A quantum-classical performance separation in nonconvex optimization

TL;DR

This paper demonstrates a quantum advantage in solving a specific family of nonconvex optimization problems with exponentially many local minima, showing quantum algorithms outperform classical ones in both theory and practice.

Contribution

The paper introduces a family of nonconvex optimization instances with exponential local minima and proves quantum algorithms can solve them efficiently, unlike classical methods.

Findings

Quantum algorithms solve the instances with polynomial quantum queries.

Classical algorithms require super-polynomial time for the same instances.

Empirical results support the theoretical quantum advantage.

Abstract

In this paper, we identify a family of nonconvex continuous optimization instances, each $d$-dimensional instance with $2^d$ local minima, to demonstrate a quantum-classical performance separation. Specifically, we prove that the recently proposed Quantum Hamiltonian Descent (QHD) algorithm [Leng et al., arXiv:2303.01471] is able to solve any $d$-dimensional instance from this family using $\widetilde{\mathcal{O}}(d^3)$ quantum queries to the function value and $\widetilde{\mathcal{O}}(d^4)$ additional 1-qubit and 2-qubit elementary quantum gates. On the other side, a comprehensive empirical study suggests that representative state-of-the-art classical optimization algorithms/solvers (including Gurobi) would require a super-polynomial time to solve such optimization instances.