Variational Neural-Network Ansatz for Steady States in Open Quantum Systems

We present a general variational approach to determine the steady state of open quantum lattice systems via a neural-network approach. The steady-state density matrix of the lattice system is constructed via a purified neural-network Ansatz in an extended Hilbert space with ancillary degrees of freedom. The variational minimization of cost functions associated to the master equation can be performed using a Markov chain Monte Carlo sampling. As a first application and proof of principle, we apply the method to the dissipative quantum transverse Ising model.

Figure 1

Graph representation of the artificial neural network used for the density matrix Ansatz. The vector $\mathit{\sigma }=({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_N)$ contains the variables of the physical system (visible layer). The vector $\mathit{a}=(a_1,a_2,\dots ,{\sigma }_{N_a})$ describes the ancillary degrees of freedom of the extended Hilbert space (ancilla layer), where a purified density matrix is considered [see Eq. (3)]. The vector $\mathit{h}=(h_1,h_2,\dots ,h_{N_h})$ contains variables of auxiliary nodes (hidden layer). The network parameters are ${\mathit{v}}_{\nu }=({\mathbf{b}}_{\nu }^{(\sigma )},{\mathit{b}}_{\nu }^{(a)},{\mathit{b}}_{\nu }^{(h)},{\mathit{W}}_{\nu },{\mathit{U}}_{\nu })$. One network is used for the representation of the amplitude ($\nu ={\mathit{v}}_A$) of the purified wave function in Eq. (3), while another independent network with the same topology is used to represent the phase ($\nu ={\mathit{v}}_{\theta }$).

Figure 2

Top panel: sketch of the considered physical system described by the dissipative quantum transverse Ising 1D model with periodic boundary conditions. The quantity $g$ denotes an applied magnetic field, $V$ the spin-spin coupling, and $\gamma$ the spin flip rate. Bottom panel: the prediction of the neural-network variational calculations (circles) are compared to the results obtained by quantum trajectory simulations of the master equation by considering the whole Hilbert space (solid lines). The top, middle, and bottom panels depict the expectation values of the three components of the averaged magnetization as a function of the applied magnetic field $g$ (in units of $\gamma$). Model parameters: $V/\gamma =2$ (spin-spin coupling), $N=16$ (number of lattice sites). Neural-network parameters: $\alpha =\beta =1$ for $g\le \gamma$ and $g\ge 2.5\gamma$ while $\alpha =1$ and $\beta =4$ for the remaining points. The parameters required for the convergence of the Monte Carlo calculations depend on the value of $g/\gamma$, with the intermediate region being the most demanding. The maximum number of accepted Monte Carlo samples is 8640 and the maximum number of steps for the stochastic gradient descent is $10^4$. For points outside the intermediate region, 3000 accepted Monte Carlo samples and $10^3$ iteration steps have been performed.

Figure 3

Relative error with respect to the exact result for the observable $⟨{\sigma }_x⟩$ as a function of $\alpha$ and $\beta$. Parameters are set as in Fig. 2 but for a fixed value $g/\gamma =1.2$.

Figure 4

Same parameters as in Fig. 2 but for a fixed value $g/\gamma =1$. Top panel: the cost function is shown as a function of the iteration steps. Bottom panel: the corresponding evolution of the $x$ component of the average magnetization during the stochastic minimization is shown.