Quantum circuit synthesis with SQiSW

TL;DR

This paper explores the use of the SQiSW gate for quantum circuit synthesis, demonstrating optimized methods for exact and approximate synthesis of multi-qubit gates, including a new scheme for the Toffoli gate, leveraging Lie theory and numerical algorithms.

Contribution

It introduces novel synthesis bounds and an efficient scheme for Toffoli gate implementation using only SQiSW gates, advancing quantum circuit optimization techniques.

Findings

Exact synthesis bounds for 3- and n-qubit gates using SQiSW.

An 8 SQiSW gate scheme for Toffoli gate synthesis.

A pruning algorithm reducing search space in numerical synthesis.

Abstract

The primary objective of quantum circuit synthesis is to efficiently and accurately realize specific quantum algorithms or operations utilizing a predefined set of quantum gates, while also optimizing the circuit size. It holds a pivotal position in Noisy Intermediate-Scale Quantum (NISQ) computation. Historically, most synthesis efforts have predominantly utilized CNOT or CZ gates as the 2-qubit gates. However, the SQiSW gate, also known as the square root of iSWAP gate, has garnered considerable attention due to its outstanding experimental performance with low error rates and high efficiency in 2-qubit gate synthesis. In this paper, we investigate the potential of the SQiSW gate in various synthesis problems by utilizing only the SQiSW gate along with arbitrary single-qubit gates, while optimizing the overall circuit size. For exact synthesis, the upper bound of SQiSW gates to synthesize arbitrary 3-qubit and $n$-qubit gates are 24 and $\frac{139}{192}4^n(1+o(1))$ respectively, which relies on the properties of SQiSW gate in Lie theory and Quantum Shannon Decomposition. We also introduce an exact synthesis scheme for Toffoli gate using only 8 SQiSW gates, which is grounded in numerical observation. More generally, with respect to numerical approximations, we provide a theoretical analysis of a pruning algorithm to reduce the size of the searching space in numerical experiment to $\frac{1}{12}+o(1)$ of previous size, helping us reach the result that 11 SQiSW gates are enough in arbitrary 3-qubit gates synthesis up to an acceptable numerical error.