Abstract Deep neural networks have been used to solve Ising models, including autoregressive neural networks, convolutional neural networks, recurrent neural networks, and graph neural networks. Learning probability distributions of energy configuration or finding ground states of disordered, fully connected Ising models is essential for statistical mechanics and NP-hard problems. Despite tremendous efforts, neural network architectures with abilities to high-accurately solve these intractable problems on larger systems remain a challenge. Here we propose a variational autoregressive architecture with a message passing mechanism, which effectively utilizes the interactions between spin variables. The architecture trained under an annealing framework outperforms existing neural network-based methods in solving several prototypical Ising spin Hamiltonians, especially for larger systems at low temperatures. The advantages also come from the great mitigation of mode collapse during training process. Considering these difficult problems to be solved, our method extends computational limits of unsupervised neural networks to solve combinatorial optimization problems.
Abstract Deep neural networks have been used to solve Ising models, including autoregressive neural networks, convolutional neural networks, recurrent neural networks, and graph neural networks. Learning probability distributions of energy configuration or finding ground states of disordered, fully connected Ising models is essential for statistical mechanics and NP-hard problems. Despite tremendous efforts, neural network architectures with abilities to high-accurately solve these intractable problems on larger systems remain a challenge. Here we propose a variational autoregressive architecture with a message passing mechanism, which effectively utilizes the interactions between spin variables. The architecture trained under an annealing framework outperforms existing neural network-based methods in solving several prototypical Ising spin Hamiltonians, especially for larger systems at low temperatures. The advantages also come from the great mitigation of mode collapse during training process. Considering these difficult problems to be solved, our method extends computational limits of unsupervised neural networks to solve combinatorial optimization problems.
We propose a general framework for solving statistical mechanics of systems with finite size. The approach extends the celebrated variational mean-field approaches using autoregressive neural networks, which support direct sampling and exact calculation of normalized probability of configurations. It computes variational free energy, estimates physical quantities such as entropy, magnetizations and correlations, and generates uncorrelated samples all at once. Training of the network employs the policy gradient approach in reinforcement learning, which unbiasedly estimates the gradient of variational parameters. We apply our approach to several classic systems, including 2D Ising models, the Hopfield model, the Sherrington-Kirkpatrick model, and the inverse Ising model, for demonstrating its advantages over existing variational mean-field methods. Our approach sheds light on solving statistical physics problems using modern deep generative neural networks.
Variational neural network models have achieved remarkable success in solving ground-state problems of quantum many-body systems. However, addressing classical and quantum spin glasses remains challenging, as disorder and energy frustration give rise to an exponentially large number of local energy minima separated by high-energy barriers, hindering the efficiency of conventional Metropolis-based Monte Carlo methods. To bridge this gap, we introduce Deep Boltzmann Quantum States, a class of neural quantum states inspired by deep Boltzmann machines that inherit efficient block Gibbs sampling. We also propose two key advances in the training algorithm. Firstly, we combine natural-gradient updates with state-of-the-art stochastic optimizers. Secondly, we gradually tune the hardness of the problem Hamiltonian by interpolating from an easy to a hard regime, without the need to closely approximate the instantaneous adiabatic state at intermediate times. We match the exact solution or the best available estimate for several instances of classical and quantum Ising spin-glass models with infinite-range interactions and hundreds of spins. We also solve instances of the NP-hard Job Shop Scheduling Problem exceeding the current limitations of quantum annealing hardware. To summarize, deep neural architectures with efficient global update rules and trained within an annealing-like scheme, provide a powerful framework for solving real-world hard combinatorial optimization and for investigating disordered quantum many-body systems.
Abstract Computing free energy is a fundamental problem in statistical physics. Recently, two distinct methods have been developed and have demonstrated remarkable success: the tensor-network-based contraction method and the neural-network-based variational method. Tensor networks are accurate, but their application is often limited to low-dimensional systems due to the high computational complexity in high-dimensional systems. The neural network method applies to systems with general topology. However, as a variational method, it is not as accurate as tensor networks. In this work, we propose an integrated approach, tensor-network-based variational autoregressive networks (TNVAN), that leverages the strengths of both tensor networks and neural networks: combining the variational autoregressive neural network’s ability to compute an upper bound on free energy and perform unbiased sampling from the variational distribution with the tensor network’s power to accurately compute the partition function for small sub-systems, resulting in a robust method for precisely estimating free energy. To evaluate the proposed approach, we conducted numerical experiments on spin glass systems with various topologies, including two-dimensional lattices, fully connected graphs, and random graphs. Our numerical results demonstrate the superior accuracy of our method compared to existing approaches. In particular, it effectively handles systems with long-range interactions and leverages GPU efficiency without requiring singular value decomposition, indicating great potential in tackling statistical mechanics problems and simulating high-dimensional complex systems through both tensor networks and neural networks.
Monte Carlo methods are essential across diverse scientific fields, yet their efficiency is frequently hampered by critical slowing down-a sharp increase in autocorrelation times near phase transitions. Although deep learning approaches, such as neural-network-based samplers, have been proposed to alleviate this issue, they face another serious problem: the difficulty of training the models. This difficulty partially stems from the overly general nature of original machine-learning architectures, which often ignore underlying physical symmetries and force networks to relearn them from scratch. In this paper, we demonstrate that incorporating physical priors into the model significantly enhances performance. Building upon existing strategies that integrate spin-spin interactions, we propose a framework that utilizes a prior probability distribution as a starting point for training. Our results for the Ising model, as well as for the Edwards-Anderson spin glass model, suggest that moving away from `blank slate' models in favor of physics-informed priors reduces the training burden and facilitates the simulation of larger system sizes in discrete spin models.
We propose a general method for solving statistical mechanics problems defined on sparse graphs, such as random graphs, real-world networks, and low-dimensional lattices. Our approach extract a small feedback vertex set of the sparse graph, converting the sparse system to a strongly correlated system with many-body and dense interactions on the feedback set, then solve it using variational method based on neural networks to estimate free energy, observables, and generate unbiased samples via direct sampling. Extensive experiments show that our approach is more accurate than existing approaches for sparse spin glass systems. On random graphs and real-world networks, our approach significantly outperforms the standard methods for sparse systems such as belief-propagation; on structured sparse systems such as two-dimensional lattices our approach is significantly faster and more accurate than recently proposed variational autoregressive networks using convolution neural networks.
Abstract We extend the ability of an unitary quantum circuit by interfacing it with a classical autoregressive neural network. The combined model parametrizes a variational density matrix as a classical mixture of quantum pure states, where the autoregressive network generates bitstring samples as input states to the quantum circuit. We devise an efficient variational algorithm to jointly optimize the classical neural network and the quantum circuit to solve quantum statistical mechanics problems. One can obtain thermal observables such as the variational free energy, entropy, and specific heat. As a byproduct, the algorithm also gives access to low energy excitation states. We demonstrate applications of the approach to thermal properties and excitation spectra of the quantum Ising model with resources that are feasible on near-term quantum computers.