Beyond Ground States: Physics-Inspired Optimization of Excited States of Classical Hamiltonians

TL;DR

This paper presents ExcLQA, a physics-inspired classical algorithm that efficiently finds excited states of Hamiltonians, demonstrated on lattice problems relevant to cryptography, outperforming traditional methods.

Contribution

Introduction of ExcLQA, a novel classical algorithm extending local quantum annealing to target excited states of Hamiltonians with tunable penalization.

Findings

Successfully solves SVP instances up to rank 46

Outperforms Metropolis-Hastings in solved ratio

Achieves better approximation factors

Abstract

We introduce excited local quantum annealing (ExcLQA), a classical, physics-inspired algorithm that extends local quantum annealing (LQA) to identify excited states of classical Ising Hamiltonians. LQA simulates quantum annealing while constraining the quantum state to remain in a product state and uses a gradient-based approach to find approximate solutions to large-scale quadratic unconstrained binary optimization problems. ExcLQA extends this framework by adding a penalty term in the cost function to target excited states, with a single hyperparameter that can be tuned via binary search to set the desired penalization level. We benchmark ExcLQA on the shortest vector problem (SVP), a fundamental lattice problem underlying the security of many postquantum cryptographic schemes. Solving an SVP instance can be mapped to identifying the first excited state of a Hamiltonian, with approximate solutions located among nearby excited states. Our results show that ExcLQA manages to solve SVP instances up to rank 46, and outperforms the Metropolis-Hastings algorithm in solved ratio, number of shots, and approximation factor in the tested instances.