- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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
  • Letter
  • Open Access

Average-computation benchmarking for local expectation values in digital quantum devices

Flavio Baccari1,2,*,†, Pavel Kos3,4,*,‡, and Georgios Styliaris3,4,*,§

  • *These authors contributed equally to this work.
  • †Contact author: flavio.baccari@unipd.it
  • ‡Contact author: pavel.kos@fmf.uni-lj.si
  • §Contact author: georgios.styliaris@mpq.mpg.de
PDF

Phys. Rev. Research 8, L012065 – Published 17 March, 2026

DOI: https://doi.org/10.1103/zn6p-c8w2

Abstract

As quantum devices progress toward a quantum advantage regime, they become harder to benchmark. A particularly relevant challenge is to assess the quality of the whole computation, beyond testing the performance of each single operation. Here, we introduce a scheme for this task that combines the target computation with variants of it, which, when averaged, allow for classically solvable correlation functions. Importantly, the variants exactly preserve the circuit architecture and depth, without simplifying the gates into a classically simulable set. The method is based on replacing each gate by an ensemble of similar gates, which when averaged together form space-time channels [P. Kos and G. Styliaris, Quantum 7, 1020 (2023)]. We introduce explicit constructions for ensembles producing such channels, all applicable to arbitrary brickwork circuits, and provide a general recipe to find new ones through semidefinite programming. The resulting average computation retains important information about the original circuit and is able to detect noise beyond a Clifford benchmarking regime. Moreover, we provide evidence that estimating average-computation expectation values requires running only a limited number of different circuit realizations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (54)

  1. A. Morvan, B. Villalonga, X. Mi, S. Mandra, A. Bengtsson, P. Klimov, Z. Chen, S. Hong, C. Erickson, I. Drozdov, et al., Phase transitions in random circuit sampling, Nature (London) 634, 328 (2024).
  2. A. D. King, A. Nocera, M. M. Rams, J. Dziarmaga, R. Wiersema, W. Bernoudy, J. Raymond, N. Kaushal, N. Heinsdorf, R. Harris, et al., Computational supremacy in quantum simulation, Science 388, 199 (2025).
  3. F. Pan, K. Chen, and P. Zhang, Solving the sampling problem of the sycamore quantum circuits, Phys. Rev. Lett. 129, 090502 (2022).
  4. D. Aharonov, X. Gao, Z. Landau, Y. Liu, and U. Vazirani, A polynomial-time classical algorithm for noisy random circuit sampling, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing (ACM Press, New York, NY, 2023), pp. 945–957.
  5. C. Oh, M. Liu, Y. Alexeev, B. Fefferman, and L. Jiang, Classical algorithm for simulating experimental Gaussian Boson sampling, Nat. Phys. 20, 1461 (2024).
  6. K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017).
  7. Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X 7, 021050 (2017).
  8. A. W. Smith, K. E. Khosla, C. N. Self, and M. Kim, Qubit readout error mitigation with bit-flip averaging, Sci. Adv. 7, eabi8009 (2021).
  9. D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, et al., Logical quantum processor based on reconfigurable atom arrays, Nature (London) 626, 58 (2024).
  10. Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2025).
  11. J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum certification and benchmarking, Nat. Rev. Phys. 2, 382 (2020).
  12. 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).
  13. E. Nielsen, J. K. Gamble, K. Rudinger, T. Scholten, K. Young, and R. Blume-Kohout, Gate set tomography, Quantum 5, 557 (2021).
  14. E. Magesan, J. M. Gambetta, and J. Emerson, Scalable and robust randomized benchmarking of quantum processes, Phys. Rev. Lett. 106, 180504 (2011).
  15. R. Harper, S. T. Flammia, and J. J. Wallman, Efficient learning of quantum noise, Nat. Phys. 16, 1184 (2020).
  16. D. C. McKay, A. W. Cross, C. J. Wood, and J. M. Gambetta, Correlated randomized benchmarking, arXiv:2003.02354.
  17. E. Van Den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Nat. Phys. 19, 1116 (2023).
  18. J. Hines, M. Lu, R. K. Naik, A. Hashim, J.-L. Ville, B. Mitchell, J. M. Kriekebaum, D. I. Santiago, S. Seritan, E. Nielsen, et al., Demonstrating scalable randomized benchmarking of universal gate sets, Phys. Rev. X 13, 041030 (2023).
  19. S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Characterizing quantum supremacy in near-term devices, Nat. Phys. 14, 595 (2018).
  20. D. Hangleiter, M. Kliesch, M. Schwarz, and J. Eisert, Direct certification of a class of quantum simulations, Quantum Sci. Technol. 2, 015004 (2017).
  21. A. Gheorghiu, T. Kapourniotis, and E. Kashefi, Verification of quantum computation: An overview of existing approaches, Theory Comput. Syst. 63, 715 (2019).
  22. E. Cruz, F. Baccari, J. Tura, N. Schuch, and J. I. Cirac, Preparation and verification of tensor network states, Phys. Rev. Res. 4, 023161 (2022).
  23. T. Proctor, S. Seritan, E. Nielsen, K. Rudinger, K. Young, R. Blume-Kohout, and M. Sarovar, Establishing trust in quantum computations, arXiv:2204.07568.
  24. T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, Measuring the capabilities of quantum computers, Nat. Phys. 18, 75 (2022).
  25. S. Ferracin, T. Kapourniotis, and A. Datta, Accrediting outputs of noisy intermediate-scale quantum computing devices, New J. Phys. 21, 113038 (2019).
  26. Z. Wang, Y. Chen, Z. Song, D. Qin, H. Li, Q. Guo, H. Wang, C. Song, and Y. Li, Scalable evaluation of quantum-circuit error loss using Clifford sampling, Phys. Rev. Lett. 126, 080501 (2021).
  27. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, et al., Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  28. D. Gottesman, The Heisenberg representation of quantum computers, arXiv:quant-ph/9807006.
  29. R. Jozsa and A. Miyake, Matchgates and classical simulation of quantum circuits, Proc. R. Soc. A 464, 3089 (2008).
  30. S. Bravyi and D. Gosset, Improved classical simulation of quantum circuits dominated by Clifford gates, Phys. Rev. Lett. 116, 250501 (2016).
  31. R. S. Bennink, E. M. Ferragut, T. S. Humble, J. A. Laska, J. J. Nutaro, M. G. Pleszkoch, and R. C. Pooser, Unbiased simulation of near-Clifford quantum circuits, Phys. Rev. A 95, 062337 (2017).
  32. J. Carrasco, M. Langer, A. Neven, and B. Kraus, Gaining confidence on the correct realization of arbitrary quantum computations, Phys. Rev. Res. 6, L032074 (2024).
  33. S. Merkel, T. Proctor, S. Ferracin, J. Hines, S. Barron, L. C. G. Govia, and D. McKay, When Clifford benchmarks are sufficient; estimating application performance with scalable proxy circuits, arXiv:2503.05943.
  34. We always assume no single-qubit gate in the layout by simply merging each of them to the nearest two-qubit gate in the circuit.
  35. L. G. Valiant, Quantum circuits that can be simulated classically in polynomial time, SIAM J. Comput. 31, 1229 (2002).
  36. B. M. Terhal and D. P. DiVincenzo, Classical simulation of noninteracting-fermion quantum circuits, Phys. Rev. A 65, 032325 (2002).
  37. B. Bertini, P. Kos, and T. Prosen, Exact correlation functions for dual-unitary lattice models in 1+1 dimensions, Phys. Rev. Lett. 123, 210601 (2019).
  38. P. Kos and G. Styliaris, Circuits of space and time quantum channels, Quantum 7, 1020 (2023).
  39. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  40. B. Kraus and J. I. Cirac, Optimal creation of entanglement using a two-qubit gate, Phys. Rev. A 63, 062309 (2001).
  41. J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A 94, 052325 (2016).
  42. P. Kos, B. Bertini, and T. Prosen, Correlations in perturbed dual-unitary circuits: Efficient path-integral formula, Phys. Rev. X 11, 011022 (2021).
  43. See Supplemental Material at http://link.aps.org/supplemental/10.1103/zn6p-c8w2 for details, which includes Refs.  [44, 45].
  44. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Transforming quantum operations: Quantum supermaps, Europhys. Lett. 83, 30004 (2008).
  45. N. Yoshioka, M. Amico, W. Kirby, P. Jurcevic, A. Dutt, B. Fuller, S. Garion, H. Haas, I. Hamamura, A. Ivrii, R. Majumdar, Z. Minev, M. Motta, B. Pokharel, P. Rivero, K. Sharma, C. J. Wood, A. Javadi-Abhari, and A. Mezzacapo, Krylov diagonalization of large many-body Hamiltonians on a quantum processor, Nat. Commun. 16, 5014 (2025).
  46. S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004).
  47. S. T. Flammia and R. O’Donnell, Pauli error estimation via population recovery, Quantum 5, 549 (2021).
  48. C. Rouzé and D. S. Franca, Efficient learning of the structure and parameters of local Pauli noise channels, arXiv:2307.02959.
  49. F. Baccari, P. Kos, and G. Styliaris, Average-computation benchmarking for local expectation values in digital quantum devices, Zenodo (2025), https://doi.org/10.5281/zenodo.16731873.
  50. L. E. Fischer, M. Leahy, A. Eddins, N. Keenan, D. Ferracin, M. A. C. Rossi, Y. Kim, A. He, F. Pietracaprina, B. Sokolov, S. Dooley, Z. Zimborás, F. Tacchino, S. Maniscalco, J. Goold, G. García-Pérez, I. Tavernelli, A. Kandala, and S. N. Filippov, Dynamical simulations of many-body quantum chaos on a quantum computer, Nat. Phys. 22, 302 (2026).
  51. For other positions, the expectation values vanish.
  52. R. M. Milbradt, L. Scheller, C. Aßmus, and C. B. Mendl, Ternary unitary quantum lattice models and circuits in 2 + 1 dimensions, Phys. Rev. Lett. 130, 090601 (2023).
  53. R. Suzuki, K. Mitarai, and K. Fujii, Computational power of one- and two-dimensional dual-unitary quantum circuits, Quantum 6, 631 (2022).
  54. C. Jonay, V. Khemani, and M. Ippoliti, Triunitary quantum circuits, Phys. Rev. Res. 3, 043046 (2021).

Outline

Information

Phys. Rev. Research 8, L012065– Published 17 March, 2026
  • Received 4 August 2025
  • Accepted 18 February 2026
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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation