Accurately Simulating the Time Evolution of an Ising Model with Echo Verified Clifford Data Regression on a Superconducting Quantum Computer

Tim Weaving; Alexis Ralli; Peter J. Love; Sauro Succi; Peter V. Coveney

doi:10.22331/q-2025-05-05-1732

Accurately Simulating the Time Evolution of an Ising Model with Echo Verified Clifford Data Regression on a Superconducting Quantum Computer

Tim Weaving¹, Alexis Ralli^1,2, Peter J. Love^2,3, Sauro Succi^4,5,6, and Peter V. Coveney^1,7,8

¹Centre for Computational Science, Department of Chemistry, University College London, WC1H 0AJ, United Kingdom
²Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA
³Computational Science Initiative, Brookhaven National Laboratory, Upton, NY 11973, USA
⁴Center for Life Nano-Neuro Science @ La Sapienza, Italian Institute of Technology, 00161 Roma, Italy
⁵Department of Mechanical Engineering, University College London, WC1E 7JE, United Kingdom
⁶Department of Physics, Harvard University, Cambridge, MA 02138, USA
⁷Advanced Research Computing Centre, University College London, WC1H 0AJ, United Kingdom
⁸Informatics Institute, University of Amsterdam, Amsterdam, 1098 XH, Netherlands

Published:	2025-05-05, volume 9, page 1732
Editor:	Mohan Sarovar
Eprint:	arXiv:2408.07439v2
Doi:	https://doi.org/10.22331/q-2025-05-05-1732
Citation:	Quantum 9, 1732 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We present an error mitigation strategy composed of Echo Verification (EV) and Clifford Data Regression (CDR), the combination of which allows one to learn the effect of the quantum noise channel to extract error mitigated estimates for the expectation value of Pauli observables. We analyse the behaviour of the method under the depolarizing channel and derive an estimator for the depolarization rate in terms of the ancilla purity and postselection probability. We also highlight the sensitivity of this probability to noise, a potential bottleneck for the technique. We subsequently consider a more general noise channel consisting of arbitrary Pauli errors, which reveals a linear relationship between the error rates and the estimation of expectation values, suggesting the learnability of noise in EV by regression techniques. Finally, we present a practical demonstration of Echo Verified Clifford Data Regression (EVCDR) on a superconducting quantum computer and observe accurate results for the time evolution of an Ising model over spin-lattices consisting of up to 35 sites and circuit depths up to 173 entangling layers.

Featured image: Overview of the Echo Verified Clifford Data Regression (EVCDR) framework. The results obtained from a standard Echo Verification routine are biased with a linear noise damping factor that is learned via the Clifford Data Regression procedure. This is achieved by randomly sampling near-Clifford training circuits by approximating all but a few of the rotation gates by their nearest Clifford counterpart (for example, by rounding gate angles to an integer multiple of $\frac{\pi}{2}$). By evaluating the training circuits both classically and on the quantum hardware we may learn the effect of the underlying noise channel and subsequently suppress error in the final energy estimate. We may then reflect estimates from the true non-Clifford circuit through the Clifford fitting curves to obtain CDR mitigated estimates that are then combined in the same way as the standard EV estimator.

► BibTeX data

@article{Weaving2025accurately,
  doi = {10.22331/q-2025-05-05-1732},
  url = {https://doi.org/10.22331/q-2025-05-05-1732},
  title = {Accurately {S}imulating the {T}ime {E}volution of an {I}sing {M}odel with {E}cho {V}erified {C}lifford {D}ata {R}egression on a {S}uperconducting {Q}uantum {C}omputer},
  author = {Weaving, Tim and Ralli, Alexis and Love, Peter J. and Succi, Sauro and Coveney, Peter V.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1732},
  month = may,
  year = {2025}
}

► References

[1] S. Bravyi, S. Sheldon, A. Kandala, D. C. Mckay, and J. M. Gambetta, Mitigating measurement errors in multiqubit experiments, Physical Review A 103, 042605 (2021).
https://doi.org/10.1103/PhysRevA.103.042605

[2] P. D. Nation, H. Kang, N. Sundaresan, and J. M. Gambetta, Scalable mitigation of measurement errors on quantum computers, PRX Quantum 2, 040326 (2021).
https://doi.org/10.1103/PRXQuantum.2.040326

[3] M. Urbanek, B. Nachman, V. R. Pascuzzi, A. He, C. W. Bauer, and W. A. de Jong, Mitigating depolarizing noise on quantum computers with noise-estimation circuits, Physical review letters 127, 270502 (2021).
https://doi.org/10.1103/PhysRevLett.127.270502

[4] M. R. Geller and Z. Zhou, Efficient error models for fault-tolerant architectures and the pauli twirling approximation, Phys. Rev. A 88, 012314 (2013).
https://doi.org/10.1103/PhysRevA.88.012314

[5] Z. Cai and S. C. Benjamin, Constructing smaller pauli twirling sets for arbitrary error channels, Scientific reports 9, 11281 (2019).
https://doi.org/10.1038/s41598-019-46722-7

[6] J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Physical Review A 94, 052325 (2016).
https://doi.org/10.1103/PhysRevA.94.052325

[7] A. Hashim, R. K. Naik, A. Morvan, J.-L. Ville, B. Mitchell, J. M. Kreikebaum, M. Davis, E. Smith, C. Iancu, K. P. O'Brien, I. Hincks, J. J. Wallman, J. Emerson, and I. Siddiqi, Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor, Phys. Rev. X 11, 041039 (2021).
https://doi.org/10.1103/PhysRevX.11.041039

[8] Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Physical Review X 7, 021050 (2017).
https://doi.org/10.1103/PhysRevX.7.021050

[9] K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Physical review letters 119, 180509 (2017).
https://doi.org/10.1103/PhysRevLett.119.180509

[10] S. Endo, S. C. Benjamin, and Y. Li, Practical quantum error mitigation for near-future applications, Physical Review X 8, 031027 (2018).
https://doi.org/10.1103/PhysRevX.8.031027

[11] A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Error mitigation extends the computational reach of a noisy quantum processor, Nature 567, 491 (2019).
https://doi.org/10.1038/s41586-019-1040-7

[12] T. Giurgica-Tiron, Y. Hindy, R. LaRose, A. Mari, and W. J. Zeng, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, 2020) pp. 306–316.
https://doi.org/10.1109/QCE49297.2020.00045

[13] A. He, B. Nachman, W. A. de Jong, and C. W. Bauer, Zero-noise extrapolation for quantum-gate error mitigation with identity insertions, Physical Review A 102, 012426 (2020).
https://doi.org/10.1103/PhysRevA.102.012426

[14] A. Mari, N. Shammah, and W. J. Zeng, Extending quantum probabilistic error cancellation by noise scaling, Physical Review A 104, 052607 (2021).
https://doi.org/10.1103/PhysRevA.104.052607

[15] O. G. Maupin, A. D. Burch, B. Ruzic, C. G. Yale, A. Russo, D. S. Lobser, M. C. Revelle, M. N. Chow, S. M. Clark, A. J. Landahl, et al., Error mitigation, optimization, and extrapolation on a trapped-ion testbed, Physical Review A 110, 032416 (2024).
https://doi.org/10.1103/PhysRevA.110.032416

[16] T. Weaving, A. Ralli, P. J. Love, S. Succi, and P. V. Coveney, Contextual subspace variational quantum eigensolver calculation of the dissociation curve of molecular nitrogen on a superconducting quantum computer, npj Quantum Information 11, 25 (2025).
https://doi.org/10.1038/s41534-024-00952-4

[17] T. E. O’Brien, S. Polla, N. C. Rubin, W. J. Huggins, S. McArdle, S. Boixo, J. R. McClean, and R. Babbush, Error mitigation via verified phase estimation, PRX Quantum 2, 020317 (2021).
https://doi.org/10.1103/PRXQuantum.2.020317

[18] Z. Cai, Resource-efficient purification-based quantum error mitigation, arXiv preprint (2021), 2107.07279.
arXiv:2107.07279

[19] T. Weaving, A. Ralli, W. M. Kirby, P. J. Love, S. Succi, and P. V. Coveney, Benchmarking noisy intermediate scale quantum error mitigation strategies for ground state preparation of the hcl molecule, Phys. Rev. Res. 5, 043054 (2023).
https://doi.org/10.1103/PhysRevResearch.5.043054

[20] O. Kiss, M. Grossi, and A. Roggero, Quantum error mitigation for fourier moment computation, Physical Review D 111, 034504 (2025).
https://doi.org/10.1103/PhysRevD.111.034504

[21] B. F. Schiffer, D. van Vreumingen, J. Tura, and S. Polla, Virtual mitigation of coherent non-adiabatic transitions by echo verification, Quantum 8, 1346 (2024).
https://doi.org/10.22331/q-2024-05-14-1346

[22] M. Huo and Y. Li, Dual-state purification for practical quantum error mitigation, Physical Review A 105, 022427 (2022).
https://doi.org/10.1103/PhysRevA.105.022427

[23] W. J. Huggins, S. McArdle, T. E. O’Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Virtual distillation for quantum error mitigation, Physical Review X 11, 041036 (2021).
https://doi.org/10.1103/PhysRevX.11.041036

[24] Z. Liu, X. Zhang, Y.-Y. Fei, and Z. Cai, Virtual channel purification, arXiv preprint (2024), 2402.07866 [quant-ph].
arXiv:2402.07866

[25] M. C. Tran, A. Y. Guo, Y. Su, J. R. Garrison, Z. Eldredge, M. Foss-Feig, A. M. Childs, and A. V. Gorshkov, Locality and digital quantum simulation of power-law interactions, Physical Review X 9, 031006 (2019).
https://doi.org/10.1103/PhysRevX.9.031006

[26] L. Leone, S. F. Oliviero, L. Cincio, and M. Cerezo, On the practical usefulness of the hardware efficient ansatz, Quantum 8, 1395 (2024).
https://doi.org/10.22331/q-2024-07-03-1395

[27] M. C. Tran, K. Sharma, and K. Temme, Locality and error mitigation of quantum circuits, arXiv preprint (2023), 2303.06496.
arXiv:2303.06496

[28] X. Mi, P. Roushan, C. Quintana, S. Mandra, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Babbush, et al., Information scrambling in quantum circuits, Science 374, 1479 (2021).
https://doi.org/10.1126/science.abg5029

[29] P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, Error mitigation with clifford quantum-circuit data, Quantum 5, 592 (2021).
https://doi.org/10.22331/q-2021-11-26-592

[30] D. Gottesman, Stabilizer codes and quantum error correction (California Institute of Technology, 1997) quant-ph/9705052.
arXiv:quant-ph/9705052

[31] A. Ralli and T. Weaving, symmer, https://github.com/UCL-CCS/symmer (2022).
https://github.com/UCL-CCS/symmer

[32] A. C. Aitken, On least squares and linear combination of observations, Proceedings of the Royal Society of Edinburgh 55, 42 (1936).
https://doi.org/10.1017/S0370164600014346

[33] M. S. ANIS et al., Qiskit: An open-source framework for quantum computing (2021).
https://doi.org/10.5281/zenodo.2573505

[34] S. Seabold and J. Perktold, in 9th Python in Science Conference (2010).
https://doi.org/10.25080/Majora-92bf1922-011

[35] T. Weaving, EVCDR code and data repository, https://github.com/TimWeaving/EVTools (2025).
https://github.com/TimWeaving/EVTools

[36] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of trotter error with commutator scaling, Physical Review X 11, 011020 (2021).
https://doi.org/10.1103/PhysRevX.11.011020

[37] S. Aaronson and Y. Zhang, On verifiable quantum advantage with peaked circuit sampling, arXiv preprint (2024), 2404.14493.
arXiv:2404.14493

[38] R. Blume-Kohout and K. C. Young, A volumetric framework for quantum computer benchmarks, Quantum 4, 362 (2020).
https://doi.org/10.22331/q-2020-11-15-362

Cited by

[1] Tim Weaving, Angus Mingare, Alexis Ralli, and Peter V. Coveney, "Selected Configuration Interaction Using Time-Evolved Population Statistics", Journal of Chemical Theory and Computation 22 9, 4315 (2026).

[2] Tim Weaving, Alexis Ralli, Peter J. Love, Sauro Succi, and Peter V. Coveney, "Contextual subspace variational quantum eigensolver calculation of the dissociation curve of molecular nitrogen on a superconducting quantum computer", npj Quantum Information 11 1, 25 (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-10 08:55:05) and SAO/NASA ADS (last updated successfully 2026-08-08 21:06:13). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-08-10 08:55:05: Cannot retrieve data from ADS due to rate limitations.

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.