Constructing neural stationary states for open quantum many-body systems

We propose a scheme based on the neural-network quantum states to simulate the stationary states of open quantum many-body systems. Using the high expressive power of the variational ansatz described by the restricted Boltzmann machines, which we dub as the neural stationary state ansatz, we compute the stationary states of quantum dynamics obeying the Lindblad master equations. The mapping of the stationary-state search problem into finding a zero-energy ground state of an appropriate Hermitian operator allows us to apply the conventional variational Monte Carlo method for the optimization. Our method is shown to simulate various spin systems efficiently, i.e., the transverse-field Ising models in both one and two dimensions and the XYZ model in one dimension.

Figure 1

Schematic illustration of our method for the case with two spins. (a) (Left) A spin model with dissipations ${\widehat{\mathrm{\Gamma }}}_i$, which are indicated by the yellow zigzag arrows, described by the Lindblad superoperator $\mathcal{L}$. The usual Hermitian interaction is denoted by the real black line. (Right) The vector representation of the Lindblad superoperator as an operator $\widehat{\mathcal{L}}$ acting on the doubled Hilbert space. The dissipations become the non-Hermitian interactions, denoted by the yellow dotted lines, between the physical and newly introduced fictitious spins. (b) The Hermitian operator ${\widehat{\mathcal{L}}}^{†}\widehat{\mathcal{L}}$ in the vector representation, whose expectation value plays a role of the cost function for the variational ansatz. (c) Our neural stationary state represented by the RBM. The black thin lines denote the non-zero interaction parameters in the ansatz between the physical spins ${\sigma }_i$ (or fictitious spins ${\tau }_i$) and hidden spins $h_j$.

Figure 2

The volume-law scaling of the operator space entanglement entropy in the random-valued NSS ansatz. Here, the number ratios of the hidden spins are taken as $({\alpha }_1,{\alpha }_2)=(2,8),(2,6),(2,4),(2,2)$. We do not observe any evident difference between finite ${\alpha }_1$, and hence we fix as ${\alpha }_1=2$. The amplitudes of the random parameters are set as $r=0.05,0.003,0.1,0.1$ for $W_{ij},W_{ik},a_i,b_k$, respectively. Note that each points show the averaged value over 10 independently generated random states.

Figure 3

(a) The real and imaginary parts of the stationary-state density matrix of the 1D transverse-field Ising model with dissipations in Eqs. (17) and (18) obtained by the Lanczos method. (b) The real and imaginary parts of the stationary-state density matrix obtained by the NSS ansatz. The fidelity $F$ between the stationary states obtained by the Lanczos method and the NSS is over 0.999. (c) Optimization of the cost function $\langle \langle {\widehat{\mathcal{L}}}^{†}\widehat{\mathcal{L}}\rangle \rangle$. The optimization works well and $\langle \langle {\widehat{\mathcal{L}}}^{†}\widehat{\mathcal{L}}\rangle \rangle$ reaches the order of ${10}^{-3}$. (d) The entropy contribution $-p_{n}ln{p}_n$, where $p_n$ is the $n-\mathrm{th}$ eigenvalue of the density matrix from the top. The blue and orange dots denote the results for the NSS ansatz and Lanczos method, respectively. The relative error for the total entropy is order of ${10}^{-3}$. For all panels, we use $V=0.3,g=1,$ and $\gamma =0.5$, and the number ratio of the spins is $\alpha =1$. The sampling number per iteration is $N_s=2000$, repeated for $N_{\mathrm{it}}=1500$ iterations. The system sizes are given as $L=4$ for (a) and (b), while (c) and (d) are calculated for $L=8$.

Figure 4

(a) Optimization of the cost function $\langle \langle {\mathcal{L}}^{†}\mathcal{L}\rangle \rangle$ for the 2D transverse-field Ising model with dissipations in Eqs. (20) and (21). (b) The entropy contribution $-p_{n}ln{p}_n$ for $n\mathrm{th}$ eigenvalue. The relative error of the total entropy is order of ${10}^{-5}$. We use the parameters $L_x=L_y=3,V=0.3,g=1$, and $\gamma =1$. The number ratio of the spins is $\alpha =4$, and the resulting fidelity is $F=0.9996$. The sampling number per iteration is $N_s=2000$, repeated for $N_{\mathrm{it}}=4000$ iterations.

Figure 5

The entropy contribution $-p_{n}ln{p}_n$ for the $n-\mathrm{th}$ eigenvalue of the stationary-state density matrix in 1D XYZ model with dissipations in Eqs. (22) and (23). The data for the NSS (blue) and the exact diagonalization (orange) agree well with each other. The fidelity of the NSS is over 0.998 and the relative error of the total entropy is order of ${10}^{-2}$. The parameters of the model is taken as $J_x=0.9,J_y=0.4,J_z=1.0,L=4,$ and $\gamma =1$. The number ratio of the spins is taken as $\alpha =8$ and the sampling number per iteration is $N_s=8000$, repeated for $N_{\mathrm{it}}=4500$ iterations.

Figure 6

The infidelity $1-F$ of the NSS fitted to a random density matrix generated by Eq. (A1). While the NSS with larger $\alpha$, or the number ratio of the spins, better approximates the random density matrices, the number of parameters and accordingly the numerical cost required to reach some fixed fidelity increase rapidly.

Figure 7

The wall times for computing the stationary state of the 1D transverse-field Ising model with $V=2,g=1$, and $\gamma =1$. The blue and orange dots are for the NSS ansatz optimization and the Lanczos method, respectively. Here, the number ratio of the spins is $\alpha =4$. The NSS ansatz exhibits lower scaling as a function of the number of physical spins. The number of sampling is $N_s=2000$ repeated for $N_{\mathrm{it}}=1500$ iterations, which we find to be sufficient for the convergence of the VMC calculation. The computation for the NSS and Lanczos is executed on 8 cores on Intel(R) Core i7-6820HQ and 12 cores on Intel(R) Xeon(R) Silver 4110, respectively.