Dissipative search of an unstructured database
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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 𝒪 (√(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 𝒪 (ln N).
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