Neural quantum states have emerged as a widely used approach to the numerical study of the ground states of nonstoquastic Hamiltonians. However, existing approaches often rely on a priori knowledge of the sign structure or require a separately pretrained phase network. We introduce a modified stochastic reconfiguration method that effectively uses differing imaginary time steps to evolve the amplitude and phase. Using a larger time step for phase optimization, this method enables a simultaneous and efficient training of phase and amplitude neural networks. The efficacy of our method is demonstrated on the Heisenberg ${J}_{1}\ensuremath{-}{J}_{2}$ model.
Neural quantum states have emerged as a widely used approach to the numerical study of the ground states of nonstoquastic Hamiltonians. However, existing approaches often rely on a priori knowledge of the sign structure or require a separately pretrained phase network. We introduce a modified stochastic reconfiguration method that effectively uses differing imaginary time steps to evolve the amplitude and phase. Using a larger time step for phase optimization, this method enables a simultaneous and efficient training of phase and amplitude neural networks. The efficacy of our method is demonstrated on the Heisenberg J1\ensuremath−J2 model.
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.
Abstract Computing the ground state of interacting quantum matter is a long-standing challenge, especially for complex two-dimensional systems. Recent developments have highlighted the potential of neural quantum states to solve the quantum many-body problem by encoding the many-body wavefunction into artificial neural networks. However, this method has faced the critical limitation that existing optimization algorithms are not suitable for training modern large-scale deep network architectures. Here, we introduce a minimum-step stochastic-reconfiguration optimization algorithm, which allows us to train deep neural quantum states with up to 10 6 parameters. We demonstrate our method for paradigmatic frustrated spin-1/2 models on square and triangular lattices, for which our trained deep networks approach machine precision and yield improved variational energies compared to existing results. Equipped with our optimization algorithm, we find numerical evidence for gapless quantum-spin-liquid phases in the considered models, an open question to date. We present a method that captures the emergent complexity in quantum many-body problems through the expressive power of large-scale artificial neural networks.
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.
Due to the strong correlations present in quantum systems, classical machine learning algorithms like stochastic gradient descent are often insufficient for the training of neural network quantum states (NQSs). These difficulties can be overcome by using physically inspired learning algorithm, the most prominent of which is the stochastic reconfiguration (SR) which mimics imaginary time evolution. Here we explore an alternative algorithms for the optimization of complex valued NQSs based on the linear method (LM), and present the explicit formulation in terms of complex valued parameters. Beyond the theoretical formulation, we present numerical evidence that the LM can be used successfully for the optimization of complex valued NQSs, to our knowledge for the first time. We compare the LM to the state-of-the-art SR algorithm and find that the LM requires up to an order of magnitude fewer iterations for convergence, albeit at a higher cost per epoch. We further demonstrate that the LM becomes the more efficient training algorithm whenever the cost of sampling is high. This advantage, however, comes at the price of a larger variance.
Variational quantum algorithms hold great promise for unlocking the power of near-term quantum processors, yet high measurement costs, barren plateaus, and challenging optimization landscapes frequently hinder them. Here, we introduce sVQNHE, a neural-guided variational quantum algorithm that decouples amplitude and sign learning across classical and quantum modules, respectively. Our approach employs shallow quantum circuits composed of commuting diagonal gates to efficiently model quantum phase information, while a classical neural network learns the amplitude distribution and guides circuit optimization in a bidirectional feedback loop. This hybrid quantum-classical synergy not only reduces measurement costs but also achieves high expressivity with limited quantum resources and improves the convergence rate of the variational optimization. We demonstrate the advancements brought by sVQNHE through extensive numerical experiments. For the 6-qubit J1-J2 model, a prototypical system with a severe sign problem for Monte Carlo-based methods, it reduces the mean absolute error by 98.9% and suppresses variance by 99.6% relative to a baseline neural network, while requiring nearly 19x fewer optimization steps than a standard hardware-efficient VQE. Furthermore, for MaxCut problems on 45-vertex Erdos-Renyi graphs, sVQNHE improves solution quality by 19% and quantum resource efficiency by 85%. Importantly, this framework is designed to be scalable and robust against hardware noise and finite-sampling uncertainty, making it well-suited for both current NISQ processors and future high-quality quantum computers. Our results highlight a promising path forward for efficiently tackling complex many-body and combinatorial optimization problems by fully exploiting the synergy between classical and quantum resources in the NISQ era and beyond.
Recent progress in the design and optimization of neural-network quantum states (NQSs) has made them an effective method to investigate ground-state properties of quantum many-body systems. In contrast to the standard approach of training a separate NQS from scratch at every point of the phase diagram, we demonstrate that the optimization of a NQS at a highly expressive point of the phase diagram (i.e., close to a phase transition) yields features that can be reused to accurately describe a wide region across the transition. We demonstrate the feasibility of our approach on different systems in one and two dimensions by initially pretraining a NQS at a given point of the phase diagram, followed by fine-tuning only the output layer for all other points. Notably, the computational cost of the fine-tuning step is very low compared to the pretraining stage. We argue that the reduced cost of this paradigm has significant potential to advance the exploration of strongly correlated systems using NQS, mirroring the success of fine-tuning in machine learning and natural language processing. Published by the American Physical Society 2024