Learning quantum dissipation by the neural ordinary differential equation

Quantum dissipation arises from the unavoidable coupling between a quantum system and its surrounding environment, which is known as a major obstacle in the quantum processing of information. Apart from its existence, examining how to trace dissipation from observational data is important and may stimulate ways to suppress the dissipation. In this paper, we propose to learn about quantum dissipation from dynamical observations using the neural ordinary differential equation, and then demonstrate this method concretely on two open quantum-spin systems: A large spin system and a spin-1/2 chain. We also investigate the learning efficiency of the dataset, which provides useful guidance for data acquisition in experiments. Our work helps to facilitate effective modeling and decoherence suppression in open quantum systems.

Figure 1

(a) Schematic of an open quantum system. (b) ODE solvers work to find solutions of the Lindblad equation when the Hamiltonian or the Liouvillian is priorly known, while the NODE works to reconstruct the Hamiltonian or Liouvillian as solutions [$\rho \left(t\right)$ or $\langle O\left(t\right)\rangle ]$ are given. (c) Mechanism of the NODE where $\rho \left(t\right)$ and $a\left(t\right)$ flow forward and backward, respectively.

Figure 2

Learning results of the spin-3/2 model with the $t$-dependent probe. (a) Dependence of the loss function $\mathcal{L}$, the relative error of $\alpha$, and the relative error of $\gamma$ on training epochs. (b) Spin dynamics where solid, dashed, and dot-dashed lines correspond to ${\langle S_x\rangle }_{\text{real}}$ in the dataset $\mathcal{D}$, ${\langle S_x\rangle }_{\text{pred}}$ before training, and ${\langle S_x\rangle }_{\text{pred}}$ after ${10}^3$ epochs of training. (c) and (d) Real parameters (solid triangles) and predictive parameters (circles with error bars) after training, respectively. Color shadings in (b) and error bars in (c) and (d) indicate the predictive fluctuations due to $N_{\text{ini}}=10$ different initializations.

Figure 3

Learning results of the spin-1/2 chain with the $t$-independent probe. (a) Dependence of $\mathcal{L}$, $\alpha$, and $\gamma$ on training epochs. (b) Total spin dynamics of ${\langle S_x\rangle }_{\text{real}}\in \mathcal{D}$, ${\langle S_x\rangle }_{\text{pred}}$ before training, and ${\langle S_x\rangle }_{\text{pred}}$ after ${10}^3$ epochs of training. (c) and (d) Real parameters (solid triangles) and predictive parameters (circles with error bars) after training.

Figure 4

(a) $\chi$ and ${\chi }^2$ as a function of $t$, where dots visually indicate the optimal strategy of data acquisition with $t_N=t_N^{\text{opt}}$. (b)–(d) Learning results of the spin-3/2 example using the $t$-dependent probe and the MSE loss function. (b) and (c) respectively show the variation of ${\mathcal{E}}_{\gamma }$ and $\eta$ on $t_N$ with $N=N_{\gamma }=15$ being fixed. (d) shows the variation of ${\mathcal{E}}_{\gamma }$ on $N$ with $t_N=t_N^{\text{opt}}=1.7t_{\text{dc}}$ being fixed. All the numerical results have been averaged on $N_{\text{ini}}=50$ random initializations.

Figure 5

First row: Learning results for the spin-3/2 model with the $t$-independent probe. (a1) Dependence of $\mathcal{L}$, $\alpha$, and $\gamma$ on training epochs. (a2) Spin dynamics of ${\langle S_x\rangle }_{\text{real}}\in \mathcal{D}$, ${\langle S_x\rangle }_{\text{pred}}$ before training and ${\langle S_x\rangle }_{\text{pred}}$ after ${10}^3$ epochs of training. (a3) and (a4) Real parameters (solid triangles) and predictive parameters (circles with error bars) after training. Second row: Learning results for the spin-1/2 chain with the $t$-dependent probe. (b1) Training processes of $\mathcal{L}$, $\alpha$, and $\gamma$ with the dataset being constructed by total spin measurements, i.e., $\mathcal{D}=\left\{\langle S_{\mu =x,y,z}({\mathcal{P}}_s,t_n)\rangle \right\}$. (b2)–(b4) Learning results on the dataset constructed by local spin measurements, i.e., $\mathcal{D}=\left\{\langle {\sigma }_{\mu =x,y,z}^{i=3}({\mathcal{P}}_s,t_n)\rangle \right\}$.