Generalized quantum-classical correspondence for random walks on graphs
Massimo Frigerio, Claudia Benedetti, Stefano Olivares, Matteo G. A. Paris · Physical Review A · 2021
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.