Dissipative search of an unstructured database

TL;DR

This paper introduces a dissipative quantum approach to unstructured database search, achieving a logarithmic time complexity by modeling the problem as a Markov process relaxing to the target state.

Contribution

It reformulates the quantum search as a dissipative Markov process, enabling faster convergence to the target element with a logarithmic time complexity.

Findings

Achieves search in time $ ext{O}( ext{ln} N)$ with dissipative dynamics.

Reformulates quantum search as a Markov process.

Shows relaxation to the target state via energy spectrum design.

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 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 $\mathcal{O} (\ln N)$.