We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schrödinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schrödinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.
We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schrödinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schrödinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.
The time-dependent Schrödinger equation (TDSE) in real space is fundamental to understanding the dynamics of many-electron quantum systems, with applications ranging from quantum chemistry to condensed matter physics and materials science. However, solving the TDSE for complex fermionic systems remains a significant challenge, particularly due to the need to capture the time-evolving many-body correlations, while the antisymmetric nature of fermionic wavefunctions complicates the function space in which these solutions must be represented. We propose a general-purpose neural network framework for solving the real-space TDSE, Fermionic Antisymmetric Spatio-Temporal Network, which treats time as an explicit input alongside spatial coordinates, enabling a unified spatiotemporal representation of complex, antisymmetric wavefunctions for fermionic systems. This approach formulates the TDSE as a global optimization problem, avoiding step-by-step propagation and supporting highly parallelizable training. The method is demonstrated on five benchmark problems, achieving excellent agreement with reference solutions across all cases. These results demonstrate the method's accuracy and flexibility within the bound-state manifold across various dimensions and interaction regimes. While the current localized Ansatz inherently restricts the description of extensive ionization and continuum states, the method demonstrates the capability to stably simulate coherent multi-electron dynamics over extended time windows. Our framework offers a highly expressive alternative to traditional basis-dependent or mean-field methods, opening new possibilities for ab initio simulations of time-dependent quantum systems, with applications in quantum dynamics, molecular control, and ultrafast spectroscopy.
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 theory of quantum dynamics is crucial for quantum science and engineering. The computation cost for exact quantum simulation is expensive due to the exponential growth of the dimension of the Hilbert space. There are recent attempts that utilize neural networks to simulate long-time quantum dynamics. We conduct a comparative study on different approaches that simulate dynamics based on parameterizing the quantum states with state-of-the-art autoregressive neural networks. Pixel Convolution Neural Networks, Recurrent Neural Networks, and Transformer models are examined for their representability and accuracy. We identify that Recurrent Neural Networks and Transformers are superior to Pixel Convolution Neural Networks. In addition, we compare the performance of different algorithms to simulate quantum dynamics, which include the Euler’s method, the forward-backward trapezoid method, the stochastic reconfiguration method, and the spacetime method. We found that the spacetime method performs better than the other algorithms, the Euler’s method and the forward-backward method are comparable, while the SR method exhibits stability issues. Our study is performed on the transverse-field Ising model on one-dimensional systems
We investigate the performance of classical, quantum, and hybrid quantum neural networks in solving partial differential equations through both supervised and unsupervised learning. The performance of each type of network is evaluated in solving the damped harmonic oscillator, the Einstein field equations, and the time-independent Schrödinger equation. We employed physics-informed supervised neural networks to solve the damped harmonic oscillator and the Einstein field equations, and developed an unsupervised neural network algorithm based on the shooting method for the time-independent Schrödinger equation. For the quantum neural networks, we introduced three variational parameterized quantum feature maps and two quantum circuits for the hybrid quantum neural networks. Each network was examined in three different configurations and evaluated across multiple random seeds. Under favorable parameter initializations, the hybrid quantum neural networks achieve higher accuracy than the classical neural networks in most cases across all tested problems. In contrast, the quantum neural networks attain the best accuracy for the damped harmonic oscillator and also perform well in the Schrödinger equation problem. Both the quantum and the hybrid quantum neural networks require fewer parameters than the classical neural networks and converge faster during optimization. All three models exhibit sensitivity to parameter initialization, with the quantum neural networks displaying the highest variability across different problems.
We develop a variational approach to simulating the dynamics of open quantum many-body systems using deep autoregressive neural networks. The parameters of a compressed representation of a mixed quantum state are adapted dynamically according to the Lindblad master equation by employing a time-dependent variational principle. We illustrate our approach by solving the dissipative quantum Heisenberg model in one dimension for up to 40 spins and in two dimensions for a 4×4 system and by applying it to the simulation of confinement dynamics in the presence of dissipation.
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.