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arXiv:1905.09168 · Chinese Physics Letters

Predicting Quantum Many-Body Dynamics with Transferable Neural Networks*

Ze-Wang Zhang, Shuo Yang, Yi-Hang Wu, Chen-Xi Liu, Yi-Min Han, Ching-Hua Lee, Zheng Sun, Guang-Jie Li, Xiao Zhang

Published Jan 1, 2020• 7 citations
arXivHTMLPDF

Abstract

Advanced machine learning (ML) approaches such as transfer learning have seldom been applied to approximate quantum many-body systems. Here we demonstrate that a simple recurrent unit (SRU) based efficient and transferable sequence learning framework is capable of learning and accurately predicting the time evolution of the one-dimensional (1D) Ising model with simultaneous transverse and parallel magnetic fields, as quantitatively corroborated by relative entropy measurements between the predicted and exact state distributions. At a cost of constant computational complexity, a larger many-body state evolution is predicted in an autoregressive way from just one 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 with knowledge only from a smaller system.

DOI
10.1088/0256-307x/37/1/018401

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Advanced machine learning (ML) approaches such as transfer learning have seldom been applied to approximate quantum many-body systems. Here we demonstrate that a simple recurrent unit (SRU) based efficient and transferable sequence learning framework is capable of learning and accurately predicting the time evolution of the one-dimensional (1D) Ising model with simultaneous transverse and parallel magnetic fields, as quantitatively corroborated by relative entropy measurements between the predicted and exact state distributions. At a cost of constant computational complexity, a larger many-body state evolution is predicted in an autoregressive way from just one 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 with knowledge only from a smaller system.
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.
Conventional approaches to simulating quantum many-body dynamics produce a single trajectory: if the Hamiltonian or the initial state is changed, the computation must be re-performed. Recent efforts toward foundation models have begun to address this limitation, yet existing methods transfer across either Hamiltonians or initial states, but not both. In this work, we introduce the Universal Neural Propagator (UNP), a single, unified model that learns the functional mapping from driving protocols to time-evolution propagators. Trained in an entirely self-supervised way, a single UNP model predicts dynamics across a function space of driving protocols and an exponentially large Hilbert space of initial states simultaneously. We benchmark on a two-dimensional driven Ising model and demonstrate the UNP's accuracy and transferability across product and entangled initial states, as well as for both in- and out-of-distribution driving protocols. The UNP remains accurate at system sizes beyond exact diagonalization, and can be efficiently fine-tuned across all initial states using observable data. By shifting the object of learning from quantum states to operators, this work opens a route toward transferable simulation of driven quantum matter.
Neural-network quantum states have shown great potential for the study of many-body quantum systems. In statistical machine learning, transfer learning designates protocols reusing features of a machine learning model trained for a problem to solve a possibly related but different problem. We propose to evaluate the potential of transfer learning to improve the scalability of neural-network quantum states. We devise and present physics-inspired transfer learning protocols, reusing the features of neural-network quantum states learned for the computation of the ground state of a small system for systems of larger sizes. We implement different protocols for restricted Boltzmann machines on general-purpose graphics processing units. This implementation alone yields a speedup over existing implementations on multicore and distributed central processing units in comparable settings. We empirically and comparatively evaluate the efficiency (time) and effectiveness (accuracy) of different transfer learning protocols as we scale the system size in different models and different quantum phases. Namely, we consider both the transverse field Ising and Heisenberg XXZ models in one dimension, as well as in two dimensions for the latter, with system sizes up to 128 and 8×8 spins. We empirically demonstrate that some of the transfer learning protocols that we have devised can be far more effective and efficient than starting from neural-network quantum states with randomly initialized parameters.
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.
A method is presented for discrete real-time learning (DRTL) of the evolution of a many-body quantum state in the presence of time-dependent control fields. The DRTL method is based on the recent adoption of the artificial neural network (ANN) ansatz within the framework of the time-dependent variational Monte Carlo method in learning large-scale quantum many-body dynamics. The method leverages accurate short-time quantum-state propagation with the Schr\"odinger equation and the efficient stochastic quantum natural gradient descent algorithm in machine learning. Importantly, the DRTL method is devoid of high numerical costs resulting from the exponentially large many-body Hilbert space by replacing the full state propagation with short-time propagations of a sequence of partial stochastic samples of the ANN state and may be considered as a synergism between the Schr\"odinger equation and the ANN that streamlines the computation of the many-body controlled quantum dynamics. Specifically, this method minimizes the normed distances between the time-dependent variational ANN ansatz and a set of partial quantum states over a sequence of stochastically sampled subspaces that are connected to each other in small discrete time steps. We demonstrate the high accuracy and stability of the DRTL method for the spin excitation dynamics of $4\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6, 8\ifmmode\times\else\texttimes\fi{}8$, and $10\ifmmode\times\else\texttimes\fi{}10$ two-dimensional (2D) Heisenberg spin-1/2 lattices driven by time-dependent control fields applied to a corner spin along with strong exchange coupling between the nearest-neighbored spins. It is found that the number of ANN parameters for properly following the dynamically occupied subspaces (i.e., starting from the pure tensor-product spin configuration with all spins down) approximately increases quadratically with the number of spins in the system. Importantly, we show that the DRTL method can accurately and efficiently follow the small, yet importantly, occupied parts of the Hilbert space of the 2D spin square lattices during the interaction with the strong time-dependent control fields over a physically significant time period. This finding provides a basis for future work to perform quantum optimal control simulations of 2D spin lattices and other multiparticle systems.