- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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

Quantum Simulation with Sum-of-Squares Spectral Amplification

Robbie King1,2,*, Guang Hao Low1, Ryan Babbush1, Rolando D. Somma1,†, and Nicholas C. Rubin1,‡

  • *Contact author: robbieking@google.com
  • †Contact author: rsomma@google.com
  • ‡Contact author: nickrubin@google.com
PDF

Phys. Rev. Lett. 136, 110601 – Published 17 March, 2026

DOI: https://doi.org/10.1103/m3fj-m4rm

Abstract

We present sum-of-squares spectral amplification (SOSSA), a framework for improving quantum simulation relevant to low-energy problems. We show how SOSSA can be applied to problems like energy and phase estimation and provide fast quantum algorithms for these problems that significantly improve over prior art. To illustrate the power of SOSSA in applications, we consider the Sachdev-Ye-Kitaev model, a representative strongly correlated system, and demonstrate asymptotic speedups over generic simulation methods by a factor of the square root of the system size. Our results reinforce those observed in [G. H. Low et al., arXiv:2502.15882], where SOSSA was used to achieve state-of-the-art gate costs for phase estimation of real-world quantum chemistry systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (45)

  1. E. Knill, G. Ortiz, and R. D. Somma, Phys. Rev. A 75, 012328 (2007).
  2. G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017).
  3. A. Y. Kitaev, arXiv:quant-ph/9511026.
  4. S. Lloyd, Science 273, 1073 (1996).
  5. N. Wiebe, D. Berry, P. Høyer, and B. C. Sanders, J. Phys. A 43, 065203 (2010).
  6. D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Phys. Rev. Lett. 114, 090502 (2015).
  7. D. W. Berry, Y. Tong, T. Khattar, A. White, T. I. Kim, G. H. Low, S. Boixo, Z. Ding, L. Lin, S. Lee, G. K.-L. Chan, R. Babbush, and N. C. Rubin, PRX Quantum 6, 020327 (2025).
  8. N. C. Rubin, D. W. Berry, F. D. Malone, A. F. White, T. Khattar, A. Eugene DePrince III, S. Sicolo, M. Küehn, M. Kaicher, J. Lee, and R. Babbush, PRX Quantum 4, 040303 (2023).
  9. R. Babbush, J. R. McClean, M. Newman, C. Gidney, S. Boixo, and H. Neven, PRX Quantum 2, 010103 (2021).
  10. I. Loaiza and A. F. Izmaylov, J. Chem. Theory Comput. 19, 8201 (2023).
  11. J. Haah, M. B. Hastings, R. Kothari, and G. H. Low, SIAM J. Comput. 52, FOCS18-i (2023).
  12. A. Zlokapa and R. D. Somma, Quantum 8, 1449 (2024).
  13. R. D. Somma and S. Boixo, SIAM J. Comput. 42, 593 (2013).
  14. G. H. Low and I. L. Chuang, arXiv:1707.05391.
  15. M. X. Goemans and D. P. Williamson, J. ACM 42, 1115 (1995).
  16. S. Pironio, M. Navascués, and A. Acin, SIAM J. Optim. 20, 2157 (2010).
  17. D. A. Mazziotti, Acc. Chem. Res. 39, 207 (2006).
  18. M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, J. Chem. Phys. 114, 8282 (2001).
  19. M. Navascués, S. Pironio, and A. Acín, New J. Phys. 10, 073013 (2008).
  20. G. H. Low, R. King, D. W. Berry, Q. Han, A. E. DePrince III, A. White, R. Babbush, R. D. Somma, and N. C. Rubin, Phys. Rev. X 15, 041016 (2025).
  21. See Supplemental Material at http://link.aps.org/supplemental/10.1103/m3fj-m4rm for quantum simulation with sum-of-squares spectral amplification.
  22. G. H. Low and I. L. Chuang, Quantum 3, 163 (2019).
  23. A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (Phoenix AZ USA, 2019), pp. 193–204.
  24. J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, PRX Quantum 2, 040203 (2021).
  25. A. M. Childs and N. Wiebe, Quantum Inf. Comput. 12, 901 (2012).
  26. D. W. Berry, M. Kieferová, A. Scherer, Y. R. Sanders, G. H. Low, N. Wiebe, C. Gidney, and R. Babbush, npj Quantum Inf. 4, 22 (2018).
  27. N. S. Mande and R. de Wolf, arXiv:2305.04908.
  28. A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Proc. Natl. Acad. Sci. U.S.A. 115, 9456 (2018).
  29. Z. Jiang, A. Kalev, W. Mruczkiewicz, and H. Neven, Quantum 4, 276 (2020).
  30. R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Phys. Rev. X 8, 041015 (2018).
  31. Y. He, M.-X. Luo, E. Zhang, H.-K. Wang, and X.-F. Wang, Int. J. Theor. Phys. 56, 2350 (2017).
  32. G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Contemp. Math. 305, 53 (2002).
  33. S. Simon, M. Degroote, N. Moll, R. Santagati, M. Streif, and N. Wiebe, arXiv:2402.14791.
  34. G. H. Low and Y. Su, arXiv:2410.18178.
  35. Y. Ge, J. Tura, and J. I. Cirac, arXiv:1712.03193.
  36. L. Lin and Y. Tong, Quantum 4, 372 (2020).
  37. J. Lee, D. W. Berry, C. Gidney, W. J. Huggins, J. R. McClean, N. Wiebe, and R. Babbush, PRX Quantum 2, 030305 (2021).
  38. V. von Burg, G. H. Low, T. Häner, D. S. Steiger, M. Reiher, M. Roetteler, and M. Troyer, Phys. Rev. Res. 3, 033055 (2021).
  39. M. B. Hastings and R. O’Donnell, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (Wien, Austria, 2022), pp. 776–789, 10.1145/3519935.3519960.
  40. B. Baumgartner, arXiv:1106.6189.
  41. N. C. Rubin, R. Babbush, and J. McClean, New J. Phys. 20, 053020 (2018).
  42. S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, England, 2004).
  43. S. Burer and R. D. Monteiro, Math. Program. 95, 329 (2003).
  44. J. Povh, F. Rendl, and A. Wiegele, Computing 78, 277 (2006).
  45. R. M. Erdahl, Int. J. Quantum Chem. 13, 697 (1978).

Outline

Information

Phys. Rev. Lett. 136, 110601– Published 17 March, 2026
  • Received 5 May 2025
  • Revised 18 September 2025
  • Accepted 12 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation