Challenges and opportunities in quantum optimization

Quantum computers have demonstrable ability to solve problems at a scale beyond brute-force classical simulation. Interest in quantum algorithms has developed in many areas, particularly in relation to mathematical optimization - a broad field with links to computer science and physics. In this Review, we aim to give an overview of quantum optimization. Provably exact, provably approximate and heuristic settings are first explained using computational complexity theory, and we highlight where quantum advantage is possible in each context. Then, we outline the core building blocks for quantum optimization algorithms, define prominent problem classes and identify key open questions that should be addressed to advance the field. We underscore the importance of benchmarking by proposing clear metrics alongside suitable optimization problems, for appropriate comparisons with classical optimization techniques, and discuss next steps to accelerate progress towards quantum advantage in optimization.