional subspace transfer are constructed.

Abstract

We study a transport phenomenon in certain coined quantum walks where a subspace of states localized at a vertex gets transferred to another vertex. We first develop characterizations for perfect and pretty good subspace state transfer using the spectral properties of a Hermitian weighted digraph obtained from the underlying graph. We then provide a polynomial-time algorithm that tests whether pointwise perfect subspace state transfer occurs at an integer step, given that the subspace and coins are rational. Finally, we construct several infinite families of examples that admit pointwise perfect $d$-dimensional subspace state transfer where $d\ge 2$.