Dissipative search of an unstructured database

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 analogue, accomplish the search asymptotically optimally in $\mathcal{O} (\sqrt{N})$ time steps. We reformulate the search problem as a dissipative 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 physically admissible, long-range transition rates between the energy levels, the system relaxes to the ground state, corresponding to the sought element, in time $\mathcal{O} (\ln N)$.