Quantum Many-Body Dynamics in Two Dimensions with Artificial Neural Networks

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.

Figure 1

(a) Schematic illustration of the artificial neural network (ANN) encoding of many-body wave functions in 2D quantum spin systems. A given spin configuration $s$, blue and red referring to the spin $\uparrow$ and $\downarrow$ state, respectively, functions as the input to an ANN whose output at the end is the corresponding wave function amplitude ${\psi }_s$. (b) Collapse and revival of the ferromagnetic order in a quantum Ising model of $8\times 8$ spins on a square lattice after quenching the transverse field from $h=0$ to $h=2.63h_c$. (c) Dynamics of the transverse magnetization $⟨{\sigma }_i^x(t)⟩$. The quantum Fisher information density $f_Q(t)$ in (d) reveals that genuine multipartite entanglement is generated by the unitary evolution.

Figure 2

Time evolution after quenching a transverse-field Ising model of size $N=10\times 10$ from the paramagnetically polarized initial state $|{\psi }_0⟩=|\to ⟩$ (a) into the paramagnetic phase at $h=2h_c$, (b) to the critical point, and (c) into the ferromagnetic phase at $h=h_c/10$. For direct comparison the top row includes data obtained with IPEPS from Ref. [26]. The agreement is very good in all cases for the networks with the smallest error $R^2(t)$ (bottom row). The second row shows space-time plots of correlation functions $⟨{\sigma }_{i,j}^z{\sigma }_{i,j+d}^z⟩$ along the lattice axis from the simulations with minimal error.