Simulating Non-Markovian Open Quantum Dynamics with Neural Quantum States

TL;DR

This work encodes environmental memory in dissipatons, yielding the dissipaton-embedded quantum master equation (DQME), yielding the resulting NQS-DQME framework, which achieves compact representation of many-body correlations and non-Markovian memory.

Abstract

Reducing computational scaling for simulating non-Markovian dissipative dynamics using artificial neural networks is both a major focus and formidable challenge in open quantum systems. To enable neural quantum states (NQSs), we encode environmental memory in dissipatons (quasiparticles with characteristic lifetimes), yielding the dissipaton-embedded quantum master equation (DQME). The resulting NQS-DQME framework achieves compact representation of many-body correlations and non-Markovian memory. Benchmarking against numerically exact hierarchical equations of motion confirms NQS-DQME maintains comparable accuracy while enhancing scalability and interpretability. This methodology opens new paths to explore non-Markovian open quantum dynamics in previously intractable systems.

Figures (5)

• Figure 1: Schematic of the fermionic DQME theory, mapping the original OQS (left) to a dissipaton-embedded system (right). Red and blue bars represent the $N_{_{\rm S}}$ system fermion energy levels and $N_{_{\rm E}}$ memory-carrying dissipaton levels, respectively. Broadening of blue bars indicates each dissipaton's decay rate (inverse lifetime).
• Figure 2: Structure of the neural network representing $\rho_{\rm pre} (\vec{n},\vec{n}';\vec{m})$. Boxes group nodes of the same type. The visible layer comprises $2N_{_{\rm S}} + N_{_{\rm E}}$ nodes, representing elements of $\vec{n}$, $\vec{n}'$, and $\vec{m}$, respectively. Connecting lines between boxes indicate that every node in one box is linked to all nodes in the other box through weighted connections.
• Figure 3: (a) Schematic of open quantum dynamics for an impurity coupled to left ($L$) and right ($R$) reservoirs with chemical potentials $\mu_L$ and $\mu_R$. Gray shaded regions in the reservoirs represent Kondo clouds screening the impurity's localized spin. (b) $I_R(t)$ at different temperatures. The NQS-DQME results (solid lines, vertically offset for clarity) are benchmarked against HEOM reference values. Inset: steady-state current versus inverse temperature. (c) Time evolution of $\Delta s^2$ for $k_{\rm B}T=0.3\,\Gamma$, computed with different $N_{\rm{h}} = N_{\rm{a}}$. Inset: $\mathcal{E}_{I_R}$ versus $N_{\rm{h}}$ with a power function fit. (d) $I_R(t)$ at $k_{\rm B}T=0.3\,\Gamma$ computed with different $N_{\rm MC}$. Inset: $\mathcal{E}_{I_R}$ versus $N_{\rm MC}$. (e) Number of dynamical variables explicitly accessed in the NQS-DQME and HEOM methods versus $N_{_{\rm E}}$, where $N_{_{\rm E}}$ increases monotonically as $T$ decreases. Dashed lines are power function fits to the scattered data with $M_{\rm max} = 3$. System parameters (in units of $\Gamma$): $\epsilon_0 = U_0/2 = 2$, $\Delta\epsilon = -7$, and $\Delta U = 6$; see the SM for further details SM.
• Figure 4: Visualization of the RBM parameters for the open quantum dynamics shown in Fig. \ref{['fig3']}. (a,b) Heatmaps of $\lvert \bm K(t_{\rm lo}) - \bm K(t_{\rm sh})\rvert$ at (a) $k_{\rm B}T = 3.0\,\Gamma$ and (b) $k_{\rm B}T = 0.3\,\Gamma$, for $t_{{\rm sh}} = 1.25/\Gamma$ and $t_{{\rm lo}} = 5/\Gamma$. The vertical axis indexes dissipaton levels (with decay rates sorted from slowest to fastest), and the horizontal axis indexes the auxiliary nodes. (c,d) Time evolution of the auxiliary-layer-averaged weight difference for the $j$th dissipaton level, $\Delta K_j = \lvert K_j(t) - K_j(t_{\rm sh})\rvert$, at (c) $k_{\rm B}T = 3.0\,\Gamma$ and (d) $k_{\rm B}T = 0.3\,\Gamma$. Here, $\gamma_j \equiv {\rm Re}(\gamma^\sigma_j)$ denotes the decay rate in unit of $1/\Gamma$. The red line tracks the slowest-decaying dissipaton level.
• Figure 5: (a) Schematic of quantum dissipative dynamics for two impurities coupled to a reservoir. Spin-exchange interaction (wavy line) and Kondo clouds (gray regions) are shown, where cloud extents indicate Kondo correlation strengths. Time evolution of (b) $\dot{S}_{12}$ and (c) $E_{\rm hyb}$ at various temperatures. NQS-DQME results (solid lines, vertically offset for clarity) are benchmarked against HEOM reference values. Inset in (c): $\Delta E_{\rm hyb} = E_{\rm hyb}(t_{\rm lo}) -E_{\rm hyb}(t_0)$ versus inverse temperature. The increase in $\Delta E_{\rm hyb}$ with decreasing temperature indicates weakened Kondo correlations in the long-time limit. System parameters (in units of $\Gamma$): $\epsilon_{0} = -U_0/2 =-6$, and $J=8$; see the SM for details SM.