- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Applied

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Practical implementation of Toffoli-based qubit rotation

Christoffer Hindlycke*, Jakov Krnic†, and Jan-Åke Larsson‡

  • *Contact author: christoffer.hindlycke@liu.se
  • †Contact author: jakov.krnic@liu.se
  • ‡Contact author: jan-ake.larsson@liu.se
PDF

Phys. Rev. Applied 25, 014050 – Published 21 January, 2026

DOI: https://doi.org/10.1103/29z2-15bm

Abstract

The Toffoli gate is an important universal quantum gate, and will alongside the Clifford gates be available in future fault-tolerant quantum computing hardware. Many quantum algorithms rely on performing arbitrarily small single-qubit rotations for their function, and these rotations may also be used to construct any unitary from a limited (but universal) gate set. How to carry out such rotations is then of significant interest. In this work, we evaluate the performance of a recently proposed single-qubit rotation algorithm using the Clifford plus Toffoli gate set by implementation of a one-shot version on both a real and a simulated quantum computer. We test the algorithm under various simulated noise levels using a per-qubit depolarizing error noise model and examine how the probabilities and process fidelities are affected. We then conduct live runs and find that the results reasonably match the simulated results. We also attempt to model the hardware noise by combining a number of noise models, matching the results to results of the live runs to approximate the hardware noise. Our results suggest that the algorithm will perform well for up to 1% noise under the noise models we chose. We further posit the use of our algorithm as a benchmark for quantum processing units, given that it has a low complexity that is easy to fine-tune in small steps. We provide details for how to do this.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (26)

  1. P. W. Shor, Algorithms for Quantum Computation: Discrete Logarithm and Factoring, in Proceedings of 35th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Santa Fe, 1994), pp. 124–134.
  2. P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484 (1997).
  3. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995).
  4. D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Experimental quantum error correction, Phys. Rev. Lett. 81, 2152 (1998).
  5. E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne, Benchmarking quantum computers: The five-qubit error correcting code, Phys. Rev. Lett. 86, 5811 (2001).
  6. J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri, and D. J. Wineland, Realization of quantum error correction, Nature 432, 602 (2004).
  7. M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Realization of three-qubit quantum error correction with superconducting circuits, Nature 482, 382 (2012).
  8. D. Nigg, M. Müller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, Quantum computations on a topologically encoded qubit, Science 345, 302 (2014).
  9. S. Bravyi, D. Gosset, and R. König, Quantum advantage with shallow circuits, Science 362, 308 (2018).
  10. J. Preskill, Reliable quantum computers, Proc. R. Soc. London. Series A: Math., Phys. Eng. Sci. 454, 385 (1998).
  11. B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, Simplifying quantum logic using higher-dimensional Hilbert spaces, Nat. Phys. 5, 134 (2009).
  12. T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Realization of the quantum Toffoli gate with trapped ions, Phys. Rev. Lett. 102, 040501 (2009).
  13. L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Demonstration of two-qubit algorithms with a superconducting quantum processor, Nature 460, 240 (2009).
  14. A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, Implementation of a Toffoli gate with superconducting circuits, Nature 481, 170 (2012).
  15. H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletić, H. Pichler, and M. D. Lukin, Parallel implementation of high-fidelity multiqubit gates with neutral atoms, Phys. Rev. Lett. 123, 170503 (2019).
  16. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2010), Vol. 10th Anniversary Edition.
  17. C. Hindlycke and J.-Å. Larsson, Single-qubit rotation algorithm with logarithmic Toffoli count and gate depth, Phys. Rev. Res. 6, L042027 (2024).
  18. J. B. Altepeter, D. F. James, and P. G. Kwiat, in Quantum State Estimation, edited by M. Paris and J. Řeháček (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004), p. 113.
  19. I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, J. Mod. Opt. 44, 2455 (1997).
  20. A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross, B. R. Johnson, and J. M. Gambetta, Quantum computing with Qiskit, arXiv:2405.08810.
  21. IBM Quantum, https://quantum.ibm.com/ (2025).
  22. M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Phys. Lett. A 303, 249 (2002).
  23. L. Viola, E. Knill, and S. Lloyd, Dynamical decoupling of open quantum systems, Phys. Rev. Lett. 82, 2417 (1999).
  24. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett. 76, 722 (1996).
  25. P. W. Shor, Fault-tolerant quantum computation, in Proceedings of 37th Conference on Foundations of Computer Science (1996), pp. 56–65.
  26. J. A. Smolin, G. Smith, and A. Vargo, Oversimplifying quantum factoring, Nature 499, 163 (2013).

Outline

Information

Phys. Rev. Applied 25, 014050– Published 21 January, 2026
  • Received 3 September 2025
  • Accepted 9 December 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation