Artificial neural networks for quantum many-body problems

It is a great challenge to accurately represent quantum many-body states. In this talk, we will show that Boltzmann machines used in machine learning can be useful for analyzing quantum many-body problems.

First, we introduce a method proposed by Carleo and Troyer in 2017 to represent quantum states using Boltzmann machines [1]. Then, we describe the progress of the neural-network wave function method for zero-temperature simulations [2-6]. Through various extensions, neural-network wave functions are beginning to be applied to challenging problems in physics, such as frustrated spin systems [5]. 

Next, we discuss two finite-temperature calculation methods using deep Boltzmann machines (DBMs) with two hidden layers [7]. Both methods use the idea of “purification,” in which a finite-temperature mixed state is represented by a pure state of an extended system. The former analytically constructs the pure state corresponding to thermal equilibrium, realizing quantum-to-classical mapping [3]. The latter method obtains the pure state by numerically optimizing the DBM parameters. This method can be applied to, e.g., frustrated systems for which the former method suffers from the negative sign problem. We will discuss the applications to the transverse-field Ising model and J₁-J₂ Heisenberg model.

These works were done in collaboration with Andrew S. Darmawan, Youhei Yamaji, Masatoshi Imada, Giuseppe Carleo, Nobuyuki Yoshioka, and Franco Nori. 

[1] G. Carleo and M. Troyer Science 355, 602 (2017)

[2] Y. Nomura, A. S. Darmawan, Y. Yamaji, and M. Imada, Phys. Rev. B 96, 205152 (2017)

[3] G. Carleo, Y. Nomura, and M. Imada, Nat. Commun. 9, 5322 (2018)

[4] Y. Nomura, J. Phys. Soc. Jpn. 89, 054706 (2020) [Editor’s choice]

[5] Y. Nomura and M. Imada, Phys. Rev. X 11, 031034 (2021)

[6] Y. Nomura, J. Phys.: Condens. Matter 33, 174003 (2021) [special issue “Emergent Leaders 2020”]

[7] Y. Nomura, N. Yoshioka, and F. Nori, Phys. Rev. Lett. 127, 060601 (2021)