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arXiv:1803.02410 · Physical review. B./Physical review. B

Quantum algorithm for energy matching in hard optimization problems

Christopher L. Baldwin, Chris R. Laumann

Published Jun 11, 2018• 46 citations
arXivHTMLPDF

Abstract

We consider the ability of local quantum dynamics to solve the ``energy-matching'' problem: given an instance of a classical optimization problem and a low-energy state, find another macroscopically distinct low-energy state. Energy matching is difficult in rugged optimization landscapes, as the given state provides little information about the distant topography. Here, we show that the introduction of quantum dynamics can provide a speedup over classical algorithms in a large class of hard optimization problems. Tunneling allows the system to explore the optimization landscape while approximately conserving the classical energy, even in the presence of large barriers. Specifically, we study energy matching in the random $p$-spin model of spin-glass theory. Using perturbation theory and exact diagonalization, we show that introducing a transverse field leads to three sharp dynamical phases, only one of which solves the matching problem: (1) a small-field ``trapped'' phase, in which tunneling is too weak for the system to escape the vicinity of the initial state; (2) a large-field ``excited'' phase, in which the field excites the system into high-energy states, effectively forgetting the initial energy; and (3) the intermediate ``tunneling'' phase, in which the system succeeds at energy matching. The rate at which distant states are found in the tunneling phase, although exponentially slow in system size, is exponentially faster than classical search algorithms.

DOI
10.1103/physrevb.97.224201

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