Recently developed tools such as neural-network based generative models and incremental tensor networks are promising methods for simplifying the learning of functions in high-dimensional space. We consider the application of these tools in two particular areas: modeling quantum states and imitation learning from high-dimensional data. In the first application, we consider the use of generative models such as normalizing flows as neural quantum states. Because quantum states are functions ψ such that |ψ|2 is a probability density, using normalizing flows as neural quantum states is a natural choice. Neural quantum states are commonly used in Variational Monte Carlo (VMC), where the parameters θ of a parameterized quantum state ψθ are varied in order to minimize some objective function. Due to the ability to exactly sample from and estimate probability densities from normalizing flows, the use of normalizing flows as neural quantum states simplifies the calculation of integrals used in Variational Monte Carlo. This is applied in two settings: first we consider using VMC for minimizing the ground state energy of high-dimensional continuous quantum states. Second, we explore using VMC for numerically solving time-dependent partial differential equations. For the second application, we explore using generative models and incremental tensor networks for extracting a low-dimensional latent space for use in downstream tasks like imitation learning. We first explore the use of neural-network architectures like Variational Autoencoders for learning a latent space that can be sampled from. Doing this allows for synthetic states to be generated, as well as predicted actions for those states. However, the use of these neural network-based feature extractors can be expensive, due to hyperparameter tuning and the gradient-based optimization methods used to find a good latent space representation. We explore the use of incremental tensor network approaches, which can extract a latent space in one pass through the training data and requires minimal hyperparameter tuning. We find that these incremental tensor network approaches learn a latent space representation that leads to similar imitation learning accuracy, but provides significant computational speedups over the neural network approaches.
Recently developed tools such as neural-network based generative models and incremental tensor networks are promising methods for simplifying the learning of functions in high-dimensional space. We consider the application of these tools in two particular areas: modeling quantum states and imitation learning from high-dimensional data. In the first application, we consider the use of generative models such as normalizing flows as neural quantum states. Because quantum states are functions ψ such that |ψ|2 is a probability density, using normalizing flows as neural quantum states is a natural choice. Neural quantum states are commonly used in Variational Monte Carlo (VMC), where the parameters θ of a parameterized quantum state ψθ are varied in order to minimize some objective function. Due to the ability to exactly sample from and estimate probability densities from normalizing flows, the use of normalizing flows as neural quantum states simplifies the calculation of integrals used in Variational Monte Carlo. This is applied in two settings: first we consider using VMC for minimizing the ground state energy of high-dimensional continuous quantum states. Second, we explore using VMC for numerically solving time-dependent partial differential equations. For the second application, we explore using generative models and incremental tensor networks for extracting a low-dimensional latent space for use in downstream tasks like imitation learning. We first explore the use of neural-network architectures like Variational Autoencoders for learning a latent space that can be sampled from. Doing this allows for synthetic states to be generated, as well as predicted actions for those states. However, the use of these neural network-based feature extractors can be expensive, due to hyperparameter tuning and the gradient-based optimization methods used to find a good latent space representation. We explore the use of incremental tensor network approaches, which can extract a latent space in one pass through the training data and requires minimal hyperparameter tuning. We find that these incremental tensor network approaches learn a latent space representation that leads to similar imitation learning accuracy, but provides significant computational speedups over the neural network approaches.
Studying the dynamics of open quantum systems can enable breakthroughs both in fundamental physics and applications to quantum engineering and quantum computation. Since the density matrix ρ, which is the fundamental description for the dynamics of such systems, is high-dimensional, customized deep generative neural networks have been instrumental in modeling ρ. However, the complex-valued nature and normalization constraints of ρ, as well as its complicated dynamics, prohibit a seamless connection between open quantum systems and the recent advances in deep generative modeling. Here we lift that limitation by utilizing a reformulation of open quantum system dynamics to a partial differential equation (PDE) for a corresponding probability distribution Q, the Husimi Q function. Thus, we model the Q function seamlessly with off-the-shelf deep generative models such as normalizing flows. Additionally, we develop novel methods for learning normalizing flow evolution governed by high-dimensional PDEs based on the Euler method and the application of the time-dependent variational principle. We name the resulting approach Q-Flow and demonstrate the scalability and efficiency of Q-Flow on open quantum system simulations, including the dissipative harmonic oscillator and the dissipative bosonic model. Q-Flow is superior to conventional PDE solvers and state-of-the-art physics-informed neural network solvers, especially in high-dimensional systems.
Artificial neural networks were recently shown to be an efficient representation of highly entangled many-body quantum states. In practical applications, neural-network states inherit numerical schemes used in variational Monte Carlo method, most notably the use of Markov-chain Monte Carlo (MCMC) sampling to estimate quantum expectations. The local stochastic sampling in MCMC caps the potential advantages of neural networks in two ways: (i) Its intrinsic computational cost sets stringent practical limits on the width and depth of the networks, and therefore limits their expressive capacity; (ii) its difficulty in generating precise and uncorrelated samples can result in estimations of observables that are very far from their true value. Inspired by the state-of-the-art generative models used in machine learning, we propose a specialized neural-network architecture that supports efficient and exact sampling, completely circumventing the need for Markov-chain sampling. We demonstrate our approach for two-dimensional interacting spin models, showcasing the ability to obtain accurate results on larger system sizes than those currently accessible to neural-network quantum states.
The representation of a quantum wave function as a neural network quantum state (NQS) provides a powerful variational ansatz for finding the ground states of many-body quantum systems. Nevertheless, due to the complex variational landscape, traditional methods often employ the computation of quantum geometric tensor, consequently complicating optimization techniques. Contributing to efforts aiming to formulate alternative methods, we introduce an approach that bypasses the computation of the metric tensor and instead relies exclusively on first-order gradient descent with Euclidean metric. This allows for the application of larger neural networks and the use of more standard optimization methods from other machine learning domains. Our approach leverages the principle of imaginary time evolution by constructing a target wave function derived from the Schrödinger equation, and then training the neural network to approximate this target. We make this method adaptive and stable by determining the optimal time step and keeping the target fixed until the energy of the NQS decreases. We demonstrate the benefits of our scheme via numerical experiments with 2D J_1-J_2 J 1 − J 2 , Heisenberg model, which showcase enhanced stability and energy accuracy in comparison to direct energy loss minimization. Importantly, our approach displays competitiveness with the well-established density matrix renormalization group method and NQS optimization with stochastic reconfiguration.
Modeling the joint distribution of high-dimensional data is a central task in unsupervised machine learning. In recent years, many interests have been attracted to developing learning models based on tensor networks, which have the advantages of a principle understanding of the expressive power using entanglement properties, and as a bridge connecting classical computation and quantum computation. Despite the great potential, however, existing tensor network models for unsupervised machine learning only work as a proof of principle, as their performance is much worse than the standard models such as restricted Boltzmann machines and neural networks. In this Letter, we present autoregressive matrix product states (AMPS), a tensor network model combining matrix product states from quantum many-body physics and autoregressive modeling from machine learning. Our model enjoys the exact calculation of normalized probability and unbiased sampling. We demonstrate the performance of our model using two applications, generative modeling on synthetic and real-world data, and reinforcement learning in statistical physics. Using extensive numerical experiments, we show that the proposed model significantly outperforms the existing tensor network models and the restricted Boltzmann machines, and is competitive with state-of-the-art neural network models.
Quantum many-body computation is a central challenge across multiple domains of scientific research, and Monte Carlo method has emerged as one of the most effective numerical techniques for investigating large-scale many-body systems. In recent years, the integration of Monte Carlo methods with machine learning has led to the development of neural quantum state approach. Quantum neural networks (QNNs) offer superior expressiveness compared to classical neural networks. In this paper, we propose a novel Quantum Neural Network Quantum State (QNNQS) method, based on the variational Monte Carlo framework. This approach leverages a quantum neural network to represent the many-body quantum state, and optimizes it iteratively through gradient descent to determine the ground-state energy. To assess the performance of this method, we conduct simulations of the one-dimensional transverse-field Ising model, investigating the influence of quantum circuit architecture on circuit expressibility, entangling capability, and computational accuracy. The results demonstrate that the QNNQS method achieves comparable simulation accuracy while utilizing significantly fewer parameters and requiring less training resources than classical network quantum state methods.
Beyond their origin in modeling many-body quantum systems, tensor networks have emerged as a promising class of models for solving machine learning problems, notably in unsupervised generative learning. While possessing many desirable features arising from their quantum-inspired nature, tensor network generative models have previously been largely restricted to binary or categorical data, limiting their utility in real-world modeling problems. We overcome this by introducing a new family of tensor network generative models for continuous data, which are capable of learning from distributions containing continuous random variables. We develop our method in the setting of matrix product states, first deriving a universal expressivity theorem proving the ability of this model family to approximate any reasonably smooth probability density function with arbitrary precision. We then benchmark the performance of this model on several synthetic and real-world datasets, finding that the model learns and generalizes well on distributions of continuous and discrete variables. We develop methods for modeling different data domains, and introduce a trainable compression layer which is found to increase model performance given limited memory or computational resources. Overall, our methods give important theoretical and empirical evidence of the efficacy of quantum-inspired methods for the rapidly growing field of generative learning.