Session Q47: Machine Learning for Quantum Matter I

Highly accurate potential energy surfaces with deep quantum Monte Carlo

The combination of deep learning with variational Monte Carlo offers a promising new direction to compute properties of quantum systems with high accuracy and acceptable computational complexity without having to specifically taylor the method to the specific molecule or material. Building on the PauliNet deep quantum Monte Carlo architecture we present new approaches and results on using fermionic networks to solve quantum chemical systems with high accuracy.   

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)   

Withdrawn

Variational approximations to the many-body Schrödinger equation can provide accurate energies and properties, but the accuracy is determined by the flexibility and representation capacity of the wavefunction form. We demonstrate that deep neural networks with physically-motivated structures offer a compact and highly accurate wavefunction Ansatz, can be efficiently optimized using variational Monte Carlo, and frequently outperform even diffusion Monte Carlo calculations using conventional wavefunctions. We demonstrate the applicability of our approach on a range of atoms, small molecules and model reactions.   

Fermionic variational wavefunctions from neural-network constrained hidden states

For the variational simulation of fermionic systems in first quantization, trial wavefunctions must be anti-symmetric functions of the particle configurations, while being able to capture correlations beyond the single-particle Slater determinants. This is typically achieved either by considering backflow transformations, or with Jastrow-like projection factors. Despite the recent success of neural-network based parametrizations, the strong coupling limit remains to be a challenging regime.

In this talk I will introduce a new and systematically improvable family of variational states consisting on the projection of uncorrelated Slater determinants in a Hilbert space augmented by hidden fermionic degrees of freedom. The ability to jointly optimize the single-particle orbitals together with the projection (parametrized by neural-networks) onto the physical Hilbert space, provides an extremely expressive family of wavefunction ansatze. We study the ground state properties of the Hubbard model in the square lattice, achieving levels of accuracy competitive with state-of-the-art computational methods.   

Bosonic Neural Quantum States in Continuous Space

Neural quantum states (NQS) have been shown to be a powerful tool in the computation of ground-states for several quantum mechanical many-body systems.

While the field has mainly focused on Hamiltonians defined on discrete lattices, there is an increasing interest in the properties of continuous systems.

Recently, results have been reported for non-periodic, continuous systems such as molecules.

In this work we will show that it is possible to use a permutation-invariant neural-network architecture to approximate the wavefunction of periodic, bosonic systems if care is taken to correctly enforce continuity at the boundaries.

The method is general and can in principle be applied to arbitrary Hamiltonians in an arbitrary number of spatial dimensions.

We will present the application of the architecture to two different interaction potentials (Helium-4 and Gaussian) in one and two spatial dimensions. We will outline the shortcomings of the current approach and ways to improve the obtained accuracy.   

Determinant-free fermionic wave function using feed-forward neural networks

In recent years, there has been a remarkable increase in research on approximating many-body wave functions by neural networks. It is, however, still challenging to find the ground state of fermionic many-body systems due to the complex sign structure arising from the anticommutation relation between fermions. The sign structure is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N³) for N particles. We here propose a framework to bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost to O(N²) or less. In addition, we device some numerical tricks for stabilization of the calculations, e.g., a reweighting method in Monte Carlo sampling. We apply our method to the Hubbard model with 6x6 lattice sites and N=10, and find that it achieves a lower energy than the many-variable variational Monte Carlo calculation.   

Equivariant Variational Monte Carlo for quantum systems

Variational Monte Carlo algorithms based on sampling basis states in a computational basis often suffer from breaking of SU(2)-symmetry when the basis states transform non-trivially under the symmetry group.  We propose two ways to solve this issue.  On one hand, we construct basis states that transform trivially under the continuous SU(2) symmetry group, and use them as input states of a suitable variational wave function.  On the other hand, we move the transformation requirements of SU(2) to the variational wave function, allowing us to use simple basis states as its input.  In this way, we reconcile SU(2) symmetry with basis states defined on the local degrees of freedom, allowing us to additionally include lattice symmetries.  We show that both constructions lead to accurate variational states on antiferromagnetic and frustrated heisenberg systems in one and two dimensions.   

Improving Convolutional Neural Network Wave Functions Optimization

Convolutional neural networks (CNN) have become a staple of modern machine learning research, where they are ideally suited for learning local features with spatial inputs such as images. This has made them an obvious candidate for variational wave functions to represent 2D systems. However, their optimization with state of the art variational Monte Carlo (VMC) methods relies on natural gradient descent, which becomes intractable for a large number of variational parameters. This is due to requiring inverting a matrix whose dimensions scale with the number of parameters. We propose an approximate natural gradient method that minimizes an upper bound on the exact natural gradient descent residual. The method relies on grouping the parameters of the CNN by dependency and inverting sub-matrices of dimension equal to the number of parameters in that group. We Implement our method on a deep CNN with Res-Block architecture and complex valued parameters. We benchmark our results on the frustrated 2D J1-J2 Heisenberg model on the square lattice.