Why and how to add direction to a quantum walk

TL;DR

This paper introduces a formal framework for directed quantum walks using Hermitian adjacency matrices, revealing unique phenomena like zero transfer between sites, and explores families of directed cycles with this property.

Contribution

It develops a novel formalism for directed quantum walks and identifies conditions for zero transfer in directed cycles, advancing understanding in quantum information and spectral graph theory.

Findings

Directed quantum walks can exhibit zero transfer between sites.

Hermitian adjacency matrices effectively encode directionality in quantum walks.

Certain directed cycles admit zero transfer phenomena.

Abstract

We formalize the treatment of directed (or chiral) quantum walks using Hermitian adjacency matrices, bridging two developing fields of research in quantum information and spectral graph theory. We display results and simulations which highlight the conceptual differences between having directions encoded in the Hamiltonians or not. This leads to a construction of a new type of quantum phenomenon: zero transfer between pairs of sites in a connected coupled network, which is only possible in the directed model we study. Our main result is a description of several families of directed cycles that admit zero transfer.