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Certificates of quantum many-body properties assisted by machine learning

Borja Requena1, Gorka Muñoz-Gil1,2, Maciej Lewenstein1,3, Vedran Dunjko4, and Jordi Tura5,6,*

  • 1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
  • 2Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
  • 3ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
  • 4LIACS, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
  • 5Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
  • 6Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

  • *Corresponding author: tura@lorentz.leidenuniv.nl
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Phys. Rev. Research 5, 013097 – Published 10 February, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.013097

Abstract

Computationally intractable tasks are often encountered in physics and optimization. They usually comprise a cost function to be optimized over a so-called feasible set, which is specified by a set of constraints. This may yield, in general, to difficult and nonconvex optimization tasks. A number of standard methods are used to tackle such problems: variational approaches focus on parametrizing a subclass of solutions within the feasible set. In contrast, relaxation techniques have been proposed to approximate it from outside, thus complementing the variational approach to provide ultimate bounds to the global optimal solution. In this paper, we propose a novel approach combining the power of relaxation techniques with deep reinforcement learning in order to find the best possible bounds within a limited computational budget. We illustrate the viability of the method in two paradigmatic problems in quantum physics and quantum information processing: finding the ground state energy of many-body quantum systems, and building energy-based entanglement witnesses of quantum local Hamiltonians. We benchmark our approach against other classical optimization algorithms such as breadth-first search or Monte Carlo, and we characterize the effect of transfer learning. We find the latter may be indicative of phase transitions with a completely autonomous approach. Finally, we provide tools to tackle other common applications in the field of quantum information processing with our method.

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References (193)

  1. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature (London) 549, 242 (2017).
  2. 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 (London) 567, 491 (2019).
  3. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213(2014).
  4. B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, Towards quantum chemistry on a quantum computer, Nat. Chem. 2, 106 (2010).
  5. C. Hempel, C. Maier, J. Romero, J. McClean, T. Monz, H. Shen, P. Jurcevic, B. P. Lanyon, P. Love, R. Babbush, A. Aspuru-Guzik, R. Blatt, and C. F. Roos, Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator, Phys. Rev. X 8, 031022(2018).
  6. P. O'Malley, R. Babbush, I. Kivlichan, J. Romero, J. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Mutus, et al., Scalable quantum simulation of molecular energies, Phys. Rev. X 6, 031007 (2016).
  7. T. E. O'Brien, B. Senjean, R. Sagastizabal, X. Bonet-Monroig, A. Dutkiewicz, F. Buda, L. DiCarlo, and L. Visscher, Calculating energy derivatives for quantum chemistry on a quantum computer, npj Quantum Inf. 5, 113 (2019).
  8. S. R. White, Density Matrix Formulation for Quantum Renormalization Groups, Phys. Rev. Lett. 69, 2863 (1992).
  9. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  10. F. Verstraete, D. Porras, and J. I. Cirac, Density Matrix Renormalization Group and Periodic Boundary Conditions: A Quantum Information Perspective, Phys. Rev. Lett. 93, 227205 (2004).
  11. A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, Time-dependent density-matrix renormalization-group using adaptive effective hilbert spaces, J. Stat. Mech.: Theory Exp. (2004) P04005.
  12. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  13. C. Bravo-Prieto, J. Lumbreras-Zarapico, L. Tagliacozzo, and J. I. Latorre, Scaling of variational quantum circuit depth for condensed matter systems, Quantum 4, 272 (2020).
  14. J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature (London) 549, 195 (2017).
  15. V. Dunjko and H. J. Briegel, Machine learning & artificial intelligence in the quantum domain: a review of recent progress, Rep. Prog. Phys. 81, 074001 (2018).
  16. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  17. C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, and P. Zoller, Self-verifying variational quantum simulation of lattice models, Nature (London) 569, 355 (2019).
  18. G. E. Crooks, Performance of the quantum approximate optimization algorithm on the maximum cut problem, arXiv:1811.08419.
  19. L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices, Phys. Rev. X 10, 021067 (2020).
  20. M. P. Harrigan, K. J. Sung, M. Neeley, K. J. Satzinger, F. Arute, K. Arya, J. Atalaya, J. C. Bardin, R. Barends, S. Boixo et al., Quantum approximate optimization of non-planar graph problems on a planar superconducting processor, Nat. Phys. 17, 332 (2021).
  21. Y. Herasymenko and T. O'Brien, A diagrammatic approach to variational quantum ansatz construction, Quantum 5, 596 (2021).
  22. A. Garcia-Saez and J. I. Latorre, Addressing hard classical problems with adiabatically assisted variational quantum eigensolvers arXiv:1806.02287.
  23. R. Sagastizabal, X. Bonet-Monroig, M. Singh, M. A. Rol, C. C. Bultink, X. Fu, C. H. Price, V. P. Ostroukh, N. Muthusubramanian, A. Bruno, M. Beekman, N. Haider, T. E. O'Brien, and L. DiCarlo, Experimental error mitigation via symmetry verification in a variational quantum eigensolver, Phys. Rev. A 100, 010302(2019).
  24. J. Tura, Quantum algorithms for near-term devices, in QuANtum SoftWare Engineering & pRogramming, edited by M. Piattini, G. Peterssen, R. Perez-Castillo, J. L. Hevia, and M. A. Serrano, CEURWorkshop Proceedings No. 2561 (Aachen, 2020), pp. 38–50.
  25. M. Benedetti, D. Garcia-Pintos, O. Perdomo, V. Leyton-Ortega, Y. Nam, and A. Perdomo-Ortiz, A generative modeling approach for benchmarking and training shallow quantum circuits, npj Quantum Inf. 5, 45(2019).
  26. A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Complete family of separability criteria, Phys. Rev. A 69, 022308 (2004).
  27. M. Navascués, S. Pironio, and A. Acín, Bounding the Set of Quantum Correlations, Phys. Rev. Lett. 98, 010401 (2007).
  28. F. Baccari, D. Cavalcanti, P. Wittek, and A. Acín, Efficient Device-Independent Entanglement Detection for Multipartite Systems, Phys. Rev. X 7, 021042 (2017).
  29. A. Peres, Separability Criterion for Density Matrices, Phys. Rev. Lett. 77, 1413 (1996).
  30. L. Gurvits, Classical deterministic complexity of Edmonds' problem and quantum entanglement, in Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing - STOC'03 (ACM Press, New York, 2003).
  31. A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-Independent Security of Quantum Cryptography against Collective Attacks, Phys. Rev. Lett. 98, 230501 (2007).
  32. R. Gallego, L. Masanes, G. D. L. Torre, C. Dhara, L. Aolita, and A. Acín, Full randomness from arbitrarily deterministic events, Nat. Commun. 4, 2654(2013).
  33. R. Augusiak, M. Demianowicz, M. Pawłowski, J. Tura, and A. Acín, Elemental and tight monogamy relations in nonsignaling theories, Phys. Rev. A 90, 052323(2014).
  34. W. Slofstra, The set of quantum correlations is not closed, Forum Math Pi 7, e1 (2019).
  35. S. Popescu and D. Rohrlich, Quantum nonlocality as an axiom, Found. Phys. 24, 379 (1994).
  36. G. Brassard, H. Buhrman, N. Linden, A. A. Méthot, A. Tapp, and F. Unger, Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial, Phys. Rev. Lett. 96, 250401 (2006).
  37. N. Linden, S. Popescu, A. J. Short, and A. Winter, Quantum Nonlocality and Beyond: Limits from Nonlocal Computation, Phys. Rev. Lett. 99, 180502 (2007).
  38. M. Navascués and H. Wunderlich, A glance beyond the quantum model, Proc. R. Soc. London A 466, 881 (2010).
  39. M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Zukowski, Information causality as a physical principle, Nature (London) 461, 1101 EP (2009).
  40. T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, Local orthogonality as a multipartite principle for quantum correlations, Nat. Commun. 4, 2263 (2013), article.
  41. R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, Quantum Correlations Require Multipartite Information Principles, Phys. Rev. Lett. 107, 210403 (2011).
  42. M. Navascués, Y. Guryanova, M. J. Hoban, and A. Acín, Almost quantum correlations, Nat. Commun. 6, 6288 (2015), article.
  43. 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).
  44. S. Pironio, M. Navascués, and A. Acín, Convergent relaxations of polynomial optimization problems with noncommuting variables, SIAM J. Optim. 20, 2157 (2010).
  45. P. W. Anderson, Limits on the energy of the antiferromagnetic ground state, Phys. Rev. 83, 1260 (1951).
  46. L. Epele, H. Fanchiotti, C. García-Canal, and A. Pacheco, Lower bound for the ground energy of spin systems, Physica A 173, 500 (1991).
  47. 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).
  48. G. Gidofalvi and D. A. Mazziotti, Boson correlation energies via variational minimization with the two-particle reduced density matrix: ExactN-representability conditions for harmonic interactions, Phys. Rev. A 69, 042511(2004).
  49. D. A. Mazziotti, Realization of Quantum Chemistry without Wave Functions through First-Order Semidefinite Programming, Phys. Rev. Lett. 93, 213001 (2004).
  50. T. J. Osborne, Lower bound for the ground-state energy density of a 1D quantum spin system, arXiv:cond-mat/0508428.
  51. D. A. Mazziotti, Variational reduced-density-matrix method using three-particle n-representability conditions with application to many-electron molecules, Phys. Rev. A 74, 032501 (2006).
  52. D. A. Mazziotti, Structure of fermionic density matrices: Complete N-Representability Conditions, Phys. Rev. Lett. 108, 263002(2012).
  53. D. A. Mazziotti, Enhanced Constraints for Accurate Lower Bounds on Many-Electron Quantum Energies from Variational Two-Electron Reduced Density Matrix Theory, Phys. Rev. Lett. 117, 153001 (2016).
  54. F. Uskov and O. Lychkovskiy, A variational lower bound on the ground state of a many-body system and the squaring parametrization of density matrices, J. Phys.: Conf. Ser. 1163, 012057 (2019).
  55. O. Vinyals, I. Babuschkin, W. M. Czarnecki, M. Mathieu, A. Dudzik, J. Chung, D. H. Choi, R. Powell, T. Ewalds, P. Georgiev et al., Grandmaster level in starcraft ii using multi-agent reinforcement learning, Nature (London) 575, 350 (2019).
  56. A. Ramesh, M. Pavlov, G. Goh, S. Gray, C. Voss, A. Radford, M. Chen, and I. Sutskever, Zero-shot text-to-image generation, in Proceedings of the 38th International Conference on Machine Learning, edited by M. Meila and T. Zhang, Proceedings of Machine Learning Research, Vol. 139 (PMLR, 2021), pp. 8821–8831.
  57. S. Reed, K. Zolna, E. Parisotto, S. G. Colmenarejo, A. Novikov, G. Barth-Maron, M. Gimenez, Y. Sulsky, J. Kay, J. T. Springenberg et al., A generalist agent, arXiv:2205.06175.
  58. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002(2019).
  59. A. Dawid, J. Arnold, B. Requena, A. Gresch, M. Płodzień, K. Donatella, K. A. Nicoli, P. Stornati, R. Koch, M. Büttner et al., Modern applications of machine learning in quantum sciences, arXiv:2204.04198.
  60. J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nat. Phys. 13, 431 (2017).
  61. P. Huembeli, A. Dauphin, and P. Wittek, Identifying quantum phase transitions with adversarial neural networks, Phys. Rev. B 97, 134109(2018).
  62. B. S. Rem, N. Käming, M. Tarnowski, L. Asteria, N. Fläschner, C. Becker, K. Sengstock, and C. Weitenberg, Identifying quantum phase transitions using artificial neural networks on experimental data, Nat. Phys. 15, 917 (2019).
  63. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  64. Y. Nomura, A. S. Darmawan, Y. Yamaji, and M. Imada, Restricted boltzmann machine learning for solving strongly correlated quantum systems, Phys. Rev. B 96, 205152 (2017).
  65. M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko, and J. Carrasquilla, Recurrent neural network wave functions, Phys. Rev. Res. 2, 023358(2020).
  66. Y. Bengio, A. Lodi, and A. Prouvost, Machine learning for combinatorial optimization: A methodological tour d'horizon, Eur. J. Oper. Res. 290, 405 (2021).
  67. O. Vinyals, M. Fortunato, and N. Jaitly, Pointer networks, in Advances in Neural Information Processing Systems, edited by C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett, Vol. 28 (Curran Associates, Inc., New York, 2015).
  68. N. Karalias and A. Loukas, Erdos goes neural: An unsupervised learning framework for combinatorial optimization on graphs, in Advances in Neural Information Processing Systems, edited by H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, Vol. 33 (Curran Associates, Inc., New York, 2020) pp. 6659–6672.
  69. N. Mazyavkina, S. Sviridov, S. Ivanov, and E. Burnaev, Reinforcement learning for combinatorial optimization: A survey, Comput. Oper. Res. 134, 105400 (2021).
  70. R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction (MIT press, Cambridge, 2018).
  71. D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Kumaran, T. Graepel et al., A general reinforcement learning algorithm that masters chess, shogi, and Go through self-play, Science 362, 1140 (2018).
  72. A. W. Senior, R. Evans, J. Jumper, J. Kirkpatrick, L. Sifre, T. Green, C. Qin, A. Žídek, A. W. Nelson, A. Bridgland et al., Improved protein structure prediction using potentials from deep learning, Nature (London) 577, 706 (2020).
  73. A. A. Melnikov, H. P. Nautrup, M. Krenn, V. Dunjko, M. Tiersch, A. Zeilinger, and H. J. Briegel, Active learning machine learns to create new quantum experiments, Proc. Natl. Acad. Sci. U.S.A. 115, 1221 (2018).
  74. M. Bukov, A. G. Day, D. Sels, P. Weinberg, A. Polkovnikov, and P. Mehta, Reinforcement Learning in Different Phases of Quantum Control, Phys. Rev. X 8, 031086 (2018).
  75. X.-M. Zhang, Z. Wei, R. Asad, X.-C. Yang, and X. Wang, When does reinforcement learning stand out in quantum control? A comparative study on state preparation, npj Quantum Inf. 5, 1 (2019).
  76. H. P. Nautrup, N. Delfosse, V. Dunjko, H. J. Briegel, and N. Friis, Optimizing quantum error correction codes with reinforcement learning, Quantum 3, 215 (2019).
  77. 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).
  78. N. Schuch, I. Cirac, and D. Pérez-García, PEPS as ground states: Degeneracy and topology, Ann. Phys. 325, 2153 (2010).
  79. N. Schuch and J. I. Cirac, Matrix product state and mean-field solutions for one-dimensional systems can be found efficiently, Phys. Rev. A 82, 012314(2010).
  80. Y. Zhou, E. M. Stoudenmire, and X. Waintal, What Limits the Simulation of Quantum Computers? Phys. Rev. X 10, 041038 (2020).
  81. J. Kempe and O. Regev, 3-local Hamitonian is QMA-complete, Quantum Info. Comput. 3, 258 (2003).
  82. J. Kempe, A. Kitaev, and O. Regev, The complexity of the local Hamiltonian problem, SIAM J. Comput. 35, 1070 (2006).
  83. D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, The power of quantum systems on a line, Commun. Math. Phys. 287, 41 (2009).
  84. N. Schuch and F. Verstraete, Computational complexity of interacting electrons and fundamental limitations of density functional theory, Nat. Phys. 5, 732 (2009).
  85. J. Czartowski, K. Szymański, B. Gardas, Y. V. Fyodorov, and K. Życzkowski, Separability gap and large-deviation entanglement criterion, Phys. Rev. A 100, 042326 (2019).
  86. R. Tarrach and R. Valent, Exact lower bounds to the ground-state energy of spin systems: The two-dimensional S=1/2 antiferromagnetic Heisenberg model, Phys. Rev. B 41, 9611 (1990).
  87. A. Chandran, D. Kaszlikowski, A. Sen(De), U. Sen, and V. Vedral, Regional Versus Global Entanglement in Resonating-Valence-Bond States, Phys. Rev. Lett. 99, 170502 (2007).
  88. F. Alet, D. Braun, and G. Misguich, Comment on “Regional Versus Global Entanglement in Resonating-Valence-Bond States”, Phys. Rev. Lett. 101, 248901(2008).
  89. A. Aloy, M. Fadel, and J. Tura, The quantum marginal problem for symmetric states: Applications to variational optimization, nonlocality and self-testing, New J. Phys. 23, 033026 (2021).
  90. M. Walter, B. Doran, D. Gross, and M. Christandl, Entanglement polytopes: Multiparticle entanglement from single-particle information, Science 340, 1205 (2013).
  91. A. Sawicki, M. Oszmaniec, and M. Kuś, Critical sets of the total variance can detect all stochastic local operations and classical communication classes of multiparticle entanglement, Phys. Rev. A 86, 040304 (2012).
  92. A. Sawicki, M. Oszmaniec, and M. Kuś, Convexity of momentum map, morse index, and quantum entanglement, Rev. Math. Phys. 26, 1450004 (2014).
  93. R. T. Grigoriy Blekherman, Pablo A. Parrilo, ed., Semidefinite Optimization and Convex Algebraic Geometry (SIAM, 2013).
  94. M. Navascués, M. Owari, and M. B. Plenio, Power of symmetric extensions for entanglement detection, Phys. Rev. A 80, 052306 (2009).
  95. M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization (Springer, Berlin, 1993).
  96. F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim. 5, 13 (1995).
  97. S. Arora, E. Hazan, and S. Kale, Fast algorithms for approximate semidefinite programming using the multiplicative weights update method, in 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) (IEEE, Piscataway, NJ, 2005).
  98. J. F. Sturm, Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones, Optim. Methods Softw. 11, 625 (1999).
  99. D. Peaucelle, D. Henrion, L. Yann, and K. Taitz, User's guide for sedumi interface 1.04 (LAAS-CNRS, Toulouse, 2002).
  100. F. G. Brandao and K. M. Svore, Quantum speed-ups for solving semidefinite programs, in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Piscataway, NJ, 2017).
  101. T. Kriváchy, Y. Cai, J. Bowles, D. Cavalcanti, and N. Brunner, High-speed batch processing of semidefinite programs with feedforward neural networks, New J. Phys. 23, 103034 (2021).
  102. C. J. C. H. Watkins, Learning from delayed rewards, Ph.D. thesis, King's College, Cambridge, 1989.
  103. V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski et al., Human-level control through deep reinforcement learning, Nature (London) 518, 529 (2015).
  104. H. v. Hasselt, A. Guez, and D. Silver, Deep reinforcement learning with double q-learning, in Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, AAAI'16 (AAAI Press, Palo Alto, CA, 2016), pp. 2094-2100.
  105. E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. 16, 407 (1961).
  106. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 2009).
  107. X. Wang, Entanglement in the quantum Heisenberg XY model, Phys. Rev. A 64, 012313 (2001).
  108. X. Wang, Thermal and ground-state entanglement in Heisenberg XX qubit rings, Phys. Rev. A 66, 034302 (2002).
  109. A. D. Pasquale, G. Costantini, P. Facchi, G. Florio, S. Pascazio, and K. Yuasa, XX model on the circle, Eur. Phys. J.: Spec. Top. 160, 127 (2008).
  110. H. Nishimori and Y. Ozeki, Ground-state long-range order in the two-dimensional xxz model, J. Phys. Soc. Jpn. 58, 1027 (1989).
  111. P. Jordan and E. Wigner, Über das paulische Äquivalenzverbot, Z. Phys. 47, 631 (1928).
  112. M. A. Nielsen, The Fermionic Canonical Commutation Relations and the Jordan-Wigner Transform, School of Physical Sciences The University of Queensland, Vol. 59, 2005.
  113. M.-C. Bañuls, J. I. Cirac, and M. M. Wolf, Entanglement in fermionic systems, Phys. Rev. A 76, 022311 (2007).
  114. J. Tura, G. De las Cuevas, R. Augusiak, M. Lewenstein, A. Acín, and J. I. Cirac, Energy as a Detector of Nonlocality of Many-Body Spin Systems, Phys. Rev. X 7, 021005 (2017).
  115. T. H. D. C. Cormen, C. E. M. Leiserson, R. L. M. Rivest, and C. C. U. Stein, Introduction to Algorithms (MIT Press, Cambridge, 2009).
  116. S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, Optimization by simulated annealing, Science 220, 671 (1983).
  117. M. E. Taylor and P. Stone, Transfer learning for reinforcement learning domains: A survey, J. Mach. Learn. Res. 10, 1633 (2009).
  118. A. Dawid, P. Huembeli, M. Tomza, M. Lewenstein, and A. Dauphin, Phase detection with neural networks: Interpreting the black box, New J. Phys. 22, 115001 (2020).
  119. E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Learning phase transitions by confusion, Nat. Phys. 13, 435 (2017).
  120. K. Kottmann, P. Huembeli, M. Lewenstein, and A. Acín, Unsupervised Phase Discovery with Deep Anomaly Detection, Phys. Rev. Lett. 125, 170603(2020).
  121. O. Gühne and G. Tóth, Entanglement detection, Phys. Rep. 474, 1 (2009).
  122. G. Tóth and O. Gühne, Detection of multipartite entanglement with two-body correlations, Appl. Phys. B 82, 237 (2006).
  123. O. Gühne, G. Tóth, and H. J. Briegel, Multipartite entanglement in spin chains, New J. Phys. 7, 229 (2005).
  124. N. Yu, Separability of a mixture of dicke states, Phys. Rev. A 94, 060101(R) (2016).
  125. J. Tura, A. Aloy, R. Quesada, M. Lewenstein, and A. Sanpera, Separability of diagonal symmetric states: A quadratic conic optimization problem, Quantum 2, 45 (2018).
  126. C. Marconi, A. Aloy, J. Tura, and A. Sanpera, Entangled symmetric states and copositive matrices, Quantum 5, 561 (2021).
  127. A. Aloy, J. Tura, F. Baccari, A. Acín, M. Lewenstein, and R. Augusiak, Device-Independent Witnesses of Entanglement Depth from Two-Body Correlators, Phys. Rev. Lett. 123, 100507(2019).
  128. J. Tura, A. Aloy, F. Baccari, A. Acín, M. Lewenstein, and R. Augusiak, Optimization of device-independent witnesses of entanglement depth from two-body correlators, Phys. Rev. A 100, 032307(2019).
  129. G. Tóth, Entanglement witnesses in spin models, Phys. Rev. A 71, 010301 (2005).
  130. B. F. Schiffer, J. Tura, and J. I. Cirac, Adiabatic spectroscopy and a variational quantum adiabatic algorithm, PRX Quantum 3, 020347 (2022).
  131. B. Requena, G. Muñoz-Gil, and J. Tura, Borjarequena/bounce, https://doi.org/10.5281/zenodo.4585623.
  132. M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, http://cvxr.com/cvx.
  133. M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs, in Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, edited by V. Blondel, S. Boyd, and H. Kimura (Springer-Verlag, Berlin, 2008), pp. 95–110, http://stanford.edu/~boyd/graph_dcp.html.
  134. J. Lofberg, YALMIP : a toolbox for modeling and optimization in MATLAB, in 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508) (IEEE, Piscataway, NJ, 2004).
  135. K. C. Toh, M. Todd, R. Tütüncü, and R. H. Tutuncu, SDPT3 - a matlab software package for semidefinite programming, Optim. Methods Softw. 11, 545 (1999).
  136. T. Baumgratz and M. B. Plenio, Lower bounds for ground states of condensed matter systems, New J. Phys. 14, 023027 (2012).
  137. D. Rosset, Characterization of correlations in quantum networks, Ph.D. thesis, University of Geneva, 2015.
  138. A. Tavakoli, D. Rosset, and M.-O. Renou, Enabling Computation of Correlation Bounds for Finite-Dimensional Quantum Systems via Symmetrization, Phys. Rev. Lett. 122, 070501(2019).
  139. K. Gatermann and P. A. Parrilo, Symmetry groups, semidefinite programs, and sums of squares, J. Pure Appl. Algebra 192, 95 (2004).
  140. M. B. Hastings, Solving gapped Hamiltonians locally, Phys. Rev. B 73, 085115(2006).
  141. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  142. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  143. M. A. Levin and X.-G. Wen, String-net condensation:   A physical mechanism for topological phases, Phys. Rev. B 71, 045110(2005).
  144. D. Gottesman, An introduction to quantum error correction and fault-tolerant quantum computation, in Quantum Information Science and its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics, Vol. 68 (2010), pp. 13–58.
  145. M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H. Briegel, Entanglement in graph states and its applications, in International School of Physics Enrico Fermi, Vol. 162 (2006).
  146. H. J. Briegel and R. Raussendorf, Persistent Entanglement in Arrays of Interacting Particles, Phys. Rev. Lett. 86, 910 (2001).
  147. A. S. Darmawan and S. D. Bartlett, Graph states as ground states of two-body frustration-free Hamiltonians, New J. Phys. 16, 073013 (2014).
  148. S. Bravyi, Efficient algorithm for a quantum analogue of 2-sat, Contemp. Math. 536, 33 (2011).
  149. J. Chen, X. Chen, R. Duan, Z. Ji, and B. Zeng, No-go theorem for one-way quantum computing on naturally occurring two-level systems, Phys. Rev. A 83, 050301(R) (2011).
  150. M. A. Nielsen, Cluster-state quantum computation, Rep. Math. Phys. 57, 147 (2006).
  151. M. V. den Nest, K. Luttmer, W. Dür, and H. J. Briegel, Graph states as ground states of many-body spin-1/2 Hamiltonians, Phys. Rev. A 77, 012301 (2008).
  152. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous Results on Valence-Bond Ground States in Antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  153. D. S. Rokhsar and S. A. Kivelson, Superconductivity and the Quantum Hard-Core Dimer Gas, Phys. Rev. Lett. 61, 2376 (1988).
  154. C. Castelnovo, C. Chamon, C. Mudry, and P. Pujol, From quantum mechanics to classical statistical physics: Generalized Rokhsar-Kivelson Hamiltonians and the stochastic matrix form decomposition, Ann. Phys. 318, 316 (2005).
  155. D. Perez-Garcia, F. Verstraete, M. Wolf, and J. Cirac, Matrix product state representations, Quantum Inf. Comput. 7, 401 (2007).
  156. D. Pérez-García, F. Verstraete, M. Wolf, and J. Cirac, Peps as unique ground states of local Hamiltonians, Quantum Inf. Comput. 8, 650 (2008).
  157. J. I. Cirac, J. Garre-Rubio, and D. Pérez-García, Mathematical open problems in projected entangled pair states, Rev. Mat. Complut. 32, 579 (2019).
  158. E. Cruz-Rico, Efficient Preparation of a Family of Ground States of 2D Local Hamiltonians and their Verification Properties, Master's thesis, Ludwig-Maximilians-Universität München, 2020.
  159. O. Sattath, S. C. Morampudi, C. R. Laumann, and R. Moessner, When a local Hamiltonian must be frustration-free, Proc. Natl. Acad. Sci. USA 113, 6433 (2016).
  160. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  161. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed Experiment to Test Local Hidden-Variable Theories, Phys. Rev. Lett. 23, 880 (1969).
  162. M. Froissart, Constructive generalization of Bell's inequalities, Il Nuovo Cimento B 64, 241 (1981).
  163. K. F. Pál and T. Vértesi, Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems, Phys. Rev. A 82, 022116 (2010).
  164. I. Šupić and J. Bowles, Self-testing of quantum systems: A review, Quantum 4, 337 (2020).
  165. A. Acín, S. Massar, and S. Pironio, Randomness Versus Nonlocality and Entanglement, Phys. Rev. Lett. 108, 100402 (2012).
  166. T. H. Yang and M. Navascués, Robust self-testing of unknown quantum systems into any entangled two-qubit states, Phys. Rev. A 87, 050102(R) (2013).
  167. C. Bamps and S. Pironio, Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing, Phys. Rev. A 91, 052111(2015).
  168. A. Coladangelo, K. T. Goh, and V. Scarani, All pure bipartite entangled states can be self-tested, Nat. Commun. 8, 15485 EP (2017), article.
  169. I. Šupić, A. Coladangelo, R. Augusiak, and A. Acín, Self-testing multipartite entangled states through projections onto two systems, New J. Phys. 20, 083041 (2018).
  170. J. Kaniewski, I. Šupić, J. Tura, F. Baccari, A. Salavrakos, and R. Augusiak, Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems, Quantum 3, 198 (2019).
  171. F. Baccari, R. Augusiak, I. Šupić, J. Tura, and A. Acín, Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing, Phys. Rev. Lett. 124, 020402(2020).
  172. D. Hilbert, Mathematical problems, Bull. Am. Math. 8, 437 (1902).
  173. T. S. Motzkin, The arithmetic-geometric inequality, in Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) (Academic Press, New York, 1967), pp. 205–224.
  174. E. Artin, Über die zerlegung definiter funktionen in quadrate, Abh. Math. Semin. Univ. Hambg. 5, 100 (1927).
  175. F. Baccari, C. Gogolin, P. Wittek, and A. Acín, Verifying the output of quantum optimizers with ground-state energy lower bounds, Phys. Rev. Res. 2, 043163(2020).
  176. L. Vandenberghe and M. S. Andersen, Chordal graphs and semidefinite optimization, Found. Trend. Optim. 1, 241 (2015).
  177. D. Cifuentes and P. A. Parrilo, Exploiting chordal structure in polynomial ideals: A Röbner bases approach, SIAM J. Discrete Math. 30, 1534 (2016).
  178. D. Cifuentes and P. A. Parrilo, Chordal networks of polynomial ideals, SIAM J. Appl. Algebra Geometry 1, 73 (2017).
  179. Y. Zheng, G. Fantuzzi, A. Papachristodoulou, P. Goulart, and A. Wynn, Chordal decomposition in operator-splitting methods for sparse semidefinite programs, Math. Program. 180, 489 (2020).
  180. R. F. Werner and M. M. Wolf, All-multipartite bell-correlation inequalities for two dichotomic observables per site, Phys. Rev. A 64, 032112(2001).
  181. M. Żukowski and C. Brukner, Bell's Theorem for General N-Qubit States, Phys. Rev. Lett. 88, 210401 (2002).
  182. C. Śliwa, Symmetries of the Bell correlation inequalities, Phys. Lett. A 317, 165 (2003).
  183. J.-D. Bancal, N. Gisin, and S. Pironio, Looking for symmetric Bell inequalities, J. Phys. A: Math. Theor. 43, 385303 (2010).
  184. J. Tura, R. Augusiak, A. B. Sainz, T. Vértesi, M. Lewenstein, and A. Acín, Detecting nonlocality in many-body quantum states, Science 344, 1256 (2014).
  185. B. Lücke, J. Peise, G. Vitagliano, J. Arlt, L. Santos, G. Tóth, and C. Klempt, Detecting Multiparticle Entanglement of Dicke States, Phys. Rev. Lett. 112, 155304(2014).
  186. R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, Operational Framework for Nonlocality, Phys. Rev. Lett. 109, 070401(2012).
  187. J.-D. Bancal, J. Barrett, N. Gisin, and S. Pironio, Definitions of multipartite nonlocality, Phys. Rev. A 88, 014102(2013).
  188. F. Baccari, J. Tura, M. Fadel, A. Aloy, J.-D. Bancal, N. Sangouard, M. Lewenstein, A. Acín, and R. Augusiak, Bell correlation depth in many-body systems, Phys. Rev. A 100, 022121(2019).
  189. V. J. L. Pastor, Exploring nonlocal correlations in many-body systems, Master's thesis, Universitat Politècnica de Catalunya, 2018.
  190. J. Tura, R. Augusiak, A. Sainz, B. Lücke, C. Klempt, M. Lewenstein, and A. Acín, Nonlocality in many-body quantum systems detected with two-body correlators, Ann. Phys. 362, 370 (2015).
  191. T. Fritz, Polyhedral duality in bell scenarios with two binary observables, J. Math. Phys. 53, 072202 (2012).
  192. L. Babai, L. Fortnow, and C. Lund, Non-deterministic exponential time has two-prover interactive protocols, Comput. Comple. 1, 3 (1991).
  193. B. Chazelle, An optimal convex hull algorithm in any fixed dimension, Discrete Comput. Geom. 10, 377 (1993).

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Phys. Rev. Research 5, 013097– Published 10 February, 2023
  • Received 23 April 2021
  • Revised 13 October 2022
  • Accepted 19 December 2022
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