Quantum many-body physics simulation has important impacts on understanding fundamental science and has applications to quantum materials design and quantum technology. However, due to the exponentially growing size of the Hilbert space with respect to the particle number, a direct simulation is intractable. While representing quantum states with tensor networks and neural networks are the two state-of-the-art methods for approximate simulations, each has its own limitations in terms of expressivity and inductive bias. To address these challenges, we develop a novel architecture, Autoregressive Neural TensorNet (ANTN), which bridges tensor networks and autoregressive neural networks. We show that Autoregressive Neural TensorNet parameterizes normalized wavefunctions, allows for exact sampling, generalizes the expressivity of tensor networks and autoregressive neural networks, and inherits a variety of symmetries from autoregressive neural networks. We demonstrate our approach on quantum state learning as well as finding the ground state of the challenging 2DJ1-J2 Heisenberg model with different systems sizes and coupling parameters, outperforming both tensor networks and autoregressive neural networks. Our work opens up new opportunities for quantum many-body physics simulation, quantum technology design, and generative modeling in artificial intelligence.

We also introduce a novel benchmark for evaluating NNPs in molecular property prediction, Hamiltonian prediction, and conformational optimization tasks.

B: Overview of the proposed PhiSNet architecture.

Usingourmethod weestablish anewbenchmark bycalculating the most accurate variational ground state energies ever published for a number of different atoms and molecules. Wesystematically break down and measure our improvements, focusing in particular on the effect of increasing physical prior knowledge.

In this work, we introduce the first type of non-Euclidean neural quantum state (NQS) ansatz, in the form of the hyperbolic GRU (a variant of recurrent neural networks (RNNs)), to be used in the Variational Monte Carlo method of approximating the ground state energy for quantum many-body systems. In particular, we examine the performances of NQS ansatzes constructed from both conventional or Euclidean RNN/GRU and from hyperbolic GRU in the prototypical settings of the one- and two-dimensional transverse field Ising models (TFIM) and the one-dimensional Heisenberg $J_1J_2$ and $J_1J_2J_3$ systems. By virtue of the fact that, for all of the experiments performed in this work, hyperbolic GRU can yield performances comparable to or better than Euclidean RNNs, which have been extensively studied in these settings in the literature, our work is a proof-of-concept for the viability of hyperbolic GRU as the first type of non-Euclidean NQS ansatz for quantum many-body systems. Furthermore, in settings where the Hamiltonian displays a clear hierarchical interaction structure, such as the 1D Heisenberg $J_1J_2$ & $J_1J_2J_3$ systems with the 1st, 2nd and even 3rd nearest neighbor interactions, our results show that hyperbolic GRU definitively outperforms its Euclidean version in almost all instances. The fact that these results are reminiscent of the established ones from natural language processing where hyperbolic GRU almost always outperforms Euclidean RNNs when the training data exhibit a tree-like or hierarchical structure leads us to hypothesize that hyperbolic GRU NQS ansatz would likely outperform Euclidean RNN/GRU NQS ansatz in quantum spin systems that involve different degrees of nearest neighbor interactions. Finally, with this work, we hope to initiate future studies of other types of non-Euclidean NQS beyond hyperbolic GRU.