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Optimising quantum circuits is generally hard

John van de Wetering, Matt Amy

TL;DR

The paper addresses the hardness of optimising gate counts in Clifford+T circuits, a core concern for fault-tolerant quantum computation. It employs SAT-based reductions, using the oracle unitary $U_f$ and the diagonal gadget $C_f$ to link satisfiability to the presence of non-Clifford gates, thereby proving NP-hardness for $T$-count, $T$-depth, Toffoli-count, CNOT-count, Hadamard-count, and $G$-count under small approximation errors. It further establishes an upper bound placing $T$-count and Toffoli-count in $ ext{NP}^{ ext{NQP}}$, with Hadamard-count sharing this bound, while noting the lack of a known general bound for $CNOT$-count. These results imply inherent challenges in resource estimation and optimization for fault-tolerant quantum circuits and open questions about the exact complexity and approximability of these problems across different gate-sets.

Abstract

In order for quantum computations to be done as efficiently as possible it is important to optimise the number of gates used in the underlying quantum circuits. In this paper we find that many gate optimisation problems for approximately universal quantum circuits are NP-hard. In particular, we show that optimising the T-count or T-depth in Clifford+T circuits, which are important metrics for the computational cost of executing fault-tolerant quantum computations, is NP-hard by reducing the problem to Boolean satisfiability. With a similar argument we show that optimising the number of CNOT gates or Hadamard gates in a Clifford+T circuit is also NP-hard. Again varying the same argument we also establish the hardness of optimising the number of Toffoli gates in a reversible classical circuit. We find an upper bound to the problems of T-count and Toffoli-count of $\text{NP}^{\text{NQP}}$. Finally, we also show that for any non-Clifford gate $G$ it is NP-hard to optimise the $G$-count over the Clifford+$G$ gate set, where we only have to match the target unitary within some small distance in the operator norm.

Optimising quantum circuits is generally hard

TL;DR

The paper addresses the hardness of optimising gate counts in Clifford+T circuits, a core concern for fault-tolerant quantum computation. It employs SAT-based reductions, using the oracle unitary and the diagonal gadget to link satisfiability to the presence of non-Clifford gates, thereby proving NP-hardness for -count, -depth, Toffoli-count, CNOT-count, Hadamard-count, and -count under small approximation errors. It further establishes an upper bound placing -count and Toffoli-count in , with Hadamard-count sharing this bound, while noting the lack of a known general bound for -count. These results imply inherent challenges in resource estimation and optimization for fault-tolerant quantum circuits and open questions about the exact complexity and approximability of these problems across different gate-sets.

Abstract

In order for quantum computations to be done as efficiently as possible it is important to optimise the number of gates used in the underlying quantum circuits. In this paper we find that many gate optimisation problems for approximately universal quantum circuits are NP-hard. In particular, we show that optimising the T-count or T-depth in Clifford+T circuits, which are important metrics for the computational cost of executing fault-tolerant quantum computations, is NP-hard by reducing the problem to Boolean satisfiability. With a similar argument we show that optimising the number of CNOT gates or Hadamard gates in a Clifford+T circuit is also NP-hard. Again varying the same argument we also establish the hardness of optimising the number of Toffoli gates in a reversible classical circuit. We find an upper bound to the problems of T-count and Toffoli-count of . Finally, we also show that for any non-Clifford gate it is NP-hard to optimise the -count over the Clifford+ gate set, where we only have to match the target unitary within some small distance in the operator norm.
Paper Structure (9 sections, 8 theorems, 10 equations)

This paper contains 9 sections, 8 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Hardness of circuit optimisation
  4. Hardness of Toffoli-count optimisation
  5. Hardness of CNOT-count optimisation
  6. Hardness of Hadamard-count optimization
  7. Hardness of general approximate circuit optimisation
  8. An upper bound to the hardness of T-count and Toffoli optimisation
  9. Conclusion and outlook

Key Result

Theorem 1

T-COUNT is NP-hard under polynomial-time Turing reductions.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Theorem 8
  • proof
  • Proposition 9