Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems

The possibility to simulate the properties of many-body open quantum systems with a large number of degrees of freedom (d.o.f.) is the premise to the solution of several outstanding problems in quantum science and quantum information. The challenge posed by this task lies in the complexity of the density matrix increasing exponentially with the system size. Here, we develop a variational method to efficiently simulate the nonequilibrium steady state of Markovian open quantum systems based on variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the quantum master equation. We test the effectiveness of the method by modeling the two-dimensional dissipative $XYZ$ spin model on a lattice.

Figure 1

Graphical representation of the neural network ansatz for the density matrix. The input states $|\mathit{\sigma }⟩,|\mathit{\eta }⟩$ are encoded in the visible layer, represented by circles. The hidden spins in the triangles encode the correlation between the physical spins in each state of the statistical mixture, while the hidden spins in the squares encode the mixture between the states. This structure is easily seen to coincide with a RBM, where the hidden layer is composed by the triangle and square nodes.

Figure 2

The steady state spin structure factor $S_{\mathrm{SS}}^{xx}(\mathit{k})$ computed as a function of $\alpha =\beta$ for a $3\times 3$ lattice and $\mathbf{k}=\mathbf{0}$ (upper panel) and $\mathbf{k}=(2\pi /3,0)$ (lower panel). The red dot-dashed line represents in both panels the exact result. The inset shows the evolution of $⟪{\mathcal{L}}_{\chi }⟫$ over the VMC run. Parameters: $J_x/\gamma =0.9$, $J_y/\gamma =1.2$, $J_z/\gamma =1.0$.

Figure 3

The magnetization $M_z$ computed as a function of the coupling $J_y/\gamma$. VMC and exact values are compared. Error bars, when not shown, are smaller than the symbol. Other parameters: $J_x/\gamma =0.9$, $J_z/\gamma =1.0$, $\alpha =\beta =3$.

Figure 4

The steady-state spin structure factor $S_{\mathrm{SS}}^{xx}(\mathbf{k}=\mathbf{0})$ computed as a function of the coupling $J_y/\gamma$. VMC and exact values are compared. Other parameters: $J_x/\gamma =0.9$, $J_z/\gamma =1.0$, $\alpha =\beta =3$.