Quantum search by local adiabatic evolution

TLDR

The adiabatic theorem has recently enabled a new class of quantum algorithms, where a slowly evolving quantum system remains near its instantaneous ground state that encodes the solution. We apply this time‑dependent Hamiltonian approach to Grover’s problem of searching a marked item in an unstructured database. By locally adjusting the evolution rate to maintain adiabaticity at every instant, the algorithm achieves a running time of order √N, matching Grover’s quantum speedup and outperforming the constant‑rate adiabatic method, which scales linearly with N.

Abstract

The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state which tends to the solution [Farhi et al., quant-ph/0001106]. We apply this time-dependent Hamiltonian approach to the Grover's problem, i. e., searching a marked item in an unstructured database. We find that, by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order $\sqrt{N}$, where $N$ is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as $N$. This is in contrast with the constant-rate adiabatic approach developed by Farhi et al., where the requirement of adiabaticity is expressed only globally, resulting in a time of order $N$.