Quantum advantage for optimization problems

Determine whether hybrid quantum-classical algorithms or purely quantum algorithms can solve optimization problems more efficiently than state-of-the-art classical solvers.

Background

The paper introduces QHyper, a software library designed to facilitate experimentation with quantum and hybrid solvers for combinatorial optimization. The motivation highlights the broader goal of establishing quantum advantage—demonstrating tasks where quantum methods surpass classical approaches. Despite significant interest, whether hybrid or purely quantum algorithms can outperform state-of-the-art classical solvers on optimization problems remains unresolved. QHyper aims to support systematic investigation of this question by providing a unified interface for problem formulation, solver execution, and (hyper)parameter optimization across quantum annealers, variational algorithms, and classical methods.

References

Therefore, determining whether hybrid or purely quantum algorithms could solve optimization problems more efficiently than state-of-the-art classical solvers would bring a major advancement and still remains an open question.

— QHyper: an integration library for hybrid quantum-classical optimization  (2409.15926 - Lamża et al., 2024) in Section 1 (Motivation and significance)

They leave open the broader question of which problems and instance classes admit a potential quantum advantage.

— Complexity Barriers to State Preparation in Quantum Approximate Optimization  (2609.31520 - Hadfield, 25 Sep 2026) in Section 7, Outlook

An important open question is whether such limited constraint information is sufficient to optimize quantum algorithms when exact solutions are unavailable.

— Reinforcement LearningtoHarness Approximation Errors for Long-Time QuantumSimulation  (2608.20139 - Shi et al., 20 Aug 2026) in Discussion and outlook section, final paragraph before Acknowledgments

Determining whether coherent learning can provide asymptotic advantages over hybrid optimization remains an important open question.

— Quantum Hamiltonian Evolution for Coherent Quantum Learning  (2609.03640 - Acedo et al., 3 Sep 2026) in Numerical demonstrations, subsection (a) Quantum classification with a data re-uploading model

Several other questions remain open. The first is to find hard optimization families with useful nonstationary initial states, such as states produced by classical approximation algorithms or states localized in metastable regions.

— Super-Quadratic Quantum Speedups for Combinatorial Optimization via Tilted Walks  (2609.40088 - Ozgul et al., 30 Sep 2026) in Section 2, Discussion, paragraph “Future Work”

It is also an interesting direction to identify different structured problem classes where the tilted walk framework yields larger advantage over the classical baselines.

— Super-Quadratic Quantum Speedups for Combinatorial Optimization via Tilted Walks  (2609.40088 - Ozgul et al., 30 Sep 2026) in Section 2, Discussion, paragraph “Future Work”