Dissipative search of an unstructured database

Abstract

The search of an unstructured database amounts to finding one element having a certain property out of $N$ elements. The classical search with an oracle checking one element at a time requires on average $N/2$ steps. The Grover algorithm for the quantum search and its unitary Hamiltonian evolution analog accomplish the search asymptotically optimally in $O(\sqrt{N})$ time steps. We reformulate the search problem as a dissipative, incoherent Markov process acting on an $N$-level system weakly coupled to a thermal bath. Assuming that the energy levels of the system represent the database elements, we show that, with a proper choice of the spectrum and long-range but bounded transition rates between the energy levels, the system relaxes to the ground state, corresponding to the sought element, in time $O(lnN)$.