  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.
