Unifying neural-network quantum states and correlator product states via tensor networks. (English) Zbl 1394.81074
The aim here is a detailed presentation of the connection between neural-network quantum state (NQS) approach and a general class of sampleable many-body quantum states, the correlator product states (CPS). The CPS have been introduced by H. J. Changlani et al. [“Approximating strongly correlated wave functions with correlator product states”, Phys. Rev. B 80, No. 24, Article ID 245116, 8 p. (2009; doi:10.1103/PhysRevB.80.245116)]. As results, in the present article, one presents a set of observations providing insights into NQS and the fact that NQS can be viewed as a special form of CPS. The second section of the paper is devoted to some background; one formally introduces the quantum many-body problem and gives an overview of approaches as TNT-tensor network theory, VMC-variational Monte Carlo and CPS. The third section presents the COPY tensor and shows how to express CPS as sampleable tensor networks. The forth section is dedicated to constructing neural-network quantum states. Examples of NQS are provided in the fifth section and a discussion on the extensions to correlator operator approach is done in the sixth section. Conclusions are presented in the seventh section. Additional details are presented in the Appendix A–E.
Reviewer: Claudia Simionescu-Badea (Wien)
MSC:
| 81P68 | Quantum computation |
| 82C32 | Neural nets applied to problems in time-dependent statistical mechanics |
| 80A10 | Classical and relativistic thermodynamics |
| 81V70 | Many-body theory; quantum Hall effect |
| 65C05 | Monte Carlo methods |
Keywords:
tensor network theory; correlator product states; neural-network quantum states; restricted Boltzmann machinesReferences:
| [1] | Läuchli, A. M.; Sudan, J.; Sørensen, E. S., Ground-state energy and spin gap of spin-\( \frac{1}{2}\) Kagomé-Heisenberg antiferromagnetic clusters: large-scale exact diagonalization results, Phys. Rev. B, 83, (2011) · doi:10.1103/PhysRevB.83.212401 |
| [2] | Eisert, J.; Cramer, M.; Plenio, M. B., Colloquium: area laws for the entanglement entropy, Rev. Mod. Phys., 82, 277, (2010) · Zbl 1205.81035 · doi:10.1103/RevModPhys.82.277 |
| [3] | Snijders, C.; Matzat, U.; Reips, U-D, ‘Big data’: big gaps of knowledge in the field of internet, Int. J. Internet Sci., 7, 1, (2012) |
| [4] | LeCun, Y.; Bengio, Y.; Hinton, G., Deep learning, Nature, 521, 436, (2015) · doi:10.1038/nature14539 |
| [5] | Goodfellow, I.; Bengio, Y.; Courville, A., Deep Learning, (2016), Cambridge, MA: MIT Press, Cambridge, MA · Zbl 1373.68009 |
| [6] | Haykin, S. S., Neural Networks, Learning Machines, (2008), London: Pearson, London |
| [7] | Bény, C., Deep learning and the renormalization group, (2013) |
| [8] | Mehta, P.; Schwab, D. J., An exact mapping between the variational renormalization group and deep learning, (2014) |
| [9] | Carrasquilla, J.; Melko, R. G., Machine learning phases of matter, Nat. Phys., 13, 43, (2017) · doi:10.1038/nphys4035 |
| [10] | van Nieuwenburg, E. P L.; Liu, Y-H; Huber, S. D., Learning phase transitions by confusion, Nat. Phys., 13, 435, (2017) · doi:10.1038/nphys4037 |
| [11] | Broecker, P.; Carrasquilla, J.; Melko, R. G.; Trebst, S., Machine learning quantum phases of matter beyond the fermion sign problem, Sci. Rep., 7, 8823, (2017) · doi:10.1038/s41598-017-09098-0 |
| [12] | Wang, L., Discovering phase transitions with unsupervised learning, Phys. Rev. B, 94, (2016) · doi:10.1103/PhysRevB.94.195105 |
| [13] | Ch’ng, K.; Carrasquilla, J.; Melko, R. G.; Khatami, E., Machine learning phases of strongly correlated fermions, Phys. Rev. X, 7, (2017) · doi:10.1103/PhysRevX.7.031038 |
| [14] | Ch’ng, K.; Vazquez, N.; Khatami, E., Unsupervised machine learning account of magnetic transitions in the Hubbard model, Phys. Rev. E, 97, (2018) · doi:10.1103/PhysRevE.97.013306 |
| [15] | Arsenault, L-F; von Lilienfeld, O. A.; Millis, A. J., Machine learning for many-body physics: efficient solution of dynamical mean-field theory, (2015) |
| [16] | Torlai, G.; Melko, R. G., Learning thermodynamics with Boltzmann machines, Phys. Rev. B, 94, (2016) · doi:10.1103/PhysRevB.94.165134 |
| [17] | Huang, L.; Wang, L., Accelerated Monte Carlo simulations with restricted Boltzmann machines, Phys. Rev. B, 95, (2017) · doi:10.1103/PhysRevB.95.035105 |
| [18] | Liu, J.; Qi, Y.; Meng, Z. Y.; Fu, L., Self-learning Monte Carlo method, Phys. Rev. B, 95, (2017) · doi:10.1103/PhysRevB.95.041101 |
| [19] | Tubman, N. M., Measuring quantum entanglement, machine learning and wave function tomography: bridging theory and experiment with the quantum gas microscope, (2016) |
| [20] | Torlai, G.; Mazzola, G.; Carrasquilla, J.; Troyer, M.; Melko, R. G.; Carleo, G., Many-body quantum state tomography with neural networks, (2017) |
| [21] | Carleo, G.; Troyer, M., Solving the quantum many-body problem with artificial neural networks, Science, 355, 602, (2017) · Zbl 1404.81313 · doi:10.1126/science.aag2302 |
| [22] | Nomura, Y.; Darmawan, A.; Yamaji, Y.; Imada, M., Restricted-Boltzman machine learning for solving strongly correlated quantum systems, Phys. Rev. B, 96, (2017) · doi:10.1103/PhysRevB.96.205152 |
| [23] | Deng, D-L; Li, X.; Sarma, S. D., Exact machine learning topological states, Phys. Rev. B, 96, (2016) · doi:10.1103/PhysRevB.96.195145 |
| [24] | Gao, X.; Duan, L-M, Efficient representation of quantum many-body states with deep neural networks, Nature Commun., 8, 662, (2017) · doi:10.1038/s41467-017-00705-2 |
| [25] | Deng, D-L; Xiaopeng, L.; Das Sarma, S., Quantum entanglement in neural network states, Phys. Rev. X, 7, (2017) · doi:10.1103/PhysRevX.7.021021 |
| [26] | Chen, J.; Cheng, S.; Xie, H.; Wang, L.; Xiang, T., On the equivalence of restricted Boltzmann machines and tensor network states, Phys. Rev. B, 97, (2018) · doi:10.1103/PhysRevB.97.085104 |
| [27] | Verstraete, F.; Murg, V.; Cirac, J. I., Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys., 57, 143, (2008) · doi:10.1080/14789940801912366 |
| [28] | Cirac, J. I.; Verstraete, F., Renormalization and tensor product states in spin chains and lattices, J. Phys. A: Math. Theor., 42, (2009) · Zbl 1181.82010 · doi:10.1088/1751-8113/42/50/504004 |
| [29] | Orus, R., A practical introduction to tensor networks: matrix product states and projected entangled pair states, Ann. Phys., 349, 117, (2014) · Zbl 1343.81003 · doi:10.1016/j.aop.2014.06.013 |
| [30] | Changlani, H. J.; Kinder, J. M.; Umrigar, C. J.; Chan, G. K L., Approximating strongly correlated wave functions with correlator product states, Phys. Rev. B, 80, (2009) · doi:10.1103/PhysRevB.80.245116 |
| [31] | Jastrow, R., Many-body problem with strong forces, Phys. Rev., 98, 1479, (1955) · Zbl 0067.44602 · doi:10.1103/PhysRev.98.1479 |
| [32] | Al-Assam, S.; Clark, S. R.; Jaksch, D., The tensor network theory library, J. Stat. Mech., (2017) · Zbl 1459.81004 · doi:10.1088/1742-5468/aa7df3 |
| [33] | Schollwöck, U., The density-matrix renormalization group in the age of matrix product states, Ann. Phys., 326, 96, (2011) · Zbl 1213.81178 · doi:10.1016/j.aop.2010.09.012 |
| [34] | Shi, Y-Y; Duan, L-M; Vidal, G., Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev. A, 74, (2006) · doi:10.1103/PhysRevA.74.022320 |
| [35] | Murg, V.; Verstraete, F.; Legeza, O.; Noack, R. M., Simulating strongly correlated quantum systems with tree tensor networks, Phys. Rev. B, 82, (2010) · doi:10.1103/PhysRevB.82.205105 |
| [36] | Evenbly, G.; Vidal, G.; Avella, A.; Mancini, F., Quantum criticality with the multi-scale entanglement renormalization Ansatz, Strongly Correlated Systems. Numerical Methods, ch 4, p 99, (2013) · doi:10.1007/978-3-642-35106-8_4 |
| [37] | Ran, S-J; Tirrito, E.; Peng, C.; Chen, X.; Su, G.; Lewenstein, M., Review of tensor network contraction approaches, (2017) |
| [38] | Vidal, G., Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett., 91, (2003) · doi:10.1103/PhysRevLett.91.147902 |
| [39] | Clark, S. R.; Jaksch, D., Dynamics of the superfluid to Mott-insulator transition in one dimension, Phys. Rev. A, 70, (2004) · doi:10.1103/PhysRevA.70.043612 |
| [40] | Bruderer, M.; Johnson, T. H.; Clark, S. R.; Jaksch, D.; Posazhennikova, A.; Belzig, W., Phonon resonances in atomic currents through Bose-Fermi mixtures in optical lattices, Phys. Rev. A, 82, (2010) · doi:10.1103/PhysRevA.82.043617 |
| [41] | Coulthard, J. R.; Clark, S. R.; Al-Assam, S.; Cavalleri, A.; Jaksch, D., Enhancement of superexchange pairing in the periodically driven Hubbard model, Phys. Rev. B, 96, (2017) · doi:10.1103/PhysRevB.96.085104 |
| [42] | Mendoza-Arenas, J. J.; Clark, S. R.; Jaksch, D., Coexistence of energy diffusion and local thermalization in nonequilibrium {\it xxz} spin chains with integrability breaking, Phys. Rev. E, 91, (2015) · doi:10.1103/PhysRevE.91.042129 |
| [43] | Mendoza-Arenas, J. J.; Clark, S. R.; Felicetti, S.; Romero, G.; Solano, E.; Angelakis, D. G.; Jaksch, D., Beyond mean-field bistability in driven-dissipative lattices: Bunching-antibunching transition and quantum simulation, Phys. Rev. A, 93, (2016) · doi:10.1103/PhysRevA.93.023821 |
| [44] | Znidaric, M.; Mendoza-Arenas, J. J.; Clark, S. R.; Goold, J., Dephasing enhanced spin transport in the ergodic phase of a many-body localizable system, Ann. Phys., 529, 201600298, (2017) · doi:10.1002/andp.201600298 |
| [45] | Johnson, T. H.; Clark, S. R.; Jaksch, D., Dynamical simulations of classical stochastic systems using matrix product states, Phys. Rev. E, 82, (2010) · doi:10.1103/PhysRevE.82.036702 |
| [46] | Johnson, T. H.; Elliott, T. J.; Clark, S. R.; Jaksch, D., Capturing exponential variance using polynomial resources: applying tensor networks to nonequilibrium stochastic processes, Phys. Rev. Lett., 114, (2015) · doi:10.1103/PhysRevLett.114.090602 |
| [47] | Foulkes, W. M C.; Mitas, L.; Needs, R. J.; Rajagopal, G., Quantum Monte Carlo simulations of solids, Rev. Mod. Phys., 73, 33, (2001) · doi:10.1103/RevModPhys.73.33 |
| [48] | Gubernatis, J.; Kawashima, N.; Werner, P., Quantum Monte Carlo Methods: Algorithms for Lattice Models, (2016), Cambridge: Cambridge University Press, Cambridge · Zbl 1366.81003 |
| [49] | Becca, F.; Sorella, S., Quantum Monte Carlo Approaches for Correlated Systems, (2017), Cambridge: Cambridge University Press, Cambridge · Zbl 1420.81003 |
| [50] | Lou, J.; Sandvik, A. W., Variational ground states of two-dimensional antiferromagnets in the valence bond basis, Phys. Rev. B, 76, (2007) · doi:10.1103/PhysRevB.76.104432 |
| [51] | Nightingale, M. P.; Melik-Alaverdian, V., Optimization of ground- and excited-state wave functions and van der Waals clusters, Phys. Rev. Lett., 87, (2001) · doi:10.1103/PhysRevLett.87.043401 |
| [52] | Toulouse, J.; Umrigar, C. J., Optimization of quantum Monte Carlo wave functions by energy minimization, J. Chem. Phys., 126, (2007) · doi:10.1063/1.2437215 |
| [53] | Sorella, S., Generalized Lanczos algorithm for variational quantum Monte Carlo, Phys. Rev. B, 64, (2001) · doi:10.1103/PhysRevB.64.024512 |
| [54] | Marti, K. H.; Bauer, B.; Reiher, M.; Troyer, M.; Verstraete, F., Complete-graph tensor network states: a new fermionic wave function ansatz for molecules, New J. Phys., 12, (2010) · doi:10.1088/1367-2630/12/10/103008 |
| [55] | Mezzacapo, F.; Schuch, N.; Boninsegni, M.; Cirac, J. I., Ground-state properties of quantum many-body systems: entangled-plaquette states and variational Monte Carlo, New J. Phys., 11, (2009) · doi:10.1088/1367-2630/11/8/083026 |
| [56] | Neuscamman, E.; Changlani, H.; Kinder, J.; Chan, G. K L., Nonstochastic algorithms for Jastrow-Slater and correlator product state wave functions, Phys. Rev. B, 84, (2011) · doi:10.1103/PhysRevB.84.205132 |
| [57] | Schuch, N.; Wolf, M. M.; Verstraete, F.; Cirac, J. I., Simulation of quantum many-body systems with strings of operators and Monte Carlo tensor contractions, Phys. Rev. Lett., 100, (2008) · Zbl 1228.81112 · doi:10.1103/PhysRevLett.100.040501 |
| [58] | Biamonte, J. D.; Clark, S. R.; Jaksch, D., Categorical tensor network states, AIP Adv., 1, (2011) · doi:10.1063/1.3672009 |
| [59] | Denny, S. J.; Biamonte, J. D.; Jaksch, D.; Clark, S. R., Algebraically contractible topological tensor network states, J. Phys. A: Math. Theor., 45, (2012) · Zbl 1238.81070 · doi:10.1088/1751-8113/45/1/015309 |
| [60] | Al-Assam, S.; Clark, S. R.; Foot, C. J.; Jaksch, D., Capturing long range correlations in two-dimensional quantum lattice systems using correlator product states, Phys. Rev. B, 84, (2011) · doi:10.1103/PhysRevB.84.205108 |
| [61] | Huse, D. A.; Elser, V., Simple variational wave functions for two-dimensional Heisenberg spin-1/2 antiferromagnets, Phys. Rev. Lett., 60, 2531, (1988) · doi:10.1103/PhysRevLett.60.2531 |
| [62] | Fischer, A.; Igel, C.; Alvarez, L., An introduction to restricted Boltzmann machines, p 14, (2012) · doi:10.1007/978-3-642-33275-3_2 |
| [63] | Hinton, G. E., Training products of experts by minimizing contrastive divergence, Technical Report 2000-004, (2000) · Zbl 1010.68111 |
| [64] | LeCun, Y.; Cortes, C.; Burges, C. J C., MNIST database, (1998) |
| [65] | Le Roux, N.; Bengio, Y., Representational power of restricted boltzmann machines and deep belief networks, Neural Comput., 20, 1631-1649, (2008) · Zbl 1140.68057 · doi:10.1162/neco.2008.04-07-510 |
| [66] | Trefethen, L. N.; Bau, D., Numerical Linear Algebra, (1997), Philadelphia: SIAM, Philadelphia · Zbl 0874.65013 |
| [67] | Hackbusch, W., Tensor Spaces and Numerical Tensor Calculus, (2012), Berlin: Springer, Berlin · Zbl 1244.65061 |
| [68] | Kolda, T. G.; Bader, B. W., Tensor decompositions and applications, SIAM Rev., 51, 455, (2009) · Zbl 1173.65029 · doi:10.1137/07070111X |
| [69] | Raussendorf, R.; Briegel, H. J., A one-way quantum computer, Phys. Rev. Lett., 86, 5188, (2001) · doi:10.1103/PhysRevLett.86.5188 |
| [70] | Clark, S. R.; Alves, C. M.; Jaksch, D., Efficient generation of graph states for quantum computation, New J. Phys., 7, 124, (2005) · doi:10.1088/1367-2630/7/1/124 |
| [71] | Hein, M.; Eisert, J.; Briegel, H. J., Multiparty entanglement in graph states, Phys. Rev. A, 69, (2004) · Zbl 1232.81007 · doi:10.1103/PhysRevA.69.062311 |
| [72] | Bondy, A.; Murty, U., Graph Theory, (2011), London: Springer, London · Zbl 1134.05001 |
| [73] | Anders, S.; Plenio, M. B.; Dür, W.; Verstraete, F.; Briegel, H-J, Ground-state approximation for strongly interacting spin systems in arbitrary spatial dimension, Phys. Rev. Lett., 97, (2006) · doi:10.1103/PhysRevLett.97.107206 |
| [74] | Laughlin, R. B., Anomalous quantum hall effect: an incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett., 50, 1395, (1983) · doi:10.1103/PhysRevLett.50.1395 |
| [75] | Kitaev, A. Y., Fault-tolerant quantum computation by anyons, Ann. Phys., 303, 2, (2003) · Zbl 1012.81006 · doi:10.1016/S0003-4916(02)00018-0 |
| [76] | Verstraete, F.; Wolf, M. M.; Perez-Garcia, D.; Cirac, J. I., Criticality, the area law, and the computational power of projected entangled pair states, Phys. Rev. Lett., 96, (2006) · Zbl 1228.81096 · doi:10.1103/PhysRevLett.96.220601 |
| [77] | Batchelor, M. T.; Blöte, H. W J.; Nienhuis, B.; Yung, C. M., Critical behaviour of the fully packed loop model on the square lattice, J. Phys. A: Math. Gen., 29, L399, (1996) · Zbl 0900.82009 · doi:10.1088/0305-4470/29/16/001 |
| [78] | Fisher, M. E., Statistical mechanics of dimers on a plane lattice, Phys. Rev., 124, 1664, (1961) · Zbl 0105.22403 · doi:10.1103/PhysRev.124.1664 |
| [79] | Rokhsar, D. S.; Kivelson, S. A., Superconductivity and the quantum hard-core dimer gas, Phys. Rev. Lett., 61, 2376, (1988) · doi:10.1103/PhysRevLett.61.2376 |
| [80] | Anderson, P. W., Resonating valence bonds: a new kind of insulator?, Mater. Res. Bull., 8, 153, (1973) · doi:10.1016/0025-5408(73)90167-0 |
| [81] | Anderson, P. W., The resonating valence bond state in La_2CuO_4 and superconductivity, Science, 235, 1196, (1987) · doi:10.1126/science.235.4793.1196 |
| [82] | Savary, L.; Balents, L., Quantum spin liquids: a review, Rep. Prog. Phys., 80, (2017) · doi:10.1088/0034-4885/80/1/016502 |
| [83] | Zhou, Y.; Kanoda, K.; Ng, T-K, Quantum spin liquid states, Rev. Mod. Phys., 89, (2017) · doi:10.1103/RevModPhys.89.025003 |
| [84] | Liang, S.; Doucot, B.; Anderson, P. W., Some new variational resonating-valence-bond-type wave functions for the spin-1/2 antiferromagnetic Heisenberg model on a square lattice, Phys. Rev. Lett., 61, 365, (1988) · doi:10.1103/PhysRevLett.61.365 |
| [85] | Schuch, N.; Poilblanc, D.; Cirac, J. I.; Pérez-García, D., Resonating valence bond states in the PEPS formalism, Phys. Rev. B, 86, (2012) · doi:10.1103/PhysRevB.86.115108 |
| [86] | Gutzwiller, M. C., Effect of correlation on the ferromagnetism of transition metals, Phys. Rev. Lett., 10, 159, (1963) · doi:10.1103/PhysRevLett.10.159 |
| [87] | Gros, C., Physics of projected wavefunctions, Ann. Phys., 189, 53, (1989) · doi:10.1016/0003-4916(89)90077-8 |
| [88] | Sikora, O.; Chang, H-W; Chou, C-P; Pollmann, F.; Kao, Y-J, Variational Monte Carlo simulations using tensor-product projected states, Phys. Rev. B, 91, (2015) · doi:10.1103/PhysRevB.91.165113 |
| [89] | Al-Assam, S.; Clark, S. R.; Jaksch, D., The Tensor Network Theory Library · Zbl 1459.81004 |
| [90] | Glasser, I.; Pancotti, N.; August, M.; Rodriguez, I. D.; Cirac, J. I., Neural networks quantum states, string-bond states and chiral topological states, Phys. Rev. X, 8, (2018) · doi:10.1103/PhysRevX.8.011006 |
| [91] | Kaubruegger, R.; Pastori, L.; Budich, J. C., Chiral topological phases from artificial neural networks, (2017) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
