Quantum circuit synthesis with SQiSW

TL;DR

This work investigates quantum circuit synthesis using only the SQiSW gate and single-qubit gates, motivated by SQiSW’s favorable experimental performance. It establishes exact-synthesis bounds, proving that an arbitrary 3-qubit gate can be realized with at most $24$ SQiSW gates and any $n$-qubit gate with at most $\frac{139}{192}4^n(1+o(1))$ SQiSW gates, plus an exact Toffoli circuit using $8$ SQiSW gates. It also develops a numerical optimization framework with a pruning strategy that reduces the search space to $\frac{1}{12}+o(1)$ of the original and demonstrates practical results: 8 and 11 SQiSW gates suffice for Toffoli and arbitrary 3-qubit gates, respectively, under acceptable error. Overall, the paper advances both exact and numerical approaches to gate-synthesis with SQiSW, offering tighter bounds and actionable schemes for implementing larger circuits on NISQ devices.

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.

Figures (6)

• Figure 1: Weyl chamber in $\mathbb{R}^3$, with some common gates and their interaction coefficients. Note that there are many ways to draw Weyl chamber, and here we adopt that in Huang2023.
• Figure 2: Region in Weyl chamber that can be spanned by 2 SQiSW gates, filled in red Huang2023.
• Figure 3: CNOT gate is locally equivalent to CZ gate, and a CZ gate is equal to a CNOT gate with $R_y$ gates on both sides of its target qubit.
• Figure 4: A quantum circuit with structure $(0,1), (1,2), (0,2)$.
• Figure 5: The logarithm of best error E in numerical optimization, using $N$ SQiSW gates in the circuit.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 1: Quantum multiplexor Shende2006
• Lemma 1: Decomposition of multiplexor rotation gates Shende2006bullock2003Mottonen2004
• Lemma 2: KAK decomposition tucci2005
• Definition 2: Local equivalence Huang2023
• Lemma 3: Synthesis of 2-qubit gates using SQiSW Huang2023
• Lemma 4: Decomposition of arbitrary 3-qubit gates chen2023