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Quantum circuit synthesis with SQiSW

Jialiang Tang, Jialin Zhang, and Xiaoming Sun

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AbstractThe 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.► BibTeX data@article{Tang2025quantumcircuit, doi = {10.22331/q-2025-10-20-1889}, url = {https://doi.org/10.22331/q-2025-10-20-1889}, title = {Quantum circuit synthesis with {SQ}i{SW}}, author = {Tang, Jialiang and Zhang, Jialin and Sun, Xiaoming}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1889}, month = oct, year = {2025} }► References [1] V.V. Shende, S.S. Bullock, and I.L. Markov. ``Synthesis of quantum-logic circuits''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). https://doi.org/10.1109/TCAD.2005.855930 [2] Mikko Möttönen, Juha J. Vartiainen, Ville Bergholm, and Martti M. Salomaa. ``Quantum circuits for general multiqubit gates''. Phys. Rev. Lett. 93, 130502 (2004). https://doi.org/10.1103/PhysRevLett.93.130502 [3] Jiaqing Jiang, Xiaoming Sun, Shang-Hua Teng, Bujiao Wu, Kewen Wu, and Jialin Zhang. ``Optimal space-depth trade-off of CNOT circuits in quantum logic synthesis''. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA). Pages 213–229. (2020). https://doi.org/10.1137/1.9781611975994.13 [4] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter. ``Elementary gates for quantum computation''. Phys. Rev. A 52, 3457–3467 (1995). https://doi.org/10.1103/PhysRevA.52.3457 [5] Alfred V. Aho and Krysta M. Svore. ``Compiling quantum circuits using the palindrome transform'' (2003). arXiv:quant-ph/0311008. arXiv:quant-ph/0311008 [6] Matthew Amy, Parsiad Azimzadeh, and Michele Mosca. ``On the controlled-not complexity of controlled-not–phase circuits''. Quantum Science and Technology 4, 015002 (2018). https://doi.org/10.1088/2058-9565/aad8ca [7] Shuai Yang, Guojing Tian, Jialin Zhang, and Xiaoming Sun. ``Quantum circuit synthesis on noisy intermediate-scale quantum devices''. Phys. Rev. A 109, 012602 (2024). https://doi.org/10.1103/PhysRevA.109.012602 [8] G. Cybenko. ``Reducing quantum computations to elementary unitary operations''. Computing in Science & Engineering 3, 27–32 (2001). https://doi.org/10.1109/5992.908999 [9] Farrokh Vatan and Colin P. Williams. ``Realization of a general three-qubit quantum gate'' (2004). arXiv:quant-ph/0401178. arXiv:quant-ph/0401178 [10] E. 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Shende, I.L. Markov, and S.S. Bullock. ``Smaller two-qubit circuits for quantum communication and computation''.

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IEEE (2023). https://doi.org/10.1109/qce57702.2023.00096 [41] Francesco Mezzadri. ``How to generate random matrices from the classical compact groups'' (2007). arXiv:math-ph/0609050. arXiv:math-ph/0609050 [42] Nengkun Yu, Runyao Duan, and Mingsheng Ying. ``Five two-qubit gates are necessary for implementing the toffoli gate''. Phys. Rev. A 88, 010304 (2013). https://doi.org/10.1103/PhysRevA.88.010304 [43] Vivek V. Shende, Igor L. Markov, and Stephen S. Bullock. ``Minimal universal two-qubit controlled-not-based circuits''. Phys. Rev. A 69, 062321 (2004). https://doi.org/10.1103/PhysRevA.69.062321Cited byOn Crossref's cited-by service no data on citing works was found (last attempt 2025-10-24 02:03:38). Could not fetch ADS cited-by data during last attempt 2025-10-24 02:03:38: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThe 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.► BibTeX data@article{Tang2025quantumcircuit, doi = {10.22331/q-2025-10-20-1889}, url = {https://doi.org/10.22331/q-2025-10-20-1889}, title = {Quantum circuit synthesis with {SQ}i{SW}}, author = {Tang, Jialiang and Zhang, Jialin and Sun, Xiaoming}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1889}, month = oct, year = {2025} }► References [1] V.V. Shende, S.S. Bullock, and I.L. Markov. ``Synthesis of quantum-logic circuits''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). https://doi.org/10.1109/TCAD.2005.855930 [2] Mikko Möttönen, Juha J. Vartiainen, Ville Bergholm, and Martti M. Salomaa. ``Quantum circuits for general multiqubit gates''. Phys. Rev. Lett. 93, 130502 (2004). https://doi.org/10.1103/PhysRevLett.93.130502 [3] Jiaqing Jiang, Xiaoming Sun, Shang-Hua Teng, Bujiao Wu, Kewen Wu, and Jialin Zhang. ``Optimal space-depth trade-off of CNOT circuits in quantum logic synthesis''. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA). Pages 213–229. (2020). https://doi.org/10.1137/1.9781611975994.13 [4] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter. ``Elementary gates for quantum computation''. Phys. Rev. A 52, 3457–3467 (1995). https://doi.org/10.1103/PhysRevA.52.3457 [5] Alfred V. Aho and Krysta M. Svore. ``Compiling quantum circuits using the palindrome transform'' (2003). arXiv:quant-ph/0311008. arXiv:quant-ph/0311008 [6] Matthew Amy, Parsiad Azimzadeh, and Michele Mosca. ``On the controlled-not complexity of controlled-not–phase circuits''. Quantum Science and Technology 4, 015002 (2018). https://doi.org/10.1088/2058-9565/aad8ca [7] Shuai Yang, Guojing Tian, Jialin Zhang, and Xiaoming Sun. ``Quantum circuit synthesis on noisy intermediate-scale quantum devices''. Phys. Rev. A 109, 012602 (2024). https://doi.org/10.1103/PhysRevA.109.012602 [8] G. Cybenko. ``Reducing quantum computations to elementary unitary operations''. Computing in Science & Engineering 3, 27–32 (2001). https://doi.org/10.1109/5992.908999 [9] Farrokh Vatan and Colin P. Williams. ``Realization of a general three-qubit quantum gate'' (2004). arXiv:quant-ph/0401178. arXiv:quant-ph/0401178 [10] E. Knill. ``Bounds for approximation in total variation distance by quantum circuits'' (1995). arXiv:quant-ph/9508007. arXiv:quant-ph/9508007 [11] Wei Zi, Junhong Nie, and Xiaoming Sun. ``Shallow quantum circuit implementation of symmetric functions with limited ancillary qubits''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 44, 3060–3072 (2025). https://doi.org/10.1109/TCAD.2025.3539002 [12] S.-B. Wang, P. Wang, G.-H. Li, et al. ``Variational quantum eigensolver with linear depth problem-inspired ansatz for solving portfolio optimization in finance''. Sci. China Inf. Sci. 68, 180504:1–180504:11 (2025). https://doi.org/10.1007/s11432-024-4185-1 [13] E. Knill. ``Approximation by quantum circuits'' (1995). arXiv:quant-ph/9508006. arXiv:quant-ph/9508006 [14] Juha J. Vartiainen, Mikko Möttönen, and Martti M. Salomaa. ``Efficient decomposition of quantum gates''. Phys. Rev. Lett. 92, 177902 (2004). https://doi.org/10.1103/PhysRevLett.92.177902 [15] V.V. Shende, I.L. Markov, and S.S. Bullock. ``Smaller two-qubit circuits for quantum communication and computation''.

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