Dissipative search of an unstructured database

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