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arXiv:2301.06788 · Physical review. B./Physical review. B

Optimizing design choices for neural quantum states

Moritz Reh, Markus Schmitt, Martin Gärttner

Published May 9, 2023• 46 citations
arXivHTMLPDF

Abstract

Neural quantum states are a new family of variational Ans\"atze for quantum-many body wave functions with advantageous properties in the notoriously challenging case of two spatial dimensions. Since their introduction, a wide variety of different network architectures have been employed to study paradigmatic models in quantum many-body physics with a particular focus on quantum spin models. Nonetheless, many questions remain about the effect that the choice of architecture has on the performance on a given task. In this work, we present a unified comparison of a selection of popular network architectures and symmetrization schemes employed for ground-state searches of prototypical spin Hamiltonians, namely, the two-dimensional transverse-field Ising model and the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. In the presence of a nontrivial sign structure of the ground states, we find that the details of symmetrization crucially influence the performance. We describe this effect in detail and discuss its consequences, especially for autoregressive models, as their direct sampling procedure is not compatible with the symmetrization procedure that we found to be optimal.

DOI
10.1103/physrevb.107.195115

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Neural quantum states are a new family of variational Ans"atze for quantum-many body wave functions with advantageous properties in the notoriously challenging case of two spatial dimensions. Since their introduction, a wide variety of different network architectures have been employed to study paradigmatic models in quantum many-body physics with a particular focus on quantum spin models. Nonetheless, many questions remain about the effect that the choice of architecture has on the performance on a given task. In this work, we present a unified comparison of a selection of popular network architectures and symmetrization schemes employed for ground-state searches of prototypical spin Hamiltonians, namely, the two-dimensional transverse-field Ising model and the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ model. In the presence of a nontrivial sign structure of the ground states, we find that the details of symmetrization crucially influence the performance. We describe this effect in detail and discuss its consequences, especially for autoregressive models, as their direct sampling procedure is not compatible with the symmetrization procedure that we found to be optimal.
Neural Quantum States (NQS) are powerful tools used to represent complex quantum many-body states in an increasingly wide range of applications. However, despite their popularity, at present only a rudimentary understanding of their limitations exists. In this work, we investigate the dependence of NQS on the choice of the computational basis, focusing on restricted Boltzmann machines. Considering a family of rotated Hamiltonians corresponding to the paradigmatic transverse-field Ising model, we discuss the properties of ground states responsible for the dependence of NQS performance, namely the presence of ground state degeneracies as well as the uniformity of amplitudes and phases, carefully examining their interplay. We identify that the basis-dependence of the performance is linked to the convergence properties of a cluster or cumulant expansion of multi-spin operators -- providing a framework to directly connect physical, basis-dependent properties, to performance itself. Our results provide insights that may be used to gauge the applicability of NQS to new problems and to identify the optimal basis for numerical computations.
Materials design is an inverse problem: finding the couplings of a microscopic Hamiltonian whose ground state exhibits a desired property. For strongly correlated quantum systems, this remains largely out of reach, as each candidate Hamiltonian requires its own costly simulation. Here we show that Foundation Neural-Network Quantum States, single wave functions that encode the ground states of an entire family of Hamiltonians, make this search efficient. Because the Hamiltonian couplings enter as neural-network inputs, any ground-state property can be optimized by gradient descent directly in coupling space. We use this approach to search for nonmagnetic phases in two-dimensional spin models in coupling spaces of increasing dimensionality. Specifically, we focus on quantum spin liquids, exotic phases of matter in which quantum fluctuations prevent magnetic order at zero temperature. In the largest search space considered, spanned by eight superexchange couplings and containing many paradigmatic frustrated spin systems, the optimization converges to the J1J_1J1​-J2J_2J2​-δδδ Heisenberg model when starting from a magnetic phase. We show that the ground state at the optimum set of couplings has neither magnetic nor valence-bond order in the thermodynamic limit, leading to a new quantum spin-liquid candidate. The approach is completely general and can be extended to other models and physical properties to optimize, demonstrating that Neural-Network Quantum States are tools not only for simulating quantum matter but also for designing it.
The exploration of neural network quantum states has become widespread in studies of complicated quantum many-body systems. However, achieving high precision remains challenging due to the exponential growth of Hilbert space size and the intricate sign structures. Utilizing symmetries of the physical system, we propose a method to evaluate and sample the variational Ansatz within a symmetric subspace. This approach isolates different symmetry sectors, reducing the relevant Hilbert space size by a factor approximately proportional to the size of the symmetry group. It is inspired by exact diagonalization techniques and the work of Choo et al., [Phys. Rev. Lett. 121, 167204 (2018)]. We validate our method using the frustrated spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg chain, and we compare its performance to the case without symmetrization. The results indicate that our symmetric subspace approach achieves a substantial improvement over the full Hilbert space on optimizing the Ansatz, reducing the energy error by orders of magnitude. We also compare the results on degenerate eigenstates with different quantum numbers, highlighting the advantage of operating within a smaller Hilbert subspace.
The use of artificial neural networks to represent quantum wave functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parametrizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper, we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ans\"atz, we study the frustrated spin-1/$2{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ans\"atz exhibits comparatively good but not the best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep-learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.
Neural-network quantum states (NQSs), variationally optimized by combining traditional methods and deep learning techniques, is a new way to find quantum many-body ground states and has gradually become a competitor of traditional variational methods. However, there are still some difficulties in the optimization of NQSs, such as local minima, slow convergence, and sign structure optimization. Here, we split a quantum many-body variational wave function into a multiplication of a real-valued amplitude neural network and a sign structure, and focus on the optimization of the amplitude network while keeping the sign structure fixed. The amplitude network is a convolutional neural network (CNN) with residual blocks, namely a residual network (ResNet). Our method is tested on three typical quantum many-body systems. The obtained ground state energies are better than or comparable to those from traditional variational Monte Carlo methods and density matrix renormalization group. Surprisingly, for the frustrated Heisenberg ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, our results are better than those of the complex-valued CNN in the literature, implying that the sign structure of the complex-valued NQS is difficult to optimize. We will study the optimization of the sign structure of NQSs in the future.
Neural quantum states have established themselves as a powerful and versatile family of ansatzes for variational Monte Carlo simulations of quantum many-body systems. Of particular prominence are autoregressive neural quantum states (ANQS), which enjoy the expressibility of deep neural networks, and are equipped with a procedure for fast and unbiased sampling. Yet, the non-selective nature of autoregressive sampling makes incorporating quantum number symmetries challenging. In this work, we develop a general framework to make the autoregressive sampling compliant with an arbitrary number of quantum number symmetries. We showcase its advantages by running electronic structure calculations for a range of molecules with multiple symmetries of this kind. We reach the level of accuracy reported in previous works with more than an order of magnitude speedup and achieve chemical accuracy for all studied molecules, which is a milestone unreported so far. Combined with the existing effort to incorporate space symmetries, our approach expands the symmetry toolbox essential for any variational ansatz and brings the ANQS closer to being a competitive choice for studying challenging quantum many-body systems.