Irrational quantum walks

TL;DR

This paper develops exact theoretical methods to analyze continuous-time quantum walks generated by integral Hamiltonians, enabling precise computation of mixing properties and state transfer, overcoming limitations of numerical approximations due to irrational eigenvalues.

Contribution

The paper introduces a novel theoretical framework for exact analysis of quantum walks with integral Hamiltonians, including mixing matrices and state transfer conditions.

Findings

Exact computation of average mixing matrices

Criteria for pretty good state transfer

Analysis of geometric properties of quantum walk matrices

Abstract

The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behaviour of the walk is typically not periodic. In consequence we can usually only compute numerical approximations to parameters of the walk. In this paper, we develop theory to exactly study any quantum walk generated by an integral Hamiltonian. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost) perfect state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix, and discuss possible applications of these results.