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arXiv:2010.14514 · Physical Review A

U(1)-symmetric recurrent neural networks for quantum state reconstruction

Stewart Morawetz, Isaac J. S. De Vlugt, Juan Carrasquilla, Roger G. Melko

Published Jul 1, 2021• 31 citations
arXivHTMLPDF

Abstract

Generative models are a promising technology for the enhancement of quantum simulators. These machine learning methods are capable of reconstructing a quantum state from experimental measurements, and can aid in the calculation of physical observables. In this paper, we employ a recurrent neural network (RNN) to reconstruct the ground state of the spin-1/2 $XY$ model, a prototypical Hamiltonian explored in trapped ion simulators. We explore its performance after enforcing a U(1) symmetry, which was recently shown by Hibat-Allah et al. [Phys. Rev. Res. 2, 023358 (2020)] to preserve the autoregressive nature of the RNN. By studying the reconstruction of the $XY$ model ground state from projective measurement data, we show that imposing U(1) symmetry on the RNN significantly increases the efficiency of learning, particularly in the early epoch regime. We argue that this performance increase may result from the tendency of the enforced symmetry to alleviate vanishing and exploding gradients, which helps stabilize the training process. Thus, symmetry-enforced RNNs may be particularly useful for applications of quantum simulators where a rapid feedback between optimization and circuit preparation is necessary, such as in hybrid classical-quantum algorithms.

DOI
10.1103/physreva.104.012401

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Generative models are a promising technology for the enhancement of quantum simulators. These machine learning methods are capable of reconstructing a quantum state from experimental measurements, and can aid in the calculation of physical observables. In this paper, we employ a recurrent neural network (RNN) to reconstruct the ground state of the spin-1/2 XYXYXY model, a prototypical Hamiltonian explored in trapped ion simulators. We explore its performance after enforcing a U(1) symmetry, which was recently shown by Hibat-Allah et al. [Phys. Rev. Res. 2, 023358 (2020)] to preserve the autoregressive nature of the RNN. By studying the reconstruction of the XYXYXY model ground state from projective measurement data, we show that imposing U(1) symmetry on the RNN significantly increases the efficiency of learning, particularly in the early epoch regime. We argue that this performance increase may result from the tendency of the enforced symmetry to alleviate vanishing and exploding gradients, which helps stabilize the training process. Thus, symmetry-enforced RNNs may be particularly useful for applications of quantum simulators where a rapid feedback between optimization and circuit preparation is necessary, such as in hybrid classical-quantum algorithms.
Abstract Neural-network quantum states (NQS) are powerful neural-network ansätzes that have emerged as promising tools for studying quantum many-body physics through the lens of the variational principle. These architectures are known to be systematically improvable by increasing the number of parameters. Here we demonstrate an Adaptive scheme to optimize NQSs, through the example of recurrent neural networks (RNNs), using a fraction of the computation cost while reducing training fluctuations and improving the quality of variational calculations targeting ground states of prototypical models in one- and two-spatial dimensions. This Adaptive framework reduces the computational cost through training small RNNs and reusing them to initialize larger RNNs. This work opens up the possibility for optimizing graphical processing unit resources deployed in large-scale NQS simulations.
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A general recurrent neural network (RNN) model is proposed for reconstructing wave functions with varying parameters. This model is suitable for both real and complex ground-state wave functions in condensed matter physics models. We propose three straightforward techniques for integrating variable parameters, enabling the reconstruction of an entire quantum state family within a unified RNN model through numerical augmentation and concatenation. This innovation significantly broadens the model's generalization capabilities. The core unit for wave function reconstruction in the RNN is the gated recurrent unit (GRU), and the RNN's autoregressive property is utilized to generate independent training samples. The training of the network employs a variational Monte Carlo (VMC) optimization strategy. The proposed RNN is applied to two classical condensed matter physics models: the one-dimensional (1D) transverse field Ising model (TFIM) and the 1D J1-J2 model, and it successfully reconstructs arbitrary wave functions. In numerical experiments conducted on a 4-qubit system, using exact diagonalization (ED) as a reference method and relative energy error as a performance indicator, our real- and complex-valued general RNNs demonstrate remarkable reconstruction and generalization performance of wave functions under different physical parameters. To further refine accuracy and reduce training time, we fine-tune the designed network using target samples from a few specific tasks. We present an efficient and accurate general generation model for reconstructing arbitrary wave functions to better accommodate the diversity and variability encountered in practical applications.
Recurrent neural networks are the foundation of many sequence-to-sequence models in machine learning, such as machine translation and speech synthesis. In contrast, applied quantum computing is in its infancy. Nevertheless there already exist quantum machine learning models such as variational quantum eigensolvers which have been used successfully e.g. in the context of energy minimization tasks. In this work we construct a quantum recurrent neural network (QRNN) with demonstrable performance on non-trivial tasks such as sequence learning and integer digit classification. The QRNN cell is built from parametrized quantum neurons, which, in conjunction with amplitude amplification, create a nonlinear activation of polynomials of its inputs and cell state, and allow the extraction of a probability distribution over predicted classes at each step. To study the model's performance, we provide an implementation in pytorch, which allows the relatively efficient optimization of parametrized quantum circuits with thousands of parameters. We establish a QRNN training setup by benchmarking optimization hyperparameters, and analyse suitable network topologies for simple memorisation and sequence prediction tasks from Elman's seminal paper (1990) on temporal structure learning. We then proceed to evaluate the QRNN on MNIST classification, both by feeding the QRNN each image pixel-by-pixel; and by utilising modern data augmentation as preprocessing step. Finally, we analyse to what extent the unitary nature of the network counteracts the vanishing gradient problem that plagues many existing quantum classifiers and classical RNNs.
Symmetries such as gauge invariance and anyonic symmetry play a crucial role in quantum many-body physics.We develop a general approach to constructing gauge-invariant or anyonic-symmetric autoregressive neural networks, including a wide range of architectures such as transformer and recurrent neural network, for quantum lattice models.These networks can be efficiently sampled and explicitly obey gauge symmetries or anyonic constraint.We prove that our methods can provide exact representation for the ground and excited states of the two-and three-dimensional toric codes, and the X-cube fracton model.We variationally optimize our symmetry-incorporated autoregressive neural networks for ground states as well as real-time dynamics for a variety of models.We simulate the dynamics and the ground states of the quantum link model of U(1) lattice gauge theory, obtain the phase diagram for the two-dimensional Z 2 gauge theory, determine the phase transition and the central charge of the SU(2) 3 anyonic chain, and also compute the ground-state energy of the SU(2) invariant Heisenberg spin chain.Our approach provides powerful tools for exploring condensed-matter physics, high-energy physics, and quantum information science.
Recent developments in quantum computing and machine learning have propelled the interdisciplinary study of quantum machine learning. Sequential modeling is an important task with high scientific and commercial value. Existing VQC or QNN-based methods require significant computational resources to perform the gradient-based optimization of a larger number of quantum circuit parameters. The major drawback is that such quantum gradient calculation requires a large amount of circuit evaluation, posing challenges in current near-term quantum hardware and simulation software. In this work, we approach sequential modeling by applying a reservoir computing (RC) framework to quantum recurrent neural networks (QRNN-RC) that are based on classical RNN, LSTM and GRU. The main idea to this RC approach is that the QRNN with randomly initialized weights is treated as a dynamical system and only the final classical linear layer is trained. Our numerical simulations show that the QRNN-RC can reach results comparable to fully trained QRNN models for several function approximation and time series prediction tasks. Since the QRNN training complexity is significantly reduced, the proposed model trains notably faster. In this work we also compare to corresponding classical RNN-based RC implementations and show that the quantum version learns faster by requiring fewer training epochs in most cases. Our results demonstrate a new possibility to utilize quantum neural network for sequential modeling with greater quantum hardware efficiency, an important design consideration for noisy intermediate-scale quantum (NISQ) computers.