Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

ANTN: Bridging Autoregressive Neural Networks and Tensor Networks for Quantum Many-Body Simulation

Zhuo Chen, Laker Newhouse, Eddie Chen, Di Luo, Marin Soljačić

TL;DR

ANTN introduces a unified architecture that bridges tensor networks and autoregressive neural networks to tackle quantum many-body simulation. By parameterizing wavefunctions with exact normalization and sampling, ANTN achieves greater expressivity than either TNs or ARNNs alone, while inheriting physics-informed inductive biases and symmetries. Theoretical results establish self-normalization, exact sampling, and a spectrum of expressivity and symmetry properties, including volume-law entanglement and global $U(1)$ symmetry. Empirically, ANTN outperforms TN and ARNN baselines in quantum-state learning and variational Monte Carlo for the 2D $J_1$-$J_2$ Heisenberg model, showing favorable scaling with system size. This framework promises advances in quantum materials design, quantum computing simulations, and potentially broader generative modeling tasks in AI.

Abstract

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 2D $J_1$-$J_2$ 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.

ANTN: Bridging Autoregressive Neural Networks and Tensor Networks for Quantum Many-Body Simulation

TL;DR

ANTN introduces a unified architecture that bridges tensor networks and autoregressive neural networks to tackle quantum many-body simulation. By parameterizing wavefunctions with exact normalization and sampling, ANTN achieves greater expressivity than either TNs or ARNNs alone, while inheriting physics-informed inductive biases and symmetries. Theoretical results establish self-normalization, exact sampling, and a spectrum of expressivity and symmetry properties, including volume-law entanglement and global symmetry. Empirically, ANTN outperforms TN and ARNN baselines in quantum-state learning and variational Monte Carlo for the 2D - Heisenberg model, showing favorable scaling with system size. This framework promises advances in quantum materials design, quantum computing simulations, and potentially broader generative modeling tasks in AI.

Abstract

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 2D - 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.
Paper Structure (29 sections, 16 theorems, 37 equations, 5 figures, 5 tables, 3 algorithms)

This paper contains 29 sections, 16 theorems, 37 equations, 5 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Quantum Preliminaries
  5. Quantum State Learning
  6. Variational Monte Carlo
  7. Method
  8. Preliminaries
  9. Autoregressive Neural TensorNet (ANTN)
  10. Theoretical Results
  11. Exact Sampling and Complexity Analysis of ANTN
  12. Expressivity Results of ANTN
  13. Symmetry Results of ANTN
  14. Experiments
  15. Quantum State Learning
  16. ...and 14 more sections

Key Result

Theorem 5.1

Autoregressive Neural TensorNet wavefunction is automatically normalized and allows for exact sampling.

Figures (5)

  • Figure 1: Diagrammatic representation of autoregressive neural network (ARNN), tensor network (TN) and our Autoregressive Neural TensorNet (ANTN).
  • Figure 1: Energy per site $\downarrow$ for $8\times8$ system with various algorithms where elementwise and blockwise are two constructions of ANTN (with PixelCNN + MPS). The bond dimensions for ANTN are labeled inside the parentheses. For each algorithm, we test it both with the sign rule (S) and without the sign rule (NS) The best energy is highlighted in boldface and the second in italic.
  • Figure 2: Fidelity $\uparrow$ on quantum state learning with 16 qubits for TN (MPS), ARNN (transformer) and ANTN (elementwise construction with transformer+MPS). (a) Learning random Bell states. (b) Learning real-valued depth-4 random circuit with and without sign rule. The error bar denotes the standard deviation (not the standard error of the mean) of the fidelities over the random states sampled from the corresponding distribution. The mean and standard deviation are calculated from 10 random states. The numbers inside the parentheses denote the bond dimension.
  • Figure 3: Energy per site difference $\downarrow$ between elementwise ANTN with bond dimension 8 and MPS with bond dimension 1024 optimized using DMRG algorithm for various $J_2$ and different system sizes from $4\times 4$ to $12\times 12$.
  • Figure 4: Energy per site vs total number of parameters and runtime (in hours) for various algorithms (MPS with DMRG algorithm, PixelCNN, elementwise (EW), and blockwise (BW) ANTNs) and system sizes ($10\times10$ and $12\times12$) for $J_2=0.5$. The ANTN construction uses PixelCNN as the underlying ARNN. The total number of parameters includes parameters from both the TN part and the ARNN part.

Theorems & Definitions (20)

  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Definition 5.1: Mask Symmetry
  • Definition 5.2: Function Symmetry
  • Theorem 5.4
  • Corollary 5.4.1: Global U(1) Symmetry
  • Corollary 5.4.2: $\mathbb{Z}_2$ Spin Flip Symmetry
  • Corollary 5.4.3: Discrete Abelian and Non-Abelian Symmetries
  • Theorem B.1
  • ...and 10 more