- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

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

Certifying Ground-State Properties of Many-Body Systems

Jie Wang1, Jacopo Surace2,3, Irénée Frérot4,5, Benoît Legat6, Marc-Olivier Renou7,8, Victor Magron9, and Antonio Acín2,10

PDF

Phys. Rev. X 14, 031006 – Published 11 July, 2024

DOI: https://doi.org/10.1103/PhysRevX.14.031006

Abstract

A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin-1/2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

The interplay of particle interactions and quantum fluctuations can stabilize intriguing phases of matter, and the resulting quantum entanglement sometimes represents a resource for quantum computations. Yet, it is one of the most difficult problems for numerical methods, as the system’s phase space grows exponentially with the number of particles. In this work, we introduce a numerical scheme to derive upper and lower bounds on arbitrary ground-state observables for many-body quantum systems. This scheme can be systematically refined to converge to the exact ground-state value.

We illustrate the potential of our numerical method in paradigmatic spin models in one and two dimensions. But the construction is much more general and can be applied to systems of both distinguishable and indistinguishable particles as well as arbitrary lattice geometries. The symmetries and sparsity of the studied model can be used to significantly improve the scalability of the computation.

In future work, our approach could be used to certify the output of quantum simulators, as well as to study phases of matter and phase transitions in complex quantum systems.

Article Text

References (73)

  1. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 2011).
  2. D. Wu, R. Rossi, F. Vicentini, N. Astrakhantsev, F. Becca, X. Cao, J. Carrasquilla, F. Ferrari, A. Georges, M. Hibat-Allah et al., Variational benchmarks for quantum many-body problems, arXiv:2302.04919.
  3. B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. K.-L. Chan, Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155 (2017).
  4. M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm, J. Chem. Phys. 114, 8282 (2001).
  5. D. A. Mazziotti and R. M. Erdahl, Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles, Phys. Rev. A 63, 042113 (2001).
  6. T. Barthel and R. Hübener, Solving condensed-matter ground-state problems by semidefinite relaxations, Phys. Rev. Lett. 108, 200404 (2012).
  7. T. Baumgratz and M. B. Plenio, Lower bounds for ground states of condensed matter systems, New J. Phys. 14, 023027 (2012).
  8. M. Navascués, S. Pironio, and A. Acín, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations, New J. Phys. 10, 073013 (2008).
  9. S. Pironio, M. Navascués, and A. Acín, Convergent relaxations of polynomial optimization problems with noncommuting variables, SIAM J. Optim. 20, 2157 (2010).
  10. J. Helton and S. McCullough, A positivstellensatz for non-commutative polynomials, Trans. Am. Math. Soc. 356, 3721 (2004).
  11. A. C. Doherty, Y.-C. Liang, B. Toner, and S. Wehner, The quantum moment problem and bounds on entangled multi-prover games, in Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity (CCC ’08) (IEEE Computer Society, College Park, MD, 2008), pp. 199–210.
  12. X. Han, Quantum many-body bootstrap, arXiv:2006.06002.
  13. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  14. U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  15. S. Östlund and S. Rommer, Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett. 75, 3537 (1995).
  16. U. Schollwöck, in January 2011 Special Issue, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (Amsterdam) 326, 96 (2011).
  17. F. Verstraete, V. Murg, and J. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  18. 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).
  19. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (Amsterdam) 349, 117 (2014).
  20. G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett. 98, 070201 (2007).
  21. G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99, 220405 (2007).
  22. G. Evenbly and G. Vidal, Algorithms for entanglement renormalization: Boundaries, impurities and interfaces, J. Stat. Phys. 157, 931 (2014).
  23. S. Liang, B. Doucot, and P. W. Anderson, Some new variational resonating-valence-bond-type wave functions for the spin-12 antiferromagnetic Heisenberg model on a square lattice, Phys. Rev. Lett. 61, 365 (1988).
  24. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  25. F. Mezzacapo, N. Schuch, M. Boninsegni, and J. I. Cirac, Ground-state properties of quantum many-body systems: Entangled-plaquette states and variational monte carlo, New J. Phys. 11, 083026 (2009).
  26. H. J. Changlani, J. M. Kinder, C. J. Umrigar, and G. K.-L. Chan, Approximating strongly correlated wave functions with correlator product states, Phys. Rev. B 80, 245116 (2009).
  27. J. Thibaut, T. Roscilde, and F. Mezzacapo, Long-range entangled-plaquette states for critical and frustrated quantum systems on a lattice, Phys. Rev. B 100, 155148 (2019).
  28. M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).
  29. X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  30. Z. Landau, U. Vazirani, and T. Vidick, A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians, Nat. Phys. 11, 566 (2015).
  31. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Computational complexity of projected entangled pair states, Phys. Rev. Lett. 98, 140506 (2007).

  32. Note that MQ forms a convex set: This property follows directly from the convexity of the set of quantum density matrices.

  33. Y.-K. Liu, M. Christandl, and F. Verstraete, Quantum computational complexity of the n-representability problem: QMA complete, Phys. Rev. Lett. 98, 110503 (2007).
  34. P. W. Anderson, Limits on the energy of the antiferromagnetic ground state, Phys. Rev. 83, 1260 (1951).
  35. W. Slofstra, The set of quantum correlations is not closed, Forum Math. Pi 7, e1 (2019).
  36. X. Han, S. A. Hartnoll, and J. Kruthoff, Bootstrapping matrix quantum mechanics, Phys. Rev. Lett. 125, 041601 (2020).
  37. J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11, 796 (2001).
  38. See Jie Wang, QMBCertify version 0.1.0 (2023), https://github.com/wangjie212/QMBCertify; See also Benoît Legat and Jacopo Surace, CondensedMatterSOS.jl, v0.1.0 (2024), https://github.com/blegat/CondensedMatterSOS.jl for other related codes.
  39. J. Wang, A more efficient reformulation of complex SDP as real SDP, arXiv:2307.11599.
  40. T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, New York, 2003).
  41. A. Haim, R. Kueng, and G. Refael, Variational-correlations approach to quantum many-body problems, arXiv:2001.06510.
  42. S. R. White and I. Affleck, Dimerization and incommensurate spiral spin correlations in the zigzag spin chain: Analogies to the Kondo lattice, Phys. Rev. B 54, 9862 (1996).
  43. C. K. Majumdar and D. K. Ghosh, On next-nearest-neighbor interaction in linear chain. I, J. Math. Phys. (N.Y.) 10, 1388 (1969).
  44. A. W. Sandvik, Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model, Phys. Rev. B 56, 11678 (1997).
  45. M. E. Zhitomirsky and K. Ueda, Valence-bond crystal phase of a frustrated spin-1/2 square-lattice antiferromagnet, Phys. Rev. B 54, 9007 (1996).
  46. L. Capriotti and S. Sorella, Spontaneous plaquette dimerization in the J1−J2 Heisenberg model, Phys. Rev. Lett. 84, 3173 (2000).
  47. H.-C. Jiang, H. Yao, and L. Balents, Spin liquid ground state of the spin-12 square J1−J2 Heisenberg model, Phys. Rev. B 86, 024424 (2012).
  48. L. Wang, D. Poilblanc, Z.-C. Gu, X.-G. Wen, and F. Verstraete, Constructing a gapless spin-liquid state for the spin-1/2 J1−J2 Heisenberg model on a square lattice, Phys. Rev. Lett. 111, 037202 (2013).
  49. S.-S. Gong, W. Zhu, D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Plaquette ordered phase and quantum phase diagram in the spin-12 J1−J2 square Heisenberg model, Phys. Rev. Lett. 113, 027201 (2014).
  50. K. Choo, T. Neupert, and G. Carleo, Two-dimensional frustrated J1−J2 model studied with neural network quantum states, Phys. Rev. B 100, 125124 (2019).
  51. L. Wang and A. W. Sandvik, Critical level crossings and gapless spin liquid in the square-lattice spin-1/2 J1−J2 Heisenberg antiferromagnet, Phys. Rev. Lett. 121, 107202 (2018).
  52. Y. Nomura and M. Imada, Dirac-type nodal spin liquid revealed by refined quantum many-body solver using neural-network wave function, correlation ratio, and level spectroscopy, Phys. Rev. X 11, 031034 (2021).
  53. F. Ferrari and F. Becca, Gapless spin liquid and valence-bond solid in the J1−J2 Heisenberg model on the square lattice: Insights from singlet and triplet excitations, Phys. Rev. B 102, 014417 (2020).
  54. W.-Y. Liu, S.-S. Gong, Y.-B. Li, D. Poilblanc, W.-Q. Chen, and Z.-C. Gu, Gapless quantum spin liquid and global phase diagram of the spin-1/2 J1−J2 square antiferromagnetic Heisenberg model, Sci. Bull. 67, 1034 (2022).
  55. M. Cho, B. Gabai, Y.-H. Lin, V. A. Rodriguez, J. Sandor, and X. Yin, Bootstrapping the Ising model on the lattice, arXiv:2206.12538.
  56. B. Requena, G. Muñoz Gil, M. Lewenstein, V. Dunjko, and J. Tura, Certificates of quantum many-body properties assisted by machine learning, Phys. Rev. Res. 5, 013097 (2023).
  57. H. Fawzi, O. Fawzi, and S. O. Scalet, Certified algorithms for equilibrium states of local quantum Hamiltonians, arXiv:2311.18706.
  58. M. Araújo, I. Klep, T. Vértesi, A. J. P. Garner, and M. Navascues, Karush-Kuhn-Tucker conditions for non-commutative optimization problems, arXiv:2311.18707.
  59. H. Fawzi, O. Fawzi, and S. O. Scalet, Entropy constraints for ground energy optimization, arXiv:2305.06855.
  60. I. Kull, N. Schuch, B. Dive, and M. Navascués, Lower bounding ground-state energies of local Hamiltonians through the renormalization group, Phys. Rev. X 14, 021008 (2024).
  61. V. Magron, X. Allamigeon, S. Gaubert, and B. Werner, Formal proofs for nonlinear optimization, J. Formaliz. Reason. 8, 1 (2015).
  62. K. Cafuta, I. Klep, and J. Povh, Rational sums of Hermitian squares of free noncommutative polynomials, Ars Math. Contemp. 9, 243 (2015).
  63. J. Wang and L. Hu, Solving low-rank semidefinite programs via manifold optimization, arXiv:2303.01722.
  64. K. F. Pál and T. Vértesi, Quantum bounds on Bell inequalities, Phys. Rev. A 79, 022120 (2009).
  65. J. B. Lasserre, Convergent SDP-relaxations in polynomial optimization with sparsity, SIAM J. Optim. 17, 822 (2006).
  66. I. Klep, V. Magron, and J. Povh, Sparse noncommutative polynomial optimization, Math. Program., 193, 789 (2021).
  67. J. Wang, V. Magron, and J.-B. Lasserre, TSSOS: A moment-SOS hierarchy that exploits term sparsity, SIAM J. Optim. 31, 30 (2021).
  68. J. Wang and V. Magron, Exploiting term sparsity in noncommutative polynomial optimization, Comput. Optim. Applic. 80, 483 (2021).
  69. V. Magron and J. Wang, Sparse Polynomial Optimization: Theory and Practice (World Scientific, Singapore, 2023).
  70. J. Lofberg, Pre-and post-processing sum-of-squares programs in practice, IEEE Trans. Autom. Control 54, 1007 (2009).
  71. T. Hrga, I. Klep, and J. Povh, Certifying optimality of Bell inequality violations: Noncommutative polynomial optimization through semidefinite programming and local optimization, SIAM J. Optim. 34, 1341 (2024).
  72. C. Riener, T. Theobald, L. J. Andrén, and J. B. Lasserre, Exploiting symmetries in SDP-relaxations for polynomial optimization, Math. Oper. Res. 38, 122 (2013).
  73. H. J. Schulz, T. A. L. Ziman, and D. Poilblanc, Magnetic order and disorder in the frustrated quantum Heisenberg antiferromagnet in two dimensions, J. Phys. I (France) 6, 675 (1996).

Outline

Information

Phys. Rev. X 14, 031006– Published 11 July, 2024
  • Received 2 November 2023
  • Accepted 9 May 2024
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