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arXiv:2104.11011

Learning Neural Network Quantum States with the Linear Method

J. Thorben Frank, Michael J. Kastoryano

Published Apr 22, 2021• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2104.11011

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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.
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 Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with 3 × 105parameters, achieving state-of-the-art ground-state energy in theJ1–J2Heisenberg model atJ2/J1 = 0.5 on the 10 × 10 square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for neural-network quantum states, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle.
Training deep neural networks remains a computationally intensive and often challenging process. Traditional optimization methods, such as stochastic gradient descent (SGD) and its variants, frequently struggle to escape local minima and converge to optimal solutions, leading to suboptimal network performance. This paper proposes a novel approach to neural network training that leverages principles inspired by quantum mechanics. Specifically, we investigate the application of quantum-inspired optimization techniques, including simulated annealing and quantum walks, to dynamically adjust the training process. The core idea is to mimic the probabilistic exploration inherent in quantum systems to effectively navigate the complex loss landscape of neural networks. We formulate the training process as a quantum-inspired optimization problem, where the network's weights are treated as quantum bits (Qubits) and the optimization algorithm seeks to find the lowest energy state, analogous to finding the minimum loss. The proposed method demonstrates the potential for accelerating convergence and improving the overall accuracy of neural networks compared to conventional optimization strategies. Our analysis focuses on the key aspects of the approach, including the mathematical formulation of the quantum-inspired optimization problem and the selection of appropriate parameters to govern the exploration and exploitation phases. We present a theoretical framework for understanding the dynamics of this approach and discuss potential avenues for further research, including hybrid quantum-classical algorithms and the integration of quantum computing hardware.
Recently, artificial intelligence for science has made significant inroads into various fields of natural science research. In the field of quantum many-body computation, researchers have developed numerous ground state solvers based on neural-network quantum states (NQSs), achieving ground state energies with accuracy comparable to or surpassing traditional methods such as variational Monte Carlo methods, density matrix renormalization group, and quantum Monte Carlo methods. Here, we combine supervised learning, variational Monte Carlo (VMC), and the Lanczos method to develop a systematic approach to improving the NQSs of many-body systems, which we refer to as the NQS Lanczos method. The algorithm mainly consists of two parts: the supervised learning part and the VMC optimization part. Through supervised learning, the Lanczos states are represented by the NQSs. Through VMC, the NQSs are further optimized. We analyze the reasons for the underfitting problem and demonstrate how the NQS Lanczos method systematically improves the energy in the highly frustrated regime of the two-dimensional Heisenberg J1J_1J1​-J2J_2J2​ model. Compared to the existing method that combines the Lanczos method with the restricted Boltzmann machine, the primary advantage of the NQS Lanczos method is its linearly increasing computational cost.
Training large-scale neural networks with traditional stochastic gradient descent (SGD) methods often faces significant challenges, including slow convergence rates and high computational costs. This paper proposes a novel approach, Quantum-Inspired Stochastic Gradient Descent (QISGD), designed to address these limitations. QISGD leverages principles inspired by quantum measurement processes, specifically employing a random projection operator to approximate the gradient. This technique aims to provide a more efficient and potentially faster way to update the network's parameters during training. The theoretical underpinnings of QISGD are presented, along with a detailed explanation of the algorithm's components. We demonstrate the potential benefits of QISGD through a conceptual analysis and outline a framework for future research and implementation. The core innovation lies in translating quantum measurement intuition into a practical optimization strategy for classical neural networks. The goal is to improve the training efficiency of large networks while retaining the benefits of SGD.
The rapid growth of machine learning (ML) has been accompanied by a significant challenge: the scaling of optimization algorithms to handle the immense parameter spaces of modern models, such as deep neural networks. Traditional optimization methods, including gradient descent and its variants, often encounter difficulties in efficiently exploring these spaces, leading to slow training times and potential stagnation. This paper investigates the application of quantum-inspired optimization techniques to address this scaling problem. We explore algorithms inspired by quantum annealing and other quantum phenomena, adapting their core principles to classical computation. Specifically, we propose a novel optimization framework utilizing a stochastic search strategy based on quantum superposition and measurement, designed to navigate high-dimensional parameter spaces more effectively. The efficacy of this approach is evaluated through theoretical analysis and conceptual simulations, demonstrating its potential to accelerate the training of large-scale machine learning models. We demonstrate how these methods can improve convergence speed and potentially escape local minima more effectively than conventional techniques. The core contribution lies in providing a theoretical foundation and a practical framework for applying quantum-inspired concepts to optimization within the context of machine learning.