Deep autoregressive models for the efficient variational simulation of many-body quantum systems (1902.04057v3)

Abstract: 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, 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 paper introduces Neural Autoregressive Quantum States (NAQS) which decompose wave-functions into conditional probabilities for exact and efficient sampling.
• The paper demonstrates that NAQS outperforms traditional VMC and tensor-network methods by accurately simulating two-dimensional Ising and Heisenberg models.
• The paper paves the way for scalable deep learning architectures in quantum simulations, expanding feasibility for larger and more complex quantum systems.

Deep Autoregressive Models for Many-Body Quantum Systems

The paper "Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems" introduces a novel approach using artificial neural networks to simulate quantum many-body systems by circumventing traditional limitations in variational methods. This research focuses on utilizing deep autoregressive models, an advancement inspired by the generative models prevalent in machine learning, to represent quantum states more efficiently.

Background

Quantum many-body systems have long posed significant challenges due to the complexity in representing their highly-entangled states accurately. Traditional methods like the Variational Monte Carlo (VMC) leveraging Jastrow wave-functions have limitations due to constrained variational freedom and limited entanglement capacity. Similarly, non-stochastic tensor-network approaches, despite improved variational capacities, also encounter numerical scalability issues.

Recent progress has shown that artificial neural networks, including restricted Boltzmann machines (RBMs) and convolutional neural networks (ConvNets), can encapsulate the entangled properties of many-body quantum states effectively. Specifically, ConvNets offer a superior theoretical framework for capturing complex entanglement. However, previous applications have been limited by the computational demands associated with deep architectures and the reliance on Markov Chain Monte Carlo (MCMC) sampling, which introduces correlation and computational overhead.

Proposed Approach

The authors propose a paradigm shift with the introduction of Neural Autoregressive Quantum States (NAQS), which rely on autoregressive models to avoid the downsides of MCMC sampling altogether. NAQS decomposes the wave-function into a product of conditional probabilities, facilitating efficient and exact sampling. This autoregressive property allows quantum states to be represented as feed-forward neural networks where sampling each quantum state is achievable in polynomial complexity relative to the network's depth.

This method draws from autoregressive density estimation techniques such as Neural Autoregressive Density Estimators (NADE), extending their application into the quantum field. This development enables the exact generation of independent and identically distributed samples from highly entangled wave functions. The separation from MCMC-based VMC allows for more expressive and scalable neural architectures, thus pushing the boundaries of sizes and complexities previously accessible in simulation tasks.

Experiments and Results

The implementation provides empirical validation through two-dimensional transverse-field Ising models and antiferromagnetic Heisenberg models. The experiments demonstrate that NAQS not only matches but often surpasses the precision of both Quantum Monte Carlo (QMC) simulations and established variational methods like Projected Entangled Pair States (PEPS).

One of the notable outcomes highlighted in the ferromagnetic phase of the Ising model is the NAQS's ability to navigate broken symmetry landscapes without suffering from ergodicity issues that challenge MCMC methods. This is significant as the direct sampling method avoids biases and improves convergence, producing estimates of ground-state properties with lower variance and higher accuracy.

Implications and Future Directions

The implications of this research extend the feasibility of using deep learning architectures for quantum simulations beyond the reach of traditional shallow neural-network approaches. By optimizing neural-network quantum states with efficient sampling architectures, the research paves the way for exploring larger quantum systems and potentially decoding intricate quantum phenomena inherent to condensed matter physics.

Future work may investigate further optimizations in network design specific to quantum state representation, refining autoregressive model architecture to suit diverse quantum systems. Emerging AI techniques could also lead to exploration in higher-dimensional systems or non-traditional quantum frameworks, expanding theoretical and practical applications in quantum mechanics, quantum information, and beyond.