From Architectures to Applications: A Review of Neural Quantum States (2402.09402v3)

Abstract: Due to the exponential growth of the Hilbert space dimension with system size, the simulation of quantum many-body systems has remained a persistent challenge until today. Here, we review a relatively new class of variational states for the simulation of such systems, namely neural quantum states (NQS), which overcome the exponential scaling by compressing the state in terms of the network parameters rather than storing all exponentially many coefficients needed for an exact parameterization of the state. We introduce the commonly used NQS architectures and their various applications for the simulation of ground and excited states, finite temperature and open system states as well as NQS approaches to simulate the dynamics of quantum states. Furthermore, we discuss NQS in the context of quantum state tomography.

• The paper reviews neural network architectures (RBMs, FFNNs, CNNs, RNNs, transformers) that overcome exponential scaling in simulating quantum many-body systems.
• It demonstrates how these methods accurately simulate ground and excited states as well as dynamic processes using techniques like variational Monte Carlo.
• The review outlines future directions for enhancing neural quantum state models through advanced optimization and deep learning strategies.

An Overview of Neural Quantum States: Exploration from Architectures to Applications

The paper "From Architectures to Applications: A Review of Neural Quantum States" provides a comprehensive survey of the rise of neural quantum states (NQS) as a potent computational framework for quantum many-body problem simulations. The challenge of simulating quantum many-body systems primarily arises from the exponential growth in the dimension of the Hilbert space with increasing system size. This paper contextualizes the significant promise that neural networks offer in this arena by efficiently overcoming the problem of exponential scaling through the compression of state information into network parameters.

Key Architectural Approaches

The discourse begins by categorizing various NQS architectures and their deployment across different quantum systems. A key class of architectures utilized involves restricted Boltzmann machines (RBMs), feedforward neural networks (FFNNs), convolutional neural networks (CNNs), recurrent neural networks (RNNs), and transformer-based models. Each architecture offers unique advantages:

1. RBMs have been prominent given their historical context in simulating spin systems and capturing volume-law entangled states.
2. FFNNs and CNNs leverage locality and translational invariance, making them particularly suitable for 2D lattices and complex many-body interactions.
3. RNNs and transformers, with their autoregressive and attention mechanisms, respectively, provide efficient means for state generation and resolution of correlations over extensive domains, making them suitable for more intricate and higher-dimensional simulations.

Significantly, the choice of architecture can greatly influence the representations' expressivity, and the manuscript details why particular architectures are picked for certain physical applications, valuing their implicit capacity to encode symmetries and correlations innate to quantum systems.

Simulation of Different Quantum States and Dynamics

The paper proceeds to discuss the applications of these architectural frameworks in the simulation of various quantum phenomena. This includes:

• Ground and Excited State Simulations: Methodologies for determining ground-state energies using variational Monte Carlo with neural network approximations are elaborated. The flexibility of NQS in representing excited states, potentially through orthogonality with lower-lying states and other advanced techniques, is highlighted.
• Dynamics of Quantum Systems: The paper outlines the application of time-dependent variational principle (TDVP)/stochastic reconfiguration methods for simulating out-of-equilibrium dynamics using NQS. Challenges such as non-convex optimization landscapes and the scaling required for long-time dynamic simulations are discussed, citing potential workarounds in recent studies, including the use of p-tVMC methods.
• Finite Temperature and Open Systems: The text also explores purification methods and the METTS framework for thermal state simulations, emphasizing the ability to handle mixed states and finite temperature effects in quantum systems.

Implications and Future Directions

The paper underscores the implications of these advancements in both practical and theoretical aspects of quantum computing and many-body physics. In practical terms, NQS offers a scalable alternative to conventional tensor network approaches, pushing the boundaries in terms of accessible system sizes and computational resources. Theoretically, NQS challenges our understanding of state representability and locality by introducing machine-learned ansatzes that encapsulate broader entanglement features beyond what many-body physicists traditionally exploit.

Future research directions hinted at in the paper involve refining the expressivity and trainability of NQS models. Enhancing architectures to tap fully into the quantum many-body landscape's complexity and overcoming the current hurdles in optimization landscapes are targeted areas. The potential to integrate advancements in deep learning, such as transfer learning and meta-learning, to further enhance the capability of NQS in dynamic and large-scale simulations is particularly exciting.

In conclusion, the paper serves as a robust review, charting the landscape of neural quantum states from their conceptual underpinning and architectural development to their applications in quantum many-body physics. With the ongoing rapid developments in AI and computational capabilities, NQS is poised to become an indispensable part of the quantum physicists' toolkit, promising advancements in the understanding and simulation of complex quantum systems.