Solving the Liouvillian Gap with Artificial Neural Networks

We propose a machine-learning inspired variational method to obtain the Liouvillian gap, which plays a crucial role in characterizing the relaxation time and dissipative phase transitions of open quantum systems. By using “spin bi-base mapping,” we map the density matrix to a pure restricted-Boltzmann-machine (RBM) state and transform the Liouvillian superoperator to a rank-two non-Hermitian operator. The Liouvillian gap can be obtained by a variational real-time evolution algorithm under this non-Hermitian operator. We apply our method to the dissipative Heisenberg model in both one and two dimensions. For the isotropic case, we find that the Liouvillian gap can be analytically obtained and in one dimension even the whole Liouvillian spectrum can be exactly solved using the Bethe ansatz method. By comparing our numerical results with their analytical counterparts, we show that the Liouvillian gap could be accessed by the RBM approach efficiently to a desirable accuracy, regardless of the dimensionality and entanglement properties.

Figure 1

A sketch of the essential idea for accessing the Liouvillian gap with artificial neural networks. (a) The restricted Boltzmann machine (RBM) representation of quantum many-body density matrices under the “spin bi-base mapping.” (b) The dissipative spin-$1/2$ XYZ model defined on a 2D square lattice. (c) A pictorial sketch of the Liouvillian spectrum. The red dot corresponds to the nonequilibrium steady state with zero Liouvillian eigenvalue, whereas the blue dots on the vertical dotted line correspond to the first decay modes. $\mathrm{\Delta }$ denotes the Liouvillian gap.

Figure 2

Numerical results for the dissipative XXZ model in one [(a) and (b)] and two [(c)] dimensions. (a) The real part of the expectation value $\mathrm{Re}(⟨\tilde{\mathcal{L}}⟩)=\mathrm{Re}(⟨{\rho }^{\prime }|\tilde{\mathcal{L}}|{\rho }^{\prime }⟩/⟨{\rho }^{\prime }|{\rho }^{\prime }⟩)$ as a function of the iteration steps for different lattice sizes $N$. The parameters are chosen as $J_x=J_y=1$, $J_z=2$, and $\gamma =3$. The red dashed horizontal line indicates the exact value of the Liouvillian gap $\mathrm{\Delta }$ obtained by exact diagonalization, which can also be derived analytically (see the main text). (b) $\mathrm{Re}(⟨\tilde{\mathcal{L}}⟩)$ as a function of iteration steps for different dissipation rates. Here, $J_x=J_y=1$, $J_z=2$, and $N=10$. The inset shows the linear dependence of the Liouvillian gap obtained by the RBM method on the dissipation rate $\gamma$. (c) $\mathrm{Re}(⟨\widehat{\mathcal{L}}⟩)$ for the 2D dissipative XXZ model with $\gamma =3$, and varying lattice sizes and coupling strengths (see Ref. [54] for details).

Figure 3

Numerical results for the dissipative XYZ model in 1D. (a) and (b) show, respectively, the fidelity between $|{\rho }^{\prime }⟩$ and $|{\rho }_1⟩$, and the expectation value $\mathrm{Re}(⟨\tilde{\mathcal{L}}⟩)$, as functions of the iteration steps for $N=4$. After around 500 iterations, $\mathrm{Re}(⟨\tilde{\mathcal{L}}⟩)$ converges to the exact value of the Liouvillian gap $\mathrm{\Delta }$ [indicated by the red dashed line in (b)] with relative error ${\epsilon }_{\mathrm{rel}}\approx {10}^{-2}$ and the fidelity approaches one ($|⟨{\rho }^{\prime }|{\rho }_1⟩|\approx 0.991$). (c) The comparison between the RBM and ED results with varying system sizes. Here, the model parameters are chosen as $J_x=4$, $J_y=0.5$, $J_z=2$, and $\gamma =1$ [54].