Inefficient quantum walks on networks: the role of the density of states

TL;DR

This paper demonstrates that the efficiency of continuous-time quantum walks on networks is heavily influenced by the density of states, with highly degenerate eigenvalues leading to inefficiency and uniform degeneracy leading to optimal transport.

Contribution

It provides a general theoretical framework linking eigenvalue degeneracy patterns to quantum walk efficiency, supported by analytical examples and discussion of complex network extensions.

Findings

Highly degenerate eigenvalues cause inefficient quantum walks.

Uniform eigenvalue degeneracy results in highly efficient transport.

Analytical solutions for simple network structures illustrate the principles.

Abstract

We show by general arguments that networks whose density of states contains few highly degenerate eigenvalues result in inefficient performances of continuous-time quantum walks (CTQW) over these networks, while systems whose eigenvalues all have the same degeneracy lead to very efficient transport. We exemplify our results by considering CTQW and, for comparison, its classical counterpart, continuous-time random walks, over simple structures, whose eigenvalues and eigenstates can be calculated analytically. Extensions to more complicated, hyper-branched networks are discussed.