Quantum algorithm for energy matching in hard optimization problems

TL;DR

This paper demonstrates that quantum tunneling can provide a computational speed-up over classical algorithms in solving the energy matching problem in complex optimization landscapes, specifically in spin glass models.

Contribution

It introduces a quantum dynamical approach to energy matching, identifying a tunneling phase that outperforms classical methods in hard optimization problems.

Findings

Quantum tunneling enables exploration of rugged landscapes.

Three dynamical phases identified in the quantum model.

Tunneling phase achieves exponential speed-up over classical algorithms.

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 speed-up over local classical algorithms in a large class of hard optimization problems. The essential intuition is that tunneling allows the system to explore the optimization landscape while approximately conserving the classical energy, even in the presence of large barriers. In particular, we study energy matching in the random p-spin model of spin glass theory. Using perturbation theory and numerical 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 low energy; and (3) the intermediate tunneling phase, in which the system succeeds at energy matching. We find that in the tunneling phase, the time required to find distant states scales exponentially with system size but is nevertheless exponentially faster than simple classical Monte Carlo.