Traditional linear approximation of quantum mechanical wave functions are not practically applicable for systems with more than 3 degrees of freedom due to the “the curse of dimensionality”. Indeed,the number of parameters required to describe a wave function in high-dimensional space grows exponentially with the number of degrees of freedom. Inevitably, strong model assumptions should be used when studying such systems numerically. There are, however, estimates of the complexity of a function reproduced by a deep neural network (DNN) that demonstrate the same exponential growth with respect to the number of the network layers. The number of parameters for DNN grows only linearly with the number of layers. This gives us a hope that application of DNN as an approximant for a wave function in high-dimensional space might moderate the computational requirements for reproducing such systems and make 4- or higher- dimensional systems feasible for direct numerical modeling. We present a study of DNN approximation properties for a multi-dimensional quantum harmonic oscillator. We demonstrate that the computational resources required to reproduce the wave function depend on the dimensionality of the problem and the quantum numbers of the state. Increasing the number of hidden layers in a fully-connected feed-forward DNN we can reproduce some excited states of a multidimensional system with computational resources comparable to low-dimensional cases. Using the DNN as an approximant for a wave function paves a way to developing a new class of computational schemes for solving the Schroedinger equation for high-dimensional systems.
Traditional linear approximation of quantum mechanical wave functions are not practically applicable for systems with more than 3 degrees of freedom due to the “the curse of dimensionality”. Indeed,the number of parameters required to describe a wave function in high-dimensional space grows exponentially with the number of degrees of freedom. Inevitably, strong model assumptions should be used when studying such systems numerically. There are, however, estimates of the complexity of a function reproduced by a deep neural network (DNN) that demonstrate the same exponential growth with respect to the number of the network layers. The number of parameters for DNN grows only linearly with the number of layers. This gives us a hope that application of DNN as an approximant for a wave function in high-dimensional space might moderate the computational requirements for reproducing such systems and make 4- or higher- dimensional systems feasible for direct numerical modeling. We present a study of DNN approximation properties for a multi-dimensional quantum harmonic oscillator. We demonstrate that the computational resources required to reproduce the wave function depend on the dimensionality of the problem and the quantum numbers of the state. Increasing the number of hidden layers in a fully-connected feed-forward DNN we can reproduce some excited states of a multidimensional system with computational resources comparable to low-dimensional cases. Using the DNN as an approximant for a wave function paves a way to developing a new class of computational schemes for solving the Schroedinger equation for high-dimensional systems.
Abstract This paper introduces a novel method for solving the Schrödinger equation through the use of deep neural networks (DNNs), presenting a significant departure from traditional techniques. Conventional approaches to solving the Schrödinger equation, such as analytical methods and numerical algorithms, often face challenges when dealing with complex quantum systems due to their inherent limitations. These traditional methods can become cumbersome or even infeasible as the complexity of the systems increases. In contrast, our approach harnesses the capabilities of deep neural networks to approximate both the wavefunction and the energy eigenvalues of quantum systems. By leveraging the flexible and powerful nature of DNNs, we provide a new pathway to solving the Schrödinger equation that can potentially overcome the constraints of classical methods. To validate the effectiveness of our approach, we apply it to the particle in a box problem – a fundamental quantum mechanics model with well-established analytical solutions. This benchmark problem serves as a useful test case, allowing us to demonstrate that DNNs can not only replicate the known results accurately but also offer insights into how these networks can handle more intricate quantum systems. Our results reveal that DNNs are capable of accurately reproducing the analytical solutions for the particle in a box, illustrating their potential as a versatile tool for quantum mechanics.
This paper introduces a novel deep-learning-based approach for numerical simulation of a time-evolving Schrödinger equation inspired by stochastic mechanics and generative diffusion models. Unlike existing approaches, which exhibit computational complexity that scales exponentially in the problem dimension, our method allows us to adapt to the latent low-dimensional structure of the wave function by sampling from the Markovian diffusion. Depending on the latent dimension, our method may have far lower computational complexity in higher dimensions. Moreover, we propose novel equations for stochastic quantum mechanics, resulting in quadratic computational complexity with respect to the number of dimensions. Numerical simulations verify our theoretical findings and show a significant advantage of our method compared to other deep-learning-based approaches used for quantum mechanics.
In this paper, we demonstrate the expressibility of artificial neural networks (ANNs) in quantum many-body physics by showing that a feed-forward neural network with a small number of hidden layers can be trained to approximate with high precision the ground states of some notable quantum many-body systems. We consider the one-dimensional free bosons and fermions, spinless fermions on a square lattice away from half-filling, as well as frustrated quantum magnetism with a rapidly oscillating ground-state characteristic function. In the latter case, an ANN with a standard architecture fails, while that with a slightly modified one successfully learns the frustration-induced complex sign rule in the ground state and approximates the ground states with high precisions. As an example of practical use of our method, we also perform the variational method to explore the ground state of an antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model.
One of the main reasons why the physics of quantum many-body systems is hard \nlies in the curse of dimensionality: \nThe number of states of such systems increases exponentially with the number of degrees of freedom involved. \nAs a result, computations for realistic systems become intractable, \nand even numerical methods are limited to comparably small system sizes. \nMany efforts in modern physics research are therefore concerned with \nfinding efficient representations of quantum states and \nclever approximations schemes that would allow them to characterize physical systems of interest. \n \nMeanwhile, Deep Learning (DL) has solved many non-scientific problems \nthat have been unaccessible to conventional methods for a similar reason. \nThe concept underlying DL is to extract knowledge from data \nby identifying patterns and regularities. \nThe remarkable success of DL has excited many physicists about \nthe prospect of leveraging its power to solve intractable problems in physics. \nAt the same time, DL turned out to be an interesting complex many-body \nproblem in itself. \nIn contrast to its widespread empirical applications, \nthe theoretical foundation of DL is strongly underdeveloped. \nIn particular, as long as its decision-making process and result interpretability remain opaque, \nDL can not claim the status of a scientific tool. \n \nIn this thesis, I explore the interface between DL and quantum many-body physics, \nand investigate DL both as a tool and as a subject of study. \nThe first project presented here is a theory-based study of \na fundamental open question about the role of width and the number of parameters in deep neural networks. \nIn this work, we consider a DL setup for the image recognition task on standard benchmarking datasets. \nWe combine controlled experiments with a theoretical analysis, including analytical calculations for a toy model. \nThe other three works focus on the application of Restricted Boltzmann Machines as generative models \nfor the task of wavefunction reconstruction from measurement data on a quantum many-body system. \nFirst, we implement this approach as a software package, making it available as a tool for experimentalists. \nFollowing the idea that physics problems can be used to characterize DL tools, \nwe then use our extensive knowledge of this setup to conduct a systematic study of \nhow the RBM complexity scales with the complexity of the physical system. \nFinally, in a follow-up study we focus on the effects of parameter pruning techniques \non the RBM and its scaling behavior.
Abstract Deep neural networks (DNNs) and auto differentiation have been widely used in computational physics to solve variational problems. When a DNN is used to represent the wave function and solve quantum many-body problems using variational optimization, various physical constraints have to be injected into the neural network by construction to increase the data and learning efficiency. We build the unitary constraint to the variational wave function using a monotonic neural network to represent the cumulative distribution function (CDF) . Using this constrained neural network to represent the variational wave function, we solve Schrodinger equations using auto-differentiation and stochastic gradient descent (SGD) by minimizing the violation of the trial wave function to the Schrodinger equation. For several classical problems in quantum mechanics, we obtain their ground state wave function and energy with very low errors. The method developed in the present paper may pave a new way for solving nuclear many-body problems in the future.
The quantum many-body problem is ultimately a curse of dimensionality: the state of a system with many particles is determined by a function with many dimensions, which rapidly becomes difficult to efficiently store, evaluate and manipulate numerically. On the other hand, modern machine learning models like deep neural networks can express highly correlated functions in extremely large-dimensional spaces, including those describing quantum mechanical problems. We show that if one represents wavefunctions as a stochastically generated set of sample points, the problem of finding ground states can be reduced to one where the most technically challenging step is that of performing regression-a standard supervised learning task. In the stochastic representation the (anti)symmetric property of fermionic/bosonic wavefunction can be used for data augmentation and learned rather than explicitly enforced. We further demonstrate that propagation of an ansatz towards the ground state can then be performed in a more robust and computationally scalable fashion than traditional variational approaches allow.