# Variational Autoencoder Reconstruction of Complex Many-Body Physics

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Transverse-Field Ising Model

## 3. Generative Model as a Quantum State

## 4. Variational Autoencoder Architecture

- We calculate the matrix of log probabilities, taking element-wise logarithm of decoder network output: $log\mathsf{\Pi}=\left[\begin{array}{cc}log{\pi}_{11}& log{\pi}_{12}\cdots log{\pi}_{1N}\\ log{\pi}_{21}& log{\pi}_{22}\cdots log{\pi}_{2N}\\ log{\pi}_{31}& log{\pi}_{32}\cdots log{\pi}_{3N}\\ log{\pi}_{41}& log{\pi}_{42}\cdots log{\pi}_{4N}\end{array}\right]$,
- We generate a matrix of samples from the standard Gumbel distribution G and sum it up element-wise with the matrix of log probabilities $log\mathsf{\Pi}$: $Z=log\mathsf{\Pi}+G$,
- Finally, we take the $\mathrm{softmax}$ function of the result from the previous step: ${x}_{\mathrm{soft}}^{\mathrm{fake}}\left(T\right)=\mathrm{softmax}(Z/T)$, where T is a temperature of softmax. The softmax functions is defined by the expression $\mathrm{softmax}\left({x}_{ij}\right)=\frac{exp\left({x}_{ij}\right)}{{\sum}_{i}exp\left({x}_{ij}\right)}$.

## 5. Results

- If one reconstructs a pure state, the VAE smooths the spectrum of the density matrix and approximates the pure state by a slightly mixed state, as illustrated with a simple example in Figure 13.
- The VAE does not account the positivity constraints, which yields negative eigenvalues for the density matrix. These negative eigenvalues even appear in the spectrum of the reduced density matrix, as shown in Figure 13.

## 6. Conclusions

- For a large system (32 spins), the VAE’s reliability is verified by comparing one- and two-point correlation functions.
- For small system (five spins), the VAE’s reliability is verified by direct comparison of mass functions.
- The VAE can capture a quantum phase transition.
- The response functions (magnetic differential susceptibility tensor) can be obtained using backpropagation through VAE.
- Despite the very good agreement between the VAE-based mass function and the true mass function, the VAE shows limited performance with the determination of the entangled entropy. This is point is the object of further development.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

VAE | Variational Autoencoder |

MPS | Matrix product state |

TFI | Transverse-field Ising |

IC | Informationally incomplete |

POVM | Positive-operator valued measure |

ELBO | Evidence lower bound |

NN | Neural network |

KL | Kullback–Leibler |

DMRG | Density matrix renormalization group |

## Appendix A. VAE: Training and Implementation Details

## Appendix B. Sampling from POVM-Induced Mass Function

## References

- Muller, I. A History of Thermodynamics; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Onsager, L. Reciprocal Relations in Irreversible Processes. I. Phys. Rev.
**1931**, 37, 405–426. [Google Scholar] [CrossRef] - De Groot, S.R. Thermodynamics of Irreversible Processes; Interscience: New York, NY, USA, 1958. [Google Scholar]
- Le Bellac, M.; Mortessagne, F.; Batrouni, G.G. Equilibrium and Non-Equilibrium Statistical Thermodynamics; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Apertet, Y.; Ouerdane, H.; Goupil, C.; Lecoeur, P. Revisiting Feynman’s ratchet with thermoelectric transport theory. Phys. Rev. E
**2014**, 90, 012113. [Google Scholar] [CrossRef] [PubMed] - Goupil, C.; Ouerdane, H.; Herbert, E.; D’Angelo, Y.; Lecoeur, P. Closed-loop approach to thermodynamics. Phys. Rev. E
**2016**, 94, 032136. [Google Scholar] [CrossRef] [Green Version] - Andresen, B. Current trends in finite-time thermodynamics. Angew. Chem.-Int. Edit.
**2011**, 50, 2690–2704. [Google Scholar] [CrossRef] [PubMed] - Ouerdane, H.; Apertet, Y.; Goupil, C.; Lecoeur, P. Continuity and boundary conditions in thermodynamics: From Carnot’s efficiency to efficiencies at maximum power. Eur. Phys. J. Spec. Top.
**2015**, 224, 839–864. [Google Scholar] [CrossRef] - Apertet, Y.; Ouerdane, H.; Goupil, C.; Lecoeur, P. True nature of the Curzon-Ahlborn efficiency. Phys. Rev. E
**2017**, 96, 022119. [Google Scholar] [CrossRef] - Boltzmann, L. Uber die beziehung dem zweiten Haubtsatze der mechanischen Warmetheorie und der Wahrscheinlichkeitsrechnung respektive den Satzen uber das Warmegleichgewicht. Wiener Berichte
**1877**, 76, 373–435. [Google Scholar] - Gibbs, J.W. Elementary Principles in Statistical Mechanics; Charles Scribner’s Sons: New York, NY, USA, 1902. [Google Scholar]
- Penrose, O. Foundations of statistical mechanics. Rep. Prog. Phys.
**1979**, 42, 1937–2006. [Google Scholar] [CrossRef] - Goldstein, S.; Lebowitz, J.L.; Zanghì, N. Gibbs and Boltzmann entropy in classical and quantum mechanics. arXiv
**2019**, arXiv:1903.11870. Available online: https://arxiv.org/abs/1903.11870 (accessed on 6 November 2019). - Shannon, C.E. A mathematical theory of communication. Bell Labs Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - von Neumann, J. Mathematical Foundations of Quantum Mechanics. New Edition; Princeton University Press: Princeton, NJ, USA, 2018. [Google Scholar]
- Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Heikillä, T.T. The Physics of Nanoelectronics; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Chomaz, P.; Colonna, M.; Randrup, J. Nuclear spinodal fragmentation. Phys. Rep.
**2004**, 389, 263–440. [Google Scholar] [CrossRef] - Bressanini, D.; Morosi, G.; Mella, M. Robust wave function optimization procedures in quantum Monte Carlo methods. J. Chem. Phys.
**2002**, 116, 5345–5350. [Google Scholar] [CrossRef] [Green Version] - Feiguin, A.E.; White, S.R. Finite-temperature density matrix renormalization using an enlarged Hilbert space. Phys. Rev. B
**2005**, 72, 220401. [Google Scholar] [CrossRef] [Green Version] - Deutsch, J.M. Quantum statistical mechanics in a closed system. Phys. Rev. A
**1991**, 43, 2046–2049. [Google Scholar] [CrossRef] - Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E
**1994**, 50, 888–901. [Google Scholar] [CrossRef] [Green Version] - Rigol, M.; Dunjko, V.; Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature
**2008**, 452, 854–858. [Google Scholar] [CrossRef] [Green Version] - Dymarsky, A.; Lashkari, N.; Liu, H. Subsystem eigenstate thermalization hypothesis. Phys. Rev. E
**2018**, 97, 012140. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dymarsky, A. Mechanism of macroscopic equilibration of isolated quantum systems. Phys. Rev. B
**2019**, 99, 224302. [Google Scholar] [CrossRef] [Green Version] - Carleo, G.; Becca, F.; Schiró, M.; Fabrizio, M. Localization and glassy dynamics of many-body quantum systems. Sci. Rep.
**2012**, 2, 243. [Google Scholar] [CrossRef] [PubMed] - Chen, L.; Gelin, M.; Zhao, Y. Dynamics of the spin-boson model: A comparison of the multiple Davydov D
_{1}, D_{1.5}, D_{2}Ansätze. Chem. Phys.**2018**, 515, 108–118. [Google Scholar] [CrossRef] - Lanyon, B.; Maier, C.; Holzäpfel, M.; Baumgratz, T.; Hempel, C.; Jurcevic, P.; Dhand, I.; Buyskikh, A.; Daley, A.; Cramer, M.; et al. Efficient tomography of a quantum many-body system. Nat. Phys.
**2017**, 13, 1158. [Google Scholar] [CrossRef] - Liao, H.J.; Liu, J.G.; Wang, L.; Xiang, T. Differentiable programming tensor networks. Phys. Rev. X
**2019**, 9, 031041. [Google Scholar] [CrossRef] - Fetter, A.L.; Walecka, J.D. Quantum Theory of Many-Particle Systems; Dover: New York, NY, USA, 2003. [Google Scholar]
- Frésard, R.; Kroha, J.; Wölfle, P. The pseudoparticle approach to strongly correlated electron systems. In Strongly Correlated Systems; Avella, A., Mancini, F., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; Volume 171. [Google Scholar]
- Georges, A.; Kotliar, G.; Krauth, W.; Rozenberg, M.J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys.
**1996**, 68, 13–125. [Google Scholar] [CrossRef] [Green Version] - Negele, J.W.; Orland, H. Quantum Many-Particle Systems; Perseus Books: New York, NY, USA, 1998. [Google Scholar]
- Foulkes, W.; Mitas, L.; Needs, R.; Rajagopal, G. Quantum Monte Carlo simulations of solids. Rev. Mod. Phys.
**2001**, 73, 33. [Google Scholar] [CrossRef] - Orús, R. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Ann. Phys.
**2014**, 349, 117–158. [Google Scholar] [CrossRef] [Green Version] - Orús, R. Tensor networks for complex quantum systems. Nat. Rev. Phys.
**2019**, 1, 538–550. [Google Scholar] [CrossRef] [Green Version] - Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys.
**2011**, 326, 96–192. [Google Scholar] [CrossRef] [Green Version] - Vidal, G. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett.
**2003**, 91, 147902. [Google Scholar] [CrossRef] - Evenbly, G.; Vidal, G. Quantum criticality with the multiscale entanglement renormalization ansatz. In Strongly Correlated Systems; Springer: Berlin/Heidelberg, Germany, 2013; pp. 99–130. [Google Scholar]
- Pollock, F.A.; Rodríguez-Rosario, C.; Frauenheim, T.; Paternostro, M.; Modi, K. Non-Markovian quantum processes: Complete framework and efficient characterization. Phys. Rev. A
**2018**, 97, 012127. [Google Scholar] [CrossRef] [Green Version] - Luchnikov, I.; Vintskevich, S.; Ouerdane, H.; Filippov, S. Simulation complexity of open quantum dynamics: Connection with tensor networks. Phys. Rev. Lett.
**2019**, 122, 160401. [Google Scholar] [CrossRef] - Taranto, P.; Pollock, F.A.; Modi, K. Memory strength and recoverability of non-Markovian quantum stochastic processes. arXiv
**2019**, arXiv:1907.12583. Available online: https://arxiv.org/abs/1907.12583 (accessed on 6 November 2019). - Milz, S.; Pollock, F.A.; Modi, K. Reconstructing non-Markovian quantum dynamics with limited control. Phys. Rev. A
**2018**, 98, 012108. [Google Scholar] [CrossRef] [Green Version] - Luchnikov, I.A.; Vintskevich, S.V.; Grigoriev, D.A.; Filippov, S.N. Machine learning of Markovian embedding for non-Markovian quantum dynamics. arXiv
**2019**, arXiv:1902.07019. Available online: https://arxiv.org/abs/1902.07019 (accessed on 6 November 2019). - 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.
**2008**, 57, 143–224. [Google Scholar] [CrossRef] [Green Version] - Levin, M.; Nave, C.P. Tensor renormalization group approach to two-dimensional classical lattice models. Phys. Rev. Lett.
**2007**, 99, 120601. [Google Scholar] [CrossRef] - Evenbly, G.; Vidal, G. Tensor network renormalization. Phys. Rev. Lett.
**2015**, 115, 180405. [Google Scholar] [CrossRef] [PubMed] - Gemmer, J.; Michel, M. Quantum Thermodynamics; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Kosloff, R. Quantum thermodynamics and open-systems modeling. J. Phys. Chem.
**2019**, 150, 204105. [Google Scholar] [CrossRef] - Allahverdyan, A.E.; Johal, R.S.; Mahler, G. Work extremum principle: Structure and function of quantum heat engines. Phys. Rev. E
**2008**, 77, 041118. [Google Scholar] [CrossRef] [Green Version] - Thomas, G.; Johal, R.S. Coupled quantum Otto cycle. Phys. Rev. E
**2011**, 83, 031135. [Google Scholar] [CrossRef] [Green Version] - Makhlin, Y.; Schön, G.; Shnirman, A. Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys.
**2001**, 73, 357–400. [Google Scholar] [CrossRef] [Green Version] - Navez, P.; Sowa, A.; Zagoskin, A. Entangling continuous variables with a qubit array. Phys. Rev. B
**2019**, 100, 144506. [Google Scholar] [CrossRef] [Green Version] - Bishop, C.M. Pattern Recognition and Machine Learning; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Turing, M.A. Computing machinery and intelligence. Mind
**1950**, 59, 433–460. [Google Scholar] [CrossRef] - Crevier, D. AI: The Tumultuous Search for Artificial Intelligence; BasicBooks: New York, NY, USA, 1993. [Google Scholar]
- Biamonte, J.; Wittek, P.; Pancotti, N.; Rebentrost, P.; Wiebe, N.; Lloyd, S. Quantum machine learning. Nature
**2017**, 549, 195–202. [Google Scholar] [CrossRef] [PubMed] - Carleo, G.; Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science
**2017**, 355, 602–606. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Torlai, G.; Mazzola, G.; Carrasquilla, J.; Troyer, M.; Melko, R.; Carleo, G. Neural-network quantum state tomography. Nat. Phys.
**2018**, 14, 447. [Google Scholar] [CrossRef] - Tiunov, E.S.; Tiunova, V.V.; Ulanov, A.E.; Lvovsky, A.I.; Fedorov, A.K. Experimental quantum homodyne tomography via machine learning. arXiv
**2019**, arXiv:1907.06589. Available online: https://arxiv.org/abs/1907.06589 (accessed on 6 November 2019). - Choo, K.; Neupert, T.; Carleo, G. Study of the two-dimensional frustrated J1-J2 model with neural network quantum states. Phys. Rev. B
**2019**, 100, 124125. [Google Scholar] [CrossRef] - Sharir, O.; Levine, Y.; Wies, N.; Carleo, G.; Shashua, A. Deep autoregressive models for the efficient variational simulation of many-body quantum systems. arXiv
**2019**, arXiv:1902.04057. Available online: https://arxiv.org/abs/1902.04057 (accessed on 6 November 2019). - Wu, D.; Wang, L.; Zhang, P. Solving statistical mechanics using variational autoregressive networks. Phys. Rev. Lett.
**2019**, 122, 080602. [Google Scholar] [CrossRef] - Kharkov, Y.A.; Sotskov, V.E.; Karazeev, A.A.; Kiktenko, E.O.; Fedorov, A.K. Revealing quantum chaos with machine learning. arXiv
**2019**, arXiv:1902.09216. Available online: https://arxiv.org/abs/1902.09216 (accessed on 6 November 2019). - Rocchetto, A.; Grant, E.; Strelchuk, S.; Carleo, G.; Severini, S. Learning hard quantum distributions with variational autoencoders. npj Quantum Inf.
**2018**, 4, 28. [Google Scholar] [CrossRef] [Green Version] - Carrasquilla, J.; Torlai, G.; Melko, R.G.; Aolita, L. Reconstructing quantum states with generative models. Nat. Mach. Intell.
**2019**, 1, 155. [Google Scholar] [CrossRef] - Generative Models for Physicists. Lecture note. Available online: http://wangleiphy.github.io/lectures/PILtutorial.pdf (accessed on 7 November 2019).
- Hewson, A.C. The Kondo Problem to Heavy Fermions; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Coleman, P. Heavy fermions: Electrons at the edge of magnetism. In Handbook of Magnetism and Advanced Magnetic Materials; Kronmúller, H., Parkin, S., Eds.; John Wiley & Sons: Chichester, UK, 2007. [Google Scholar]
- Sachdev, S. Quantum Phase Transitions; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Coleman, P.; Schofield, A. Quantum criticality. Nature
**2000**, 433, 226–229. [Google Scholar] [CrossRef] [PubMed] - Anderson, P.W. Localized Magnetic States in Metals. Phys. Rev.
**1961**, 124, 41–53. [Google Scholar] [CrossRef] - Frésard, R.; Ouerdane, H.; Kopp, T. Slave bosons in radial gauge: a bridge between path integral and Hamiltonian language. Nucl. Phys. B
**2007**, 785, 286–306. [Google Scholar] [CrossRef] - Frésard, R.; Ouerdane, H.; Kopp, T. Barnes slave-boson approach to the two-site single-impurity Anderson model with non-local interaction. EPL
**2008**, 82, 31001. [Google Scholar] [CrossRef] - Diu, B.; Guthmann, C.; Lederer, D.; Roulet, B. Physique Statistique; Éditions Hermann: Paris, France, 1996. [Google Scholar]
- Mila, F. Frustrated spin systems. In Many-Body Physics: From Kondo to Hubbard; Pavarini, E., Koch, E., Coleman, P., Eds.; Verlag des Forschungszentrum Jülich: Kreis Düren, Rheinland, 2015. [Google Scholar]
- Refael, G.; Moore, J.E. Entanglement Entropy of Random Quantum Critical Points in One Dimension. Phys. Rev. Lett.
**2004**, 93. [Google Scholar] [CrossRef] - Schollwóck, U. The density-matrix renormalization group. Rev. Mod. Phys.
**2005**, 77, 259–315. [Google Scholar] [CrossRef] [Green Version] - Ising, E. Beitrag zur Theorie des Ferromagnetismus. Z. Phys.
**1925**, 31, 253–258. [Google Scholar] [CrossRef] - Kramers, H.A.; Wannier, G.H. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev.
**1941**, 60, 252–262. [Google Scholar] [CrossRef] - Ovchinnikov, A.A.; Dmitriev, D.V.; Krivnov, V.Y.; Cheranovskii, V.O. Antiferromagnetic Ising chain in a mixed transverse and longitudinal magnetic field. Phys. Rev. B
**2003**, 68, 214406. [Google Scholar] [CrossRef] [Green Version] - Coldea, R.; Tennant, D.A.; Wheeler, E.M.; Wawrzynska, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; K, K. Quantum criticality in an Ising chain: Experimental evidence for emergent E
_{8}symmetry. Science**2010**, 327, 177–180. [Google Scholar] [CrossRef] [PubMed] - Sachdev, S.; Keimer, B. Quantum criticality. Phys. Today
**2011**, 64, 29–35. [Google Scholar] [CrossRef] [Green Version] - Matsubara, T. A new approach to quantum statistical mechanics. Prog. Theor. Exp.
**1955**, 14, 351–378. [Google Scholar] [CrossRef] - Kogut, J.B. An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys.
**1979**, 51, 659–713. [Google Scholar] [CrossRef] - Krizhevsky, A.; Sutskever, I.; Hinton, G.E. Imagenet classification with deep convolutional neural networks. In Proceedings of the NIPS: Advances in Neural Information Processing Systems 25, Stateline, NV, USA, 3–8 December 2012; pp. 1097–1105. [Google Scholar]
- Holevo, A.S. Probabilistic and Statistical Aspects of Quantum Theory; Springer: Berlin/Heidelberg, Germany, 2011; Volume 1. [Google Scholar]
- Filippov, S.N.; Man’ko, V.I. Inverse spin-s portrait and representation of qudit states by single probability vectors. J. Russ. Laser Res.
**2010**, 31, 32–54. [Google Scholar] [CrossRef] [Green Version] - Appleby, M.; Fuchs, C.A.; Stacey, B.C.; Zhu, H. Introducing the Qplex: A novel arena for quantum theory. Eur. Phys. J. D
**2017**, 71, 197. [Google Scholar] [CrossRef] - Caves, C.M. Symmetric informationally complete POVMs - UNM Information Physics Group internal report (1999). Available online: http://info.phys.unm.edu/~caves/reports/infopovm.pdf (accessed on 7 November 2019).
- Myung, I.J. Tutorial on maximum likelihood estimation. J. Math. Psychol.
**2003**, 47, 90–100. [Google Scholar] [CrossRef] - Filippov, S.N.; Man’ko, V.I. Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics. J. Russ. Laser Res.
**2010**, 31, 211–231. [Google Scholar] [CrossRef] [Green Version] - mpnum: A Matrix Product Representation Library for Python. Available online: https://mpnum.readthedocs.io/en/latest/ (accessed on 7 November 2019).
- Sohn, K.; Lee, H.; Yan, X. Learning structured output representation using deep conditional generative models. In Proceedings of the NIPS: Advances in Neural Information Processing Systems 28, Montreal, QC, Canada, 7–12 December 2015; pp. 3483–3491. [Google Scholar]
- Kingma, D.P.; Welling, M. Auto-encoding variational Bayes. arXiv
**2013**, arXiv:1312.6114. Available online: https://arxiv.org/abs/1312.6114 (accessed on 6 November 2019). - Rezende, D.J.; Mohamed, S.; Wierstra, D. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31st International Conference on Machine Learning (ICML), Beijing, China, 21–26 June 2014; Volume 32. [Google Scholar]
- Jang, E.; Gu, S.; Poole, B. Categorical reparameterization with Gumbel-softmax. arXiv
**2016**, arXiv:1611.01144. Available online: https://arxiv.org/abs/1611.01144 (accessed on 6 November 2019). - Kusner, M.J.; Hernández-Lobato, J.M. Gans for sequences of discrete elements with the Gumbel-softmax distribution. arXiv
**2016**, arXiv:1611.04051. Available online: https://arxiv.org/abs/1611.04051 (accessed on 6 November 2019). - Maddison, C.J.; Mnih, A.; Teh, Y.W. The concrete distribution: A continuous relaxation of discrete random variables. arXiv
**2016**, arXiv:1611.00712. Available online: https://arxiv.org/abs/1611.00712 (accessed on 6 November 2019). - Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys.
**1953**, 21, 1087–1092. [Google Scholar] [CrossRef] [Green Version] - Das, A.; Chakrabarti, B.K. Colloquium: Quantum annealing and analog quantum computation. Rev. Mod. Phys.
**2008**, 80, 1061–1081. [Google Scholar] [CrossRef] - Yavorsky, A.; Markovich, L.A.; Polyakov, E.A.; Rubtsov, A.N. Highly parallel algorithm for the Ising ground state searching problem. arXiv
**2019**, arXiv:1907.05124. Available online: https://arxiv.org/abs/1907.05124 (accessed on 6 November 2019). - Verstraete, F.; Cirac, J.I. Matrix product states represent ground states faithfully. Phys. Rev. B
**2006**, 73, 094423. [Google Scholar] [CrossRef] [Green Version] - Eisert, J.; Cramer, M.; Plenio, M.B. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys.
**2010**, 82, 277–306. [Google Scholar] [CrossRef] - Deng, D.-L.; Li, X.; Das Sarma, S. Quantum entanglement in neural network states. Phys. Rev. X
**2017**, 7, 021021. [Google Scholar] [CrossRef] - Bhattacharyya, A. On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc.
**1943**, 35, 99–109. [Google Scholar] - Boixo, S.; Ronnow, T.F.; Isakov, S.V.; Wang, Z.; Wecker, D.; Lidar, D.A.; Martinis, J.M.; Troyer, M. Evidence for quantum annealing with more than one hundred qubits. Nat. Phys.
**2014**, 10, 218–224. [Google Scholar] [CrossRef] [Green Version] - Denchev, V.S.; Boixo, S.; Isakov, S.V.; Ding, N.; Babbush, R.; Smelyanskiy, V.; Martinis, J.; Neven, H. What is the computational value of finite-range tunneling? Phys. Rev. X
**2016**, 6, 031015. [Google Scholar] [CrossRef] - Navez, P.; Tsironis, G.P.; Zagoskin, A.M. Propagation of fluctuations in the quantum Ising model. Phys. Rev. B
**2017**, 95, 064304. [Google Scholar] [CrossRef] [Green Version] - Volkov, A.A.; Artemov, V.G.; Pronin, A.V. A radically new suggestion about the electrodynamics of water: Can the pH index and the Debye relaxation be of a common origin? EPL
**2014**, 106, 46004. [Google Scholar] [CrossRef] - Artemov, V.G. A unified mechanism for ice and water electrical conductivity from direct current to terahertz. Phys. Chem. Chem. Phys.
**2019**, 21, 8067–8072. [Google Scholar] [CrossRef] [PubMed] - Github Repository with Code. Available online: https://github.com/LuchnikovI/Representation-of-quantum-many-body-states-via-VAE (accessed on 7 November 2019).
- Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv
**2014**, arXiv:1412.6980. Available online: https://arxiv.org/abs/1412.6980 (accessed on 6 November 2019).

**Figure 1.**Tensor diagrams for (

**a**) building blocks, (

**b**) matrix product state (MPS) representation of measurement outcome probability, and (

**c**) its subtensor.

**Figure 2.**Tensor diagrams for (

**a**) building blocks and (

**b**) inverse transformation from a mass function to a density matrix.

**Figure 5.**Two-point correlation function $\langle {\sigma}_{1}^{z}{\sigma}_{n}^{z}\rangle $ for different values of external magnetic field ${h}_{x}$.

**Figure 6.**Two-point correlation function $\langle {\sigma}_{1}^{x}{\sigma}_{n}^{x}\rangle $ for different values of external magnetic field ${h}_{x}$.

**Figure 7.**Average magnetization per site along x for different values of external magnetic field ${h}_{x}$.

**Figure 8.**Total magnetization along x and z axes for different values of external magnetic field ${h}_{x}$. The location of the critical region is slightly shifted towards smaller values of ${h}_{x}$ due to the finite size of the chain.

**Figure 9.**Backpropagation-based and numerical-based (central differences) values of ${\chi}_{xx}$ and ${\chi}_{zx}$ for different values of external magnetic field ${h}_{x}$. Both derivatives slightly fluctuate due to VAE error.

**Figure 10.**Comparison of two positive-operator valued measure (POVM)-induced mass functions ($P\left[\alpha \right]=\mathrm{Tr}\left(\rho {M}^{\alpha}\right)$) for a chain of size 5: numerically exact mass function and reconstructed from VAE samples mass function. A sequence of indices $\alpha $ has been transformed into a single multi-index. Indices have been ordered to put numerically exact probability in descending order. A good agreement between the mass functions is observed.

**Figure 11.**Dependence of the classical fidelity on the external magnetic field. A high predictive accuracy is demonstrated for the whole set of fields.

**Figure 12.**Comparison of the numerically exact Rényi entropy and that reconstructed from the VAE samples for different values of n.

**Figure 13.**Comparison of numerically exact spectra of density matrices and VAE-estimated spectra. The ground state spectra of the spin chain of size 5 with an external magnetic field $h=0.9$ is shown on the right panel, and the spectra of the reduced density matrix (last 3 spins) are shown on the left panel.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Luchnikov, I.A.; Ryzhov, A.; Stas, P.-J.; Filippov, S.N.; Ouerdane, H.
Variational Autoencoder Reconstruction of Complex Many-Body Physics. *Entropy* **2019**, *21*, 1091.
https://doi.org/10.3390/e21111091

**AMA Style**

Luchnikov IA, Ryzhov A, Stas P-J, Filippov SN, Ouerdane H.
Variational Autoencoder Reconstruction of Complex Many-Body Physics. *Entropy*. 2019; 21(11):1091.
https://doi.org/10.3390/e21111091

**Chicago/Turabian Style**

Luchnikov, Ilia A., Alexander Ryzhov, Pieter-Jan Stas, Sergey N. Filippov, and Henni Ouerdane.
2019. "Variational Autoencoder Reconstruction of Complex Many-Body Physics" *Entropy* 21, no. 11: 1091.
https://doi.org/10.3390/e21111091