Generalized quantum-classical correspondence for random walks on graphs

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 electromagnetic vector potential for a charged walker.

Automated summary

The study introduces a minimal set of physically motivated postulates that the Hamiltonian of a continuous-time quantum walk should satisfy for properly representing the quantum counterpart of a classical random walk on a given graph. It is found that there are infinitely many quantum Hamiltonians that meet these conditions, which offer novel degrees of freedom for quantum enhanced protocols. The on-site energies and phases of the off-diagonal elements in the Hamiltonian are shown to be unconstrained on the quantum side, providing potential for control and manipulation in quantum walks.