Qubit-count optimization using ZX-calculus

TL;DR

This paper introduces methods for reducing qubit counts in quantum circuits using ZX-calculus, including reversing gadgetization of Hadamard gates and graph-based path-decomposition, with proven NP-hardness and broad applicability.

Contribution

It presents novel algorithms for qubit optimization in quantum circuits via ZX-calculus, including NP-hardness proof and a general graph-theoretic approach applicable to various models.

Findings

Reversing Hadamard gadgetization can reduce qubits.

The qubit optimization problem is NP-hard.

Graph-based methods can optimize qubits across models.

Abstract

We propose several methods for optimizing the number of qubits in a quantum circuit while preserving the number of non-Clifford gates. One of our approaches consists in reversing, as much as possible, the gadgetization of Hadamard gates, which is a procedure used by some $T$-count optimizers to circumvent Hadamard gates at the expense of additional qubits. We prove the NP-hardness of this problem and we present an algorithm for solving it. We also propose a more general approach to optimize the number of qubits by showing how it relates to the problem of finding a minimal-width path-decomposition of the graph associated with a given ZX-diagram. This approach can be used to optimize the number of qubits for any computational model that can natively be depicted in ZX-calculus, such as the Pauli Fusion computational model which can represent lattice surgery operations. We also show how this method can be used to efficiently optimize the number of qubits in a quantum circuit by using the ZX-calculus as an intermediate representation.