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arXiv:2407.10171

Qubit-count optimization using ZX-calculus

Vivien Vandaele

Published Jul 14, 2024• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2407.10171

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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 TTT-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.
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The Clifford+ T gate set is commonly used to perform universal quantum computation. In such setup the T gate is typically much more expensive to implement in a fault-tolerant way than Clifford gates. To improve the feasibility of fault-tolerant quantum computing it is then crucial to minimize the number of T gates. Many algorithms, yielding effective results, have been designed to address this problem. It has been demonstrated that performing a pre-processing step consisting of reducing the number of Hadamard gates in the circuit can help to exploit the full potential of these algorithms and thereby lead to a substantial T -count reduction. Moreover, minimizing the number of Hadamard gates also restrains the number of additional qubits and operations resulting from the gadgetization of Hadamard gates, a procedure used by some compilers to further reduce the number of T gates. In this work we tackle the Hadamard gate reduction problem, and propose an algorithm for synthesizing a sequence of π /4 Pauli rotations with a minimal number of Hadamard gates. Based on this result, we present an algorithm which optimally minimizes the number of Hadamard gates lying between the first and the last T gate of the circuit.
Optimising quantum circuits to minimise resource usage is crucial, especially with near-term hardware limited by quantum volume. This paper introduces an optimisation algorithm aiming to minimise non-Clifford gate count and two-qubit gate count by building on ZX-calculus-based strategies. By translating a circuit into a ZX-diagram it can be simplified before being extracted back into a circuit. We assert that simplifications preserve a graph-theoretic property called causal flow. This has the advantage that qubit lines are well defined throughout, permitting a trivial extraction procedure and in turn enabling the calculation of an individual transformation's impact on the resulting circuit. A general procedure for a decision strategy is introduced, inspired by an existing heuristic based method. Both phase teleportation and the neighbour unfusion rule are generalised. In particular, allowing unfusion of multiple neighbours is shown to lead to significant improvements in optimisation. When run on a set of benchmark circuits, the algorithm developed reduces the two-qubit gate count by an average of 19.8%, beating both the previous best ZX-based strategy (14.6%) and non-ZX strategy (18.5%) at the time of publication. This lays a foundation for multiple avenues of improvement. A particularly effective strategy for optimising QFT circuits is also noted, resulting in exactly one two-qubit gate per non-Clifford gate.
Parametrised quantum circuits contain phase gates whose phase is determined by a classical algorithm prior to running the circuit on a quantum device. Such circuits are used in variational algorithms like QAOA and VQE. In order for these algorithms to be as efficient as possible it is important that we use the fewest number of parameters. We show that, while the general problem of minimising the number of parameters is NP-hard, when we restrict to circuits that are Clifford apart from parametrised phase gates and where each parameter is used just once, we can efficiently find the optimal parameter count. We show that when parameter transformations are required to be sufficiently well-behaved, the only rewrites that reduce parameters correspond to simple `fusions'. Using this we find that a previous circuit optimisation strategy by some of the authors [Kissinger, van de Wetering. PRA (2019)] finds the optimal number of parameters. Our proof uses the ZX-calculus. We also prove that the standard rewrite rules of the ZX-calculus suffice to prove any equality between parametrised Clifford circuits.