Optimising quantum circuits is generally hard

TL;DR

This paper demonstrates that optimizing various gate counts in quantum circuits, such as T-count and CNOT gates, is NP-hard, highlighting the computational difficulty of quantum circuit optimization.

Contribution

It establishes the NP-hardness of multiple quantum circuit optimization problems, including T-count, CNOT, Hadamard, and Toffoli gate counts, using reductions to Boolean satisfiability.

Findings

Optimizing T-count in Clifford+T circuits is NP-hard.

Optimizing CNOT and Hadamard gate counts is NP-hard.

Hardness results extend to Toffoli gates and general non-Clifford gates.

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.