Real time evolution with neural-network quantum states

Irene López Gutiérrez and Christian B. Mendl

Technische Universität München, Department of Informatics and Institute for Advanced Study, Boltzmannstraße 3, 85748 Garching, Germany
Technische Universität Dresden, Institute of Scientific Computing, Zellescher Weg 12-14, 01069 Dresden, Germany

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A promising application of neural-network quantum states is to describe the time dynamics of many-body quantum systems. To realize this idea, we employ neural-network quantum states to approximate the implicit midpoint rule method, which preserves the symplectic form of Hamiltonian dynamics. We ensure that our complex-valued neural networks are holomorphic functions, and exploit this property to efficiently compute gradients. Application to the transverse-field Ising model on a one- and two-dimensional lattice exhibits an accuracy comparable to the stochastic configuration method proposed in [Carleo and Troyer, Science 355, 602-606 (2017)], but does not require computing the (pseudo-)inverse of a matrix.

Numerical simulation plays an important role for the analysis and understanding of quantum systems. An inherent difficulty stems from the exponential growth of the system's Hilbert space dimension with respect to its size. In particular, the long time dynamics of a quantum system, and the associated entanglement growth, precludes the efficient representation of the statevector by tensor network methods in general.

In this work, we approach the problem using machine learning methods, as proposed in [Carleo and Troyer, Science 355, 602-606 (2017)]. The statevector is approximated by a neural network, and the time evolution of the system is realized by changes in the network parameters. Our main contribution is a translation of well-established numerical integration methods to the optimization of the network weights. We construct a cost function based on the implicit midpoint method as a concrete example, and minimize it with gradient descent. Compared with the previously proposed stochastic reconfiguration method, our approach does not require a possibly ill-conditioned matrix inversion.

The paper describes the mathematical details of the optimization and provides numerical examples of the time dynamics governed by the Ising model, on a 1D lattice with 20 sites and a small 2D lattice. We find the accuracies achieved by our method to be on par with stochastic reconfiguration. The remaining error in the simulations points towards limits of the expressibility of the neural network ansatze themselves, which could be improved in future works.

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