Abstract
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.
Highlights
The main difficulty in simulating strongly interacting many-body quantum systems on classical computers stems from the curse of dimensionality
In view of real time evolution, this could turn out to be a considerable advantage compared to established tensor network methods [10, 26, 27, 33, 34], since the increase of entanglement with time demands an exponential increase of virtual bond dimensions, limiting the applicability of tensor network methods to relatively short time intervals [1, 4]
The neural-network architecture proposed as a wavefunction ansatz in [5] is the restricted Boltzmann machine (RBM)
Summary
The main difficulty in simulating strongly interacting many-body quantum systems on classical computers stems from the curse of dimensionality. The recent successes of artificial neural network techniques have entailed a large interest in applying them to quantum many-body systems, in particular as ansatz for the wavefunction of (strongly correlated) quantum systems [5, 13, 21, 22] Such neuralnetwork quantum states have the principal capability to describe systems hosting chiral topological phases [8, 13, 17], or to handle large entanglement [11, 12, 19]. Time-dependent variational Monte Carlo (tdVMC) [5, 6, 29] combines the Dirac-Frenkel principle with Monte Carlo sampling and exploits the locality of typical quantum Hamiltonians This involves the application of the (pseudo-) inverse of a covariance matrix to evolve the variational parameters in time. We propose and explore an alternative approach, namely directly approximating a time step of a conventional ordinary differential equation (ODE) method by “training” the neural-network quantum state at the time step using (variations of) stochastic gradient descent
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