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Quantum Dynamic Simulations with Autoregressive Neural Networks

Wang. Jiangran

Published May 1, 2021• 0 citations
OpenAlex

Abstract

The theory of quantum dynamics is crucial for quantum science and engineering. The computation cost for exact quantum simulation is expensive due to the exponential growth of the dimension of the Hilbert space. There are recent attempts that utilize neural networks to simulate long-time quantum dynamics. We conduct a comparative study on different approaches that simulate dynamics based on parameterizing the quantum states with state-of-the-art autoregressive neural networks. Pixel Convolution Neural Networks, Recurrent Neural Networks, and Transformer models are examined for their representability and accuracy. We identify that Recurrent Neural Networks and Transformers are superior to Pixel Convolution Neural Networks. In addition, we compare the performance of different algorithms to simulate quantum dynamics, which include the Euler’s method, the forward-backward trapezoid method, the stochastic reconfiguration method, and the spacetime method. We found that the spacetime method performs better than the other algorithms, the Euler’s method and the forward-backward method are comparable, while the SR method exhibits stability issues. Our study is performed on the transverse-field Ising model on one-dimensional systems

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•May 22, 2019•0 citations
The theory of quantum dynamics is crucial for quantum science and engineering. The computation cost for exact quantum simulation is expensive due to the exponential growth of the dimension of the Hilbert space. There are recent attempts that utilize neural networks to simulate long-time quantum dynamics. We conduct a comparative study on different approaches that simulate dynamics based on parameterizing the quantum states with state-of-the-art autoregressive neural networks. Pixel Convolution Neural Networks, Recurrent Neural Networks, and Transformer models are examined for their representability and accuracy. We identify that Recurrent Neural Networks and Transformers are superior to Pixel Convolution Neural Networks. In addition, we compare the performance of different algorithms to simulate quantum dynamics, which include the Euler’s method, the forward-backward trapezoid method, the stochastic reconfiguration method, and the spacetime method. We found that the spacetime method performs better than the other algorithms, the Euler’s method and the forward-backward method are comparable, while the SR method exhibits stability issues. Our study is performed on the transverse-field Ising model on one-dimensional systems
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 J1J_1J1​-J2J_2J2​ 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.
Neural quantum states (NQSs) have emerged as a powerful ansatz for variational quantum Monte Carlo studies of strongly correlated systems. Here, we apply recurrent neural networks (RNNs) and autoregressive transformer neural networks to the Fermi-Hubbard and the (non-Hermitian) Hatano-Nelson-Hubbard models in one and two dimensions. In both cases, we observe that the convergence of the RNN ansatz is challenged when increasing the interaction strength. We present a physically motivated and easy-to-implement strategy for improving the optimization, namely, by ramping of the model parameters. Furthermore, we investigate the advantages and disadvantages of the autoregressive sampling property of both network architectures. For the Hatano-Nelson-Hubbard model, we identify convergence issues that stem from the autoregressive sampling scheme in combination with the non-Hermitian nature of the model. Our findings provide insights into the challenges of the NQS approach and make the first step towards exploring strongly correlated electrons using this ansatz.
The theory of open quantum systems lays the foundation for a substantial part of modern research in quantum science and engineering. Rooted in the dimensionality of their extended Hilbert spaces, the high computational complexity of simulating open quantum systems calls for the development of strategies to approximate their dynamics. In this Letter, we present an approach for tackling open quantum system dynamics. Using an exact probabilistic formulation of quantum physics based on positive operator-valued measure, we compactly represent quantum states with autoregressive neural networks; such networks bring significant algorithmic flexibility due to efficient exact sampling and tractable density. We further introduce the concept of string states to partially restore the symmetry of the autoregressive neural network and improve the description of local correlations. Efficient algorithms have been developed to simulate the dynamics of the Liouvillian superoperator using a forward-backward trapezoid method and find the steady state via a variational formulation. Our approach is benchmarked on prototypical one-dimensional and two-dimensional systems, finding results which closely track the exact solution and achieve higher accuracy than alternative approaches based on using Markov chain Monte Carlo method to sample restricted Boltzmann machines. Our Letter provides general methods for understanding quantum dynamics in various contexts, as well as techniques for solving high-dimensional probabilistic differential equations in classical setups.
Understanding the long-time dynamics of strongly correlated quantum matter remains a major challenge: as quantum entanglement grows during time evolution, conventional numerical methods become increasingly costly, limiting access to the long-range correlations and low-energy excitations that govern gapless systems. Here we show that these dynamical correlations can be extended beyond the accessible numerical window without explicitly evolving the underlying many-body wavefunction. We introduce an autoregressive machine-learning framework that enables the extrapolation of dynamical spin correlations in both time and space beyond the reach of conventional numerical methods. Trained on time-dependent density matrix renormalization group simulations of the gapless XXZ model, our approach is benchmarked against exact solutions available for this analytically solvable system. Combined with physics-informed spatial extension, a multi-layer perceptron model using ReLU activation functions has been shown to be superior to convolutional neural networks and linear regressions for longer time extrapolation. A study of error accumulation further demonstrates that our autoregressive neural network extrapolations are highly robust to perturbations, suggesting stable and reliable predictions. This work offers a complementary route to studying the dynamics of gapless quantum many-body systems, in which machine learning extends and complements the capabilities of state-of-the-art numerical approaches.
The efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter constitutes a key challenge for current computational methods. This holds in particular in the regime of two spatial dimensions, whose experimental exploration is currently pursued with strong efforts in quantum simulators. In this work we present a versatile and efficient machine learning inspired approach based on a recently introduced artificial neural network encoding of quantum many-body wave functions. We identify and resolve key challenges for the simulation of time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, we study the dynamics of the paradigmatic two-dimensional transverse-field Ising model, as recently also realized experimentally in systems of Rydberg atoms. Calculating the nonequilibrium real-time evolution across a broad range of parameters, we, for instance, observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached timescales are comparable to or exceed the capabilities of state-of-the-art tensor network methods.
Machine learning (ML) architectures such as convolutional neural networks (CNNs) have garnered considerable recent attention in the study of quantum many-body systems. However, advanced ML approaches such as gated recurrent neural networks (RNNs) have seldom been applied to such contexts. Here we demonstrate that a special class of RNNs known as long short-term memory (LSTM) networks is capable of learning and accurately predicting the time evolution of one-dimensional (1D) Ising model with simultaneous transverse and parallel magnetic fields, as quantitatively corroborated by relative entropy measurements and magnetization between the predicted and exact state distributions. In this unsupervised learning task, the many-body state evolution was predicted in an autoregressive way from an initial state, without any guidance or knowledge of any Hamiltonian. Our work paves the way for future applications of advanced ML methods in quantum many-body dynamics without relying on the explicit form of the Hamiltonian.