Generalized quantum-classical correspondence for random walks on graphs

TL;DR

This paper establishes a set of physically motivated conditions for quantum Hamiltonians to accurately represent classical random walks on graphs, revealing many possible Hamiltonians with tunable parameters for quantum protocols.

Contribution

It introduces minimal postulates for quantum Hamiltonians to mirror classical random walks, uncovering a vast family of Hamiltonians with controllable parameters for quantum applications.

Findings

Many Hamiltonians satisfy the postulates, offering new degrees of freedom.

Diagonal elements can be controlled via scalar fields.

Off-diagonal phases can be tuned by gauge fields.

Abstract

We introduce a minimal set of physically motivated postulates that the Hamiltonian H of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e. the diagonal elements of H, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential energy landscape for the quantum walk, and may be controlled by the interaction with a classical scalar field, whereas, for regular lattices in generic dimension, the off-diagonal phases of H may be tuned by the interaction with a classical gauge field residing on the edges, e.g., the electro-magnetic vector potential for a charged walker.