Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact Quantum Circuit Optimization is co-NQP-hard

Adam Husted Kjelstrøm, Andreas Pavlogiannis, Jaco van de Pol

TL;DR

This work shows that exact quantum circuit optimization across fixed gate-sets that can implement $H$ and $TOF$ is $ ext{co-} extbf{NQP}$-hard, placing these optimization tasks outside the Polynomial Hierarchy unless $ extbf{PH}$ collapses. The authors introduce a Deutsch-Josza-based gadget and a $ extsc{IsBalanced}$–to–$ extsc{Promise-IsIdentity}( ext{A})$ reduction to prove $ ext{co-} extbf{NQP}$-completeness for the identity-promise problem under gate-sets like $ ext{H}+ ext{TOF}$ and Clifford+$ ext{T}$. They show that minimizing total gates, non-Clifford gates, superposition gates, or entanglement gates all inherit this hardness, thereby tightening prior NP-hardness results and bridging to the known $ extbf{NP}^{ extbf{NQP}}$ upper bound. The results have broad implications for exact circuit canonicalization and database-style optimization in quantum compilation, especially for hardware platforms supporting $H$ and $TOF$-type operations.

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 $\text{co-NQP}$, and hence outside the Polynomial Hierarchy, unless the Polynomial Hierarchy collapses. This strengthens recent results in the literature which established an $\text{NP}$-hardness lower bound, 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.

Exact Quantum Circuit Optimization is co-NQP-hard

TL;DR

This work shows that exact quantum circuit optimization across fixed gate-sets that can implement and is -hard, placing these optimization tasks outside the Polynomial Hierarchy unless collapses. The authors introduce a Deutsch-Josza-based gadget and a –to– reduction to prove -completeness for the identity-promise problem under gate-sets like and Clifford+. They show that minimizing total gates, non-Clifford gates, superposition gates, or entanglement gates all inherit this hardness, thereby tightening prior NP-hardness results and bridging to the known upper bound. The results have broad implications for exact circuit canonicalization and database-style optimization in quantum compilation, especially for hardware platforms supporting and -type operations.

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 , computing an equivalent circuit 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 is expressed over any gate set that can implement the H and TOF gates exactly, each of the above optimization problems is hard for , and hence outside the Polynomial Hierarchy, unless the Polynomial Hierarchy collapses. This strengthens recent results in the literature which established an -hardness lower bound, and tightens the gap to the corresponding upper bound known for cases (i)-(iii) over Clifford+T and (i)-(iv) over H+TOF circuits.
Paper Structure (6 sections, 3 theorems, 14 equations, 3 figures)

This paper contains 6 sections, 3 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. A Deutsch-Josza-Based Gadget
  4. A Proof of Theorem 1
  5. Conclusion
  6. Acknowledgements

Key Result

Theorem 1

For any gate-set $\mathcal{A}$ that can implement $\text{H}$ and $\text{TOF}$ exactly, $\textsc{Promise-IsIdentity}(\mathcal{A})$ is $\text{co-}\mathbf{NQP}$-complete.

Figures (3)

  • Figure 1: A circuit implementing the Deutsch-Josza algorithm DeutschJozsa1992RapidSolutionProblemsQuantumComputation.
  • Figure 2: The gadget $Q$, parameterized by $G$ and $\varphi$. The gadget uses $n$ ancillae and a controlled gate $\Lambda^n G$.
  • Figure 3: A circuit describing $C$.

Theorems & Definitions (8)

  • Theorem 1
  • Corollary 1
  • Definition 1: $\mathbf{NQP}$
  • Definition 2: $\mathbf{C}_{=}\mathbf{P}$
  • Definition 3: $\textsc{IsBalanced}$
  • Definition 4: $\mathbf{PH}$
  • Lemma 1
  • proof