Time-Dependent Variational Principle for Open Quantum Systems with Artificial Neural Networks

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 $4\times 4$ system and by applying it to the simulation of confinement dynamics in the presence of dissipation.

Figure 1

Illustration of the variational approach to OQS dynamics. On the left, the standard density matrix formalism is shown along with the equivalent probabilistic formulation using POVMs. The right-hand side shows the variational approach, in which an artificial neural network is used as an ansatz function for the probability distribution over POVM outcomes. The main contribution of this work is the TDVP illustrated in the bottom box which leads to a general and accurate scheme for updating the network parameters according to the dynamics dictated by the master equation.

Figure 2

(a) and (b) Mean magnetizations and next-nearest neighbor connected correlation functions [e.g., $C_{XX}(d=2)={\sum }_i⟨{\widehat{X}}_i{\widehat{X}}_{i+2}{⟩}^c/N$] as a function of time in the anisotropic 1D Heisenberg model for $N=40$ spins starting in the product state $⟨\widehat{Y}⟩=-1$. Nearest neighbor couplings are given by $\overrightarrow{J}/\gamma =(2,0,1)$, $h_z/\gamma =1$ and the dissipation channel is $\widehat{L}={\widehat{\sigma }}^{-}=\frac{1}{2}(\widehat{X}-i\widehat{Y})$. The exact data are obtained for $N=10$ spins. (c) and (d) Mean $z$ magnetizations and nearest neighbor connected correlation functions (for $J_y/\gamma =1.8$) in a $4\times 4$ anisotropic 2D Heisenberg lattice with nearest neighbor couplings $\overrightarrow{J}/\gamma =[0.9,1.0(1.8),1.0]$ and the same decay as in (a) and (b), starting in the product state $⟨\widehat{Z}⟩=1$.

Figure 3

(a) Mean magnetizations in a spin chain of length $N=32$ with the quench parameters $h_x/J_z=0.25$, $h_z/J_z=0.05$ and the dissipation channel $\widehat{L}=\widehat{Z}$ with relative strength $\gamma /J_z=0.25$ compared to MCWF-data for $N=16$ spins starting in the product state $⟨\widehat{Z}⟩=1$. (b) Spreading of correlations in the spin chain. Top: Dissipative system with $\gamma =0.25$, Bottom: MPS simulation of the unitary system where $\gamma =0.0$. After an initial linear light-cone spreading, the nature of the dissipative propagation grows more diffusive, before all correlations eventually vanish. Notice that the slight deviations in panel (a) coincide with the time at which the dissipative correlations cross the MCWF system size boundary.