Toward Quantum-Aware Machine Learning: Improved Prediction of Quantum Dissipative Dynamics via Complex Valued Neural Networks

TL;DR

It is demonstrated that CVNNs outperform RVNNs in convergence speed, training stability, and physical fidelity -- including significantly improved trace conservation and Hermiticity, establishing CVNNs as a robust, scalable, quantum-aware classical approach for simulating open quantum systems in the pre-fault-tolerant quantum era.

Abstract

Accurately modeling quantum dissipative dynamics remains challenging due to environmental complexity and non-Markovian memory effects. Although machine learning provides a promising alternative to conventional simulation techniques, most existing models employ real-valued neural networks (RVNNs) that inherently mismatch the complex-valued nature of quantum mechanics. By decoupling the real and imaginary parts of the density matrix, RVNNs can obscure essential amplitude-phase correlations, compromising physical consistency. Here, we introduce complex-valued neural networks (CVNNs) as a physics-consistent framework for learning quantum dissipative dynamics. CVNNs operate directly on complex-valued inputs, preserve the algebraic structure of quantum states, and naturally encode quantum coherences. Through numerical benchmarks on the spin-boson model and several variants of the Fenna-Matthews-Olson complex, we demonstrate that CVNNs outperform RVNNs in convergence speed, training stability, and physical fidelity -- including significantly improved trace conservation and Hermiticity. These advantages increase with system size and coherence complexity, establishing CVNNs as a robust, scalable, quantum-aware classical approach for simulating open quantum systems in the pre-fault-tolerant quantum era.

Figures (5)

• Figure 1: A schematic flowchart illustrating the general training process for both CVNN and RVNN models.
• Figure 2: A comparison of training and validation loss curves versus epoch number for the CVNN and RVNN models. Results are shown for the SB model and all three prototypes of the FMO complex.
• Figure 3: Comparison of trace conservation in predicted dissipative dynamics. Results for the CVNN (left) and RVNN (right) are shown for, from top to bottom, the SB model and the 4-site, 7-site, and 8-site FMO complexes (site-1). All predictions correspond to trajectories not seen during training. Parameters are: SB model—$\varepsilon/\Delta=0.0$, $\gamma/\Delta=9.0$, $\lambda/\Delta=6.0$, $\beta\Delta=1.0$; FMO complexes—4-site ($\gamma=250~\mathrm{cm}^{-1}$, $\lambda=70~\mathrm{cm}^{-1}$, $T=130~\mathrm{K}$), 7-site ($\gamma=350~\mathrm{cm}^{-1}$, $\lambda=70~\mathrm{cm}^{-1}$, $T=30~\mathrm{K}$), and 8-site ($\gamma=400~\mathrm{cm}^{-1}$, $\lambda=250~\mathrm{cm}^{-1}$, $T=30~\mathrm{K}$).
• Figure 4: Evaluation of positive semi-definiteness of predicted RDMs via eigenvalue spectra for CVNN and RVNN models across all considered systems. Sparse positive eigenvalues are shown as blue dots for clarity, while negative eigenvalues are indicated in red, with their counts labeled for each model and system. Simulation parameters are consistent with those in Fig. \ref{['fig:trace']}.
• Figure 5: Predicted time evolution of RDM elements. Panels (A, C, E, G) show CVNN results and panels (B, D, F, H) show RVNN results for the following systems: (A, B) SB model (populations and coherences); (C, D) populations of the 4-site FMO complex; (E, F) populations of the 7-site FMO complex; (G, H) populations of the 8-site FMO complex. Reference data are shown as dots. A vertical dashed line in each panel separates the provided input dynamics from the predicted dynamics. Coherence evolution for the FMO complexes is provided in the Supporting Information. Simulation parameters match those in Fig. \ref{['fig:trace']}.