Circumventing superexponential runtimes for hard instances of quantum adiabatic optimization

TL;DR

This paper investigates the superexponential difficulty of solving certain instances of the maximum independent set problem with quantum adiabatic optimization and explores quantum quenches as a potential workaround to this challenge.

Contribution

It constructs specific problem instances with superexponentially small gaps and demonstrates how quantum quenches can bypass this limitation by exploiting quantum many-body scars.

Findings

Superexponentially small gaps identified in certain problem instances.

Quantum quenches can reveal signatures of quantum many-body scars.

Quenching from suboptimal states improves ground state overlap.

Abstract

Classical optimization problems can be solved by adiabatically preparing the ground state of a quantum Hamiltonian that encodes the problem. The performance of this approach is determined by the smallest gap encountered during the evolution. Here, we consider the maximum independent set problem, which can be efficiently encoded in the Hamiltonian describing a Rydberg atom array. We present a general construction of instances of the problem for which the minimum gap decays superexponentially with system size, implying a superexponentially large time to solution via adiabatic evolution. The small gap arises from locally independent choices, which cause the system to initially evolve and localize into a configuration far from the solution in terms of Hamming distance. We investigate remedies to this problem. Specifically, we show that quantum quenches in these models can exhibit signatures of quantum many-body scars, which in turn, can circumvent the superexponential gaps. By quenching from a suboptimal configuration, states with a larger ground state overlap can be prepared, illustrating the utility of quantum quenches as an algorithmic tool.