Exact Quantum Circuit Optimization is co-NQP-hard

Abstract:As quantum computing resources remain scarce and error rates high, minimizing the resource consumption of quantum circuits is essential for achieving practical quantum advantage. Here we consider the natural problem of, given a circuit $C$, computing a circuit $C'$ that behaves equivalently on a desired subspace, and that minimizes a quantum resource type, expressed as the count or depth of (i) arbitrary gates, or (ii) non-Clifford gates, or (iii) superposition gates, or (iv) entanglement gates. We show that, when $C$ is expressed over any gate-set that can implement the H and TOF gates exactly, a standard property of approximately universal gate-sets, each of the above optimization problems is hard for co-NQP, and hence outside the Polynomial Hierarchy, unless the Polynomial Hierarchy collapses. This complements a recent result of Van de Wetering and Amy (arXiv, 2023) which establishes an NP-hardness lower bound when equivalence is over the full state space, and tightens the gap to the corresponding $\text{NP}^\text{NQP}$ upper bound known for cases (i)-(iii) over Clifford+T and (i)-(iv) over H+TOF circuits. Our result also generalizes a result by Tanaka (Int. J. Quantum Inf., 2010) to other gate-sets and quantum resources.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2510.16420 [quant-ph]

(or arXiv:2510.16420v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2510.16420 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.