Quantum algorithms and lower bounds for convex optimization

TL;DR

This paper introduces a quantum algorithm that significantly improves the efficiency of convex optimization over classical methods and establishes fundamental lower bounds on quantum query complexity for such problems.

Contribution

It presents a quantum algorithm with near-linear query complexity for convex optimization and proves lower bounds demonstrating the limits of quantum speedup.

Findings

Quantum algorithm achieves near-linear query complexity.

Lower bounds show quantum speedup is limited to square root factors.

Quadratic improvement over classical algorithms in convex optimization.

Abstract

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an $n$-dimensional convex body using $\tilde{O}(n)$ queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires $\tilde{\Omega}(\sqrt n)$ evaluation queries and $\Omega(\sqrt{n})$ membership queries.