**6. Conclusions**

The thermodynamic study of complex many-body quantum systems still requires the development of new methods, including those that may stem from machine learning. The quantum Ising model, which is of particular importance for practical purposes [107,108], provides a rich framework to test these new methods that are also useful to obtain deeper physical insight into its nonequilibrium dynamics properties such as, e.g., quantum fluctuations propagation [109]. In the present work, we studied the ability of a VAE to reconstruct the physics of quantum many-body systems, using the transverse-field Ising model as a nontrivial example. We used the IC POVM to map the quantum problem onto a probabilistic domain and vice versa. We trained the VAE on a set of samples from the transformed quantum problem, and our numerical experiments show the following results.


Our method can be extended to any other thermodynamic system by introduction of the temperature as an external parameter, thereby considering also thermal phase transitions. As one can calculate different thermodynamic quantities by applying backpropagation through VAE, a worthwhile and highly complex system to study would be water under its difference phases, so as to test recent new ideas and models [110,111].

Our code for our numerical experiments is available on the GitHub repository website [112]. **Author Contributions:** Conceptualization, I.A.L., S.N.F., and H.O.; methodology, I.A.L. and A.R.; software, I.A.L., A.R., and P.J.S.; validation, all authors; writing-original draft preparation, I.A.L., A.R., P.J.S., and H.O; writing-review and editing, S.N.F. and H.O.

**Funding:** This research was supported by the Russian Foundation for Basic Research grants under the Project No. 18-37-00282 and the Project No. 18-37-20073. This research was also partially supported by the Skoltech NGP Program (Skoltech-MIT joint project).

**Acknowledgments:** The authors thank Stepan Vintskevich for fruitful discussions. The authors also thank Google Colaboratory for providing access to GPU for the acceleration of computations.

**Conflicts of Interest:** The authors declare no conflicts of interest.
