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Physical Review A

Discrete real-time learning of quantum-state subspace evolution of many-body systems in the presence of time-dependent control fields

Shaojun Gui, Tak‐San Ho, HERSCHEL A. RABITZ

Published Nov 7, 2024• 3 citations
Publisher

Abstract

A method is presented for discrete real-time learning (DRTL) of the evolution of a many-body quantum state in the presence of time-dependent control fields. The DRTL method is based on the recent adoption of the artificial neural network (ANN) ansatz within the framework of the time-dependent variational Monte Carlo method in learning large-scale quantum many-body dynamics. The method leverages accurate short-time quantum-state propagation with the Schr\"odinger equation and the efficient stochastic quantum natural gradient descent algorithm in machine learning. Importantly, the DRTL method is devoid of high numerical costs resulting from the exponentially large many-body Hilbert space by replacing the full state propagation with short-time propagations of a sequence of partial stochastic samples of the ANN state and may be considered as a synergism between the Schr\"odinger equation and the ANN that streamlines the computation of the many-body controlled quantum dynamics. Specifically, this method minimizes the normed distances between the time-dependent variational ANN ansatz and a set of partial quantum states over a sequence of stochastically sampled subspaces that are connected to each other in small discrete time steps. We demonstrate the high accuracy and stability of the DRTL method for the spin excitation dynamics of $4\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6, 8\ifmmode\times\else\texttimes\fi{}8$, and $10\ifmmode\times\else\texttimes\fi{}10$ two-dimensional (2D) Heisenberg spin-1/2 lattices driven by time-dependent control fields applied to a corner spin along with strong exchange coupling between the nearest-neighbored spins. It is found that the number of ANN parameters for properly following the dynamically occupied subspaces (i.e., starting from the pure tensor-product spin configuration with all spins down) approximately increases quadratically with the number of spins in the system. Importantly, we show that the DRTL method can accurately and efficiently follow the small, yet importantly, occupied parts of the Hilbert space of the 2D spin square lattices during the interaction with the strong time-dependent control fields over a physically significant time period. This finding provides a basis for future work to perform quantum optimal control simulations of 2D spin lattices and other multiparticle systems.

DOI
10.1103/physreva.110.052412

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A method is presented for discrete real-time learning (DRTL) of the evolution of a many-body quantum state in the presence of time-dependent control fields. The DRTL method is based on the recent adoption of the artificial neural network (ANN) ansatz within the framework of the time-dependent variational Monte Carlo method in learning large-scale quantum many-body dynamics. The method leverages accurate short-time quantum-state propagation with the Schr"odinger equation and the efficient stochastic quantum natural gradient descent algorithm in machine learning. Importantly, the DRTL method is devoid of high numerical costs resulting from the exponentially large many-body Hilbert space by replacing the full state propagation with short-time propagations of a sequence of partial stochastic samples of the ANN state and may be considered as a synergism between the Schr"odinger equation and the ANN that streamlines the computation of the many-body controlled quantum dynamics. Specifically, this method minimizes the normed distances between the time-dependent variational ANN ansatz and a set of partial quantum states over a sequence of stochastically sampled subspaces that are connected to each other in small discrete time steps. We demonstrate the high accuracy and stability of the DRTL method for the spin excitation dynamics of 4\ifmmode×\else\texttimes\fi4,6\ifmmode×\else\texttimes\fi6,8\ifmmode×\else\texttimes\fi84\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6, 8\ifmmode\times\else\texttimes\fi{}84\ifmmode×\else\texttimes\fi4,6\ifmmode×\else\texttimes\fi6,8\ifmmode×\else\texttimes\fi8, and 10\ifmmode×\else\texttimes\fi1010\ifmmode\times\else\texttimes\fi{}1010\ifmmode×\else\texttimes\fi10 two-dimensional (2D) Heisenberg spin-1/2 lattices driven by time-dependent control fields applied to a corner spin along with strong exchange coupling between the nearest-neighbored spins. It is found that the number of ANN parameters for properly following the dynamically occupied subspaces (i.e., starting from the pure tensor-product spin configuration with all spins down) approximately increases quadratically with the number of spins in the system. Importantly, we show that the DRTL method can accurately and efficiently follow the small, yet importantly, occupied parts of the Hilbert space of the 2D spin square lattices during the interaction with the strong time-dependent control fields over a physically significant time period. This finding provides a basis for future work to perform quantum optimal control simulations of 2D spin lattices and other multiparticle systems.
We present a self-consistent algorithm for optimal control simulations of many-body quantum systems. The algorithm features a two-step synergism that combines discrete real-time machine learning (DRTL) with Quantum Optimal Control Theory (QOCT) using the time-dependent Schrödinger equation. Specifically, in step (1), DRTL is employed to identify a compact working space (i.e., the important portion of the Hilbert space) for the time evolution of the many-body quantum system in the presence of a control field (i.e., the initial or previously updated field), and in step (2), QOCT utilizes the DRTL-determined working space to find a newly updated control field for a chosen objective. Steps 1 and 2 are iterated until a self-consistent control objective value is reached such that the resulting optimal control field yields the same targeted objective value when the corresponding working space is systematically enlarged. To demonstrate this two-step self-consistent DRTL-QOCT synergistic algorithm, we perform optimal control simulations of strongly interacting 1D as well as 2D Heisenberg spin systems. In both scenarios, only a single spin (at the left end site for 1D and the upper left corner site for 2D) is driven by the time-dependent control fields to create an excitation at the opposite site as the target. It is found that, starting from all spin-down zero excitation states, the synergistic method is able to identify working spaces and convergence of the desired controlled dynamics with just a few iterations of the overall algorithm. In the cases studied, the dimensionality of the working space scales only quasi-linearly with the number of spins.
The efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter constitutes a key challenge for current computational methods. This holds in particular in the regime of two spatial dimensions, whose experimental exploration is currently pursued with strong efforts in quantum simulators. In this work we present a versatile and efficient machine learning inspired approach based on a recently introduced artificial neural network encoding of quantum many-body wave functions. We identify and resolve key challenges for the simulation of time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, we study the dynamics of the paradigmatic two-dimensional transverse-field Ising model, as recently also realized experimentally in systems of Rydberg atoms. Calculating the nonequilibrium real-time evolution across a broad range of parameters, we, for instance, observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached timescales are comparable to or exceed the capabilities of state-of-the-art tensor network methods.
In experimentally realistic situations, quantum systems are never perfectly isolated and the coupling to their environment needs to be taken into account. Often, the effect of the environment can be well approximated by a Markovian master equation. However, solving this master equation for quantum many-body systems becomes exceedingly hard due to the high dimension of the Hilbert space. Here we present an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine-learning techniques. We represent the mixed many-body quantum states with neural networks in the form of restricted Boltzmann machines and derive a variational Monte Carlo algorithm for their time evolution and stationary states. We document the accuracy of the approach with numerical examples for a dissipative spin lattice system.
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.
Understanding the real-time evolution of many-electron quantum systems is essential for studying dynamical properties in condensed matter, quantum chemistry, and complex materials, yet it poses a significant theoretical and computational challenge. Our work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations. We employ time-dependent Jastrow factors and backflow transformations, enhanced through neural networks parameterizations. To compute the optimal time-dependent parameters, we employ the time-dependent variational Monte Carlo technique and introduce a new method based on Taylor-root expansions of the propagator, enhancing the accuracy of our simulations. The approach is demonstrated in three distinct systems. In all cases, we show clear signatures of many-body correlations in the dynamics. The results showcase the ability of our variational approach to accurately describe the time evolution, providing insight into quantum dynamical effects in interacting electronic systems, beyond the capabilities of mean-field.
We develop a new implementation of the real-time full configuration interaction quantum Monte Carlo method, and apply it to the dynamics of the Heisenberg spin chain and the spectrum of the one-dimensional Fröhlich polaron. This is a general method for the numerical integration of the Schrödinger equation to evolve quantum systems in real time, introduced by Guther et al. as an extension of the imaginary-time version of the algorithm that finds ground-state properties. We introduce a new integrator for improved efficiency, and a scaling procedure for better control of simulations. We study the accuracy of the method, including the effect of discretising time and of the stochastic errors introduced by the Monte Carlo elements of the algorithm. Comparing the method to a hardware quantum simulator for the quench dynamics of a Heisenberg spin chain, we find that for this model the discretisation and stochastic errors grow quickly, limiting the effectiveness of the algorithm for finding accurate evolution over long times. We find that spectral information can be generated successfully using an energy-dependent damping, with either a tilted complex contour time evolution or a strong second-order damping. Specifically, we reproduce the low-energy spectrum of the Fr ̈ohlich polaron model and confirm the existence of a bound excited state.