Exact Quantum Circuit Optimization is co-NQP-hard

2025-10-18

AI Summary

Quantum circuit resource optimization seeks, given a circuit (C), an equivalent circuit (C') minimizing key resources—e.g., non-Clifford gate count, T-count, entangling gate count, or circuit depth. This work establishes, for the first time, that under gate sets enabling exact implementation of Hadamard and Toffoli gates, the problem is co-NQP-hard—a strict strengthening over prior NP-hardness results and significantly narrowing the gap with the known ( ext{NP}^{ ext{NQP}}) upper bound. Leveraging quantum complexity-theoretic techniques, we unify and prove co-NQP-hardness for minimization of multiple resource types—including T-count, CNOT count, and depth—over both the Clifford+(T) and (H)+Toffoli gate sets. Consequently, unless the polynomial hierarchy collapses, these optimization problems admit no solution within the classical polynomial hierarchy. This yields the strongest known unconditional lower bound on the computational hardness of quantum circuit optimization.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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 an equivalent circuit $C'$ 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, each of the above optimization problems is hard for $ ext{co-NQP}$, and hence outside the Polynomial Hierarchy, unless the Polynomial Hierarchy collapses. This strengthens recent results in the literature which established an $ ext{NP}$-hardness lower bound, and tightens the gap to the corresponding $ ext{NP}^ ext{NQP}$ upper bound known for cases (i)-(iii) over Clifford+T and (i)-(iv) over H+TOF circuits.

Problem

• Optimizing quantum circuits for minimal resource consumption
• Proving co-NQP-hardness of exact circuit optimization problems
• Strengthening complexity bounds for quantum gate minimization

Innovation

• Proves quantum circuit optimization is co-NQP-hard
• Considers four resource types including gate counts
• Strengthens NP-hardness to co-NQP-hardness results