- 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

Beating the Natural Grover Bound for Low-Energy Estimation and State Preparation

Harry Buhrman1,2,3,*, Sevag Gharibian4,†, Zeph Landau5,‡, François Le Gall6,§, Norbert Schuch7,8,∥, and Suguru Tamaki9,¶

  • 1Quantinuum, Terrington House 13-15 Hills Road Cambridge CB2 1NL, United Kingdom
  • 2University of Amsterdam, Amsterdam, The Netherlands
  • 3QuSoft, Science Park 123, 1098 XG Amsterdam, The Netherlands
  • 4Paderborn University, Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Warburger Strasse 100, 33098 Paderborn, Germany
  • 5University of California, Berkeley, Department of Computer Science, Berkeley, California 94706, USA
  • 6Nagoya University, Graduate School of Mathematics, Furocho, Chikusaku, 464-8602 Nagoya, Japan
  • 7University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
  • 8University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria
  • 9University of Hyogo, Graduate School of Information Science, 8-2-1 Gakuennishi-machi, Nishi-ku, 651-2197 Kobe, Japan

  • *Contact author: hbuhrman@gmail.com
  • †Contact author: sevag.gharibian@upb.de
  • ‡Contact author: zeph.landau@gmail.com
  • §Contact author: legall@math.nagoya-u.ac.jp
  • ∥Contact author: norbert.schuch@univie.ac.at
  • Contact author: suguru.tamaki@gmail.com
PDF

Phys. Rev. Lett. 135, 030601 – Published 15 July, 2025

DOI: https://doi.org/10.1103/29qw-bssx

Abstract

Estimating ground state energies of many-body Hamiltonians is a central task in many areas of quantum physics. In this Letter, we give quantum algorithms which, given any k-body Hamiltonian H, compute an estimate for the ground state energy and prepare a quantum state achieving said energy, respectively. Specifically, for any ϵ>0, our algorithms return, with high probability, an estimate of the ground state energy of H within additive error ϵM, or a quantum state with the corresponding energy. Here, M is the total strength of all interaction terms, which in general is extensive in the system size. Our approach makes no assumptions about the geometry or spatial locality of interaction terms of the input Hamiltonian and thus handles even long-range or all-to-all interactions, such as in quantum chemistry, where lattice-based techniques break down. In this fully general setting, the run-time of our algorithms scales as 2cn/2 for c<1, yielding the first quantum algorithms for low-energy estimation breaking a standard square root Grover speedup for unstructured search. The core of our approach is remarkably simple, and relies on showing that an extensive fraction of the interactions can be neglected with a controlled error. What this ultimately implies is that even arbitrary k-local Hamiltonians have structure in their low energy space, in the form of an exponential-dimensional low energy subspace.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (46)

  1. A. Szabo and N. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Books on Chemistry (Dover Publications, New York, 1996).
  2. T. Helgaker, P. Jorgensen, and J. Olsen, Molecular Electronic-Structure Theory (Wiley, New York, 2014).
  3. M. A. Ruderman and C. Kittel, Indirect exchange coupling of nuclear magnetic moments by conduction electrons, Phys. Rev. 96, 99 (1954).
  4. M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  5. R. Islam, C. Senko, W. Campbell, S. Korenblit, J. Smith, A. Lee, E. Edwards, C.-C. Wang, J. Freericks, and C. Monroe, Emergence and frustration of magnetic order with variable-range interactions in a trapped ion quantum simulator, Science 340, 583 (2013).
  6. R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012).
  7. B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Observation of dipolar spin-exchange interactions with lattice-confined polar molecules, Nature (London) 501, 521 (2013).
  8. D. Otten, S. Rubbert, J. Ulrich, and F. Hassler, Universal power-law decay of electron-electron interactions due to nonlinear screening in a Josephson junction array, Phys. Rev. B 94, 115403 (2016).
  9. S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, Frustration and glassiness in spin models with cavity-mediated interactions, Phys. Rev. Lett. 107, 277201 (2011).
  10. A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation, Graduate Studies in Mathematics (American Mathematical Society, Providence, 2002), Vol. 47.
  11. J. Kempe, A. Y. Kitaev, and O. Regev, The complexity of the local Hamiltonian problem, SIAM J. Comput. 35, 1070 (2006).
  12. N. Bansal, S. Bravyi, and B. M. Terhal, Classical approximation schemes for the ground-state energy of quantum and classical Ising spin Hamiltonians on planar graphs, Quantum Inf. Comput. 9, 701 (2009).
  13. D. Aharonov, I. Arad, and T. Vidick, Guest column: The quantum PCP conjecture, ACM SIGACT News 44, 47 (2013).
  14. R. Oliveira and B. M. Terhal, The complexity of quantum spin systems on a two-dimensional square lattice, Quantum Inf. Comput. 8, 0900 (2008).
  15. T. Cubitt and A. Montanaro, Complexity classification of local Hamiltonian problems, SIAM J. Comput. 45, 268 (2016).
  16. S. Piddock and A. Montanaro, The complexity of antiferromagnetic interactions and 2D lattices, Quantum Inf. Comput. 17, 636 (2017).
  17. S. Arora and S. Safra, Probabilistic checking of proofs: A new characterization of NP, ACM J. Emerging Technol. Comput. Syst. 45, 70 (1998).
  18. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and the hardness of approximation problems, ACM J. Emerging Technol. Comput. Syst. 45, 501 (1998).
  19. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. 45, 255 (1950).
  20. J. Kuczyński and H. Woźniakowski, Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start, SIAM J. Matrix Anal. Appl. 13, 1094 (1992).
  21. R. Impagliazzo and R. Paturi, On the complexity of k-SAT, J. Comput. Syst. Sci. 62, 367 (2001).
  22. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the 28th Annual ACM Symposium on Theory of Computing (1996), pp. 212–219.
  23. D. Poulin and P. Wocjan, Preparing ground states of quantum many-body systems on a quantum computer, Phys. Rev. Lett. 102, 130503 (2009).
  24. J. van Apeldoorn, A. Gilyén, S. Gribling, and R. de Wolf, Quantum SDP-solvers: Better upper and lower bounds, Quantum 4, 230 (2020).
  25. A. Kerzner, V. Gheorghiu, M. Mosca, T. Guilbaud, F. Carminati, F. Fracas, and L. Dellantonio, A square-root speedup for finding the smallest eigenvalue, arXiv:2311.04379.
  26. Y. Ge, J. Tura, and J. I. Cirac, Faster ground state preparation and high-precision ground energy estimation with fewer qubits, J. Math. Phys. (N.Y.) 60, 022202 (2019).
  27. L. Lin and Y. Tong, Near-optimal ground state preparation, Quantum 4, 372 (2020).
  28. S. Lee, J. Lee, H. Zhai, Y. Tong, A. M. Dalzell, A. Kumar, P. Helms, J. Gray, Z.-H. Cui, W. Liu, M. Kastoryano, R. Babbush, J. Preskill, D. R. Reichman, E. T. Campbell, E. F. Valeev, L. Lin, and G. K.-L. Chan, Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry, Nat. Commun. 14, 1952 (2023).

  29. Interactions acting on less qubits can be “padded” by adding further qubits on which they act trivially.

  30. See Supplemental Material at http://link.aps.org/supplemental/10.1103/29qw-bssx for improved runtime analyses, guiding state analyses, and relationship with Hirsch’s work.

  31. While any value of δ>0 can be chosen, when δ is very small we obtain r=0, which does not give any speedup (since we are left with the original Hamiltonian).

  32. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, England, 1985).

  33. Note that Cubitt and González-Guillén [34] previously gave exponential lower bounds on the low-energy spaces of local Hamiltonians, but only for history state Hamiltonians (such as Kitaev’s circuit-to-Hamiltonian construction [10]). Our result, in contrast, applies to any k-local Hamiltonian.

  34. C. E. González-Guillén and T. S. Cubitt, History-state Hamiltonians are critical, arxiv:1810.06528.
  35. G. Brassard, P. Høyer, M. Mosca, and A. Tapp, Quantum amplitude amplification and estimation, Contemp. Math. 305, 53 (2002).
  36. A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019), pp. 193–204.
  37. J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, Grand unification of quantum algorithms, PRX Quantum 2, 040203 (2021).
  38. E. A. Hirsch, Worst-case study of local search for MAX-k-SAT, Discrete Appl. Math. 130, 173 (2003).
  39. B. Escoffier, V. T. Paschos, and E. Tourniaire, Approximating MAX SAT by moderately exponential and parameterized algorithms, Theor. Comput. Sci. 560, 147 (2014).
  40. J. Alman, T. M. Chan, and R. R. Williams, Faster deterministic and las vegas algorithms for offline approximate nearest neighbors in high dimensions, in Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (2020), pp. 637–649.
  41. A. Drucker, An improved exponential-time approximation algorithm for fully-alternating games against nature, in Proceedings of the 61st IEEE Annual Symposium on Foundations of Computer Science (2020), pp. 1081–1090.
  42. T. Korhonen, A single-exponential time 2-approximation algorithm for treewidth, in Proceedings of the 62nd IEEE Annual Symposium on Foundations of Computer Science (2021), pp. 184–192.
  43. B. C. Esmer, A. Kulik, D. Marx, D. Neuen, and R. Sharma, Optimally repurposing existing algorithms to obtain exponential-time approximations, in Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms (2024), pp. 314–345.
  44. I. Dinur, Mildly exponential reduction from gap-3SAT to polynomial-gap label-cover, Electronic colloquium on computational complexity ECCC; research reports, surveys and books in computational complexity (2016).
  45. P. Manurangsi and P. Raghavendra, A birthday repetition theorem and complexity of approximating dense CSPs, in Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 80, edited by I. Chatzigiannakis, P. Indyk, F. Kuhn, and A. Muscholl (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2017), pp. 78:1–78:15.
  46. H. Buhrman, S. Patro, and F. Speelman, A framework of quantum strong exponential-time hypotheses, in Proceedings of the 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 187, edited by M. Bläser and B. Monmege (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2021), pp. 19:1–19:19.

Outline

Information

Phys. Rev. Lett. 135, 030601– Published 15 July, 2025
  • Received 4 July 2024
  • Revised 4 February 2025
  • Accepted 28 May 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