From short-sighted to far-sighted: A comparative study of recursive machine learning approaches for open quantum systems

Open quantum systems describe quantum systems interacting with their environment, playing a fundamental role in quantum computing, quantum memory, quantum transport, proton tunneling in DNA and energy transfer in photosynthesis.[1, 2, 3, 4, 5] Their dynamics are captured by the reduced density matrix (RDM), which evolves under both the system internal dynamics and the influence of its environment.

Modeling the influence of environment is challenging due to its high-dimensional nature. Mixed quantum-classical methods[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] simplify the problem by treating the system quantum mechanically while approximating the environment classically, significantly reducing computational cost. However, these methods often struggle to capture detailed balance[18, 19, 20] or subtle quantum correlations.[21] Fully quantum approaches, including path-integral[22, 23, 24, 25, 26, 27, 28, 29, 23, 30] and quantum master equation-based methods,[31, 32, 33, 34, 35, 36, 37, 38] provide more accurate descriptions but are computationally expensive, particularly in regimes with strong system-environment coupling or where fine discretization is needed for numerical stability.

Recently, machine learning (ML) has emerged as a promising tool for learning complex spatiotemporal dynamics in high-dimensional systems.[39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] One widely used ML strategy is the recursive approach, where the future evolution of a quantum state is predicted iteratively based on a short history of past evolution. This method has been successfully applied to the relaxation dynamics of the two-state spin-boson (SB) model,[40, 43, 44, 60] even enabling extrapolation beyond the trained time window.[40] However, previous applications have been limited to predicting a single observable—such as the population difference in the SB model—and have relied solely on single-step prediction models.

In this work, we extend recursive ML approaches to more complex quantum systems, focusing on predicting the full RDM rather than just a single observable. We examine four physics-informed neural network (PINN)-based architectures: (i) the single-RDM-predicting PINN (SR-PINN), (ii) the SR-PINN with simulation parameters (PSR-PINN), (iii) the multi-RDMs-predicting PINN (MR-PINN), and (iv) the MR-PINN with simulation parameters (PMR-PINN). These architectures are tested on the relaxation dynamics of the SB model and the exciton energy transfer (EET) process in the Fenna-Matthews-Olson (FMO) complex.

From our results, we underscore the limitations of short-sighted, single-RDM-predicting models (SR-PINN and PSR-PINN) in capturing long-term system dynamics, especially in systems with intricate behavior. These models, constrained by a narrow history window, fail to predict long-term quantum evolution accurately, as they cannot fully capture the complexity of system evolution. In contrast, far-sighted models—such as MR-PINN and PMR-PINN—overcome these limitations by extending the forecast horizon, allowing them to incorporate long-range temporal correlations and achieve more stable predictions.

Although we initially anticipated that incorporating simulation parameters such as reorganization energy ($\lambda$), characteristic frequency ($\gamma$), and temperature ($T$) would improve accuracy, our findings show that these parameters do not consistently enhance performance and, in some instances, actually degrade it. This suggests that the relevant effects of these parameters are already implicitly encoded in the RDM evolution, making their explicit inclusion unnecessary in certain cases.

To build our case, let’s consider an open quantum system (S), consisting of $\ignorespaces n$ states interacting with an external environment (E). As stated before, the dynamics of the system is governed by the RDM, which evolves non-unitarily due to environmental effects. While the full system follows unitary evolution described by the Liouville–von Neumann equation, tracing out the environmental degrees of freedom introduces a superoperator $\mathcal{\mathbf{R}}$ that encodes dissipation and decoherence. Under the assumption that the initial state is separable between the system and environment ($\mathbf{\rho}(0)=\mathbf{\rho}_{\rm S}(0)\otimes\mathbf{\rho}_{\rm E}(0)$), mathematically it can be described as

where $\mathbf{\rho}_{\rm S}(t)$ is the RDM of the system at time $t$, $\mathbf{Tr}_{\rm E}$ denotes the partial trace over the environment, $\mathcal{\mathbf{R}}$ is a superoperator that encodes the effects of the environment and $\mathbf{U}(t,0)$ and $\mathbf{U}^{\dagger}(t,0)$ are the forward and backward time-evolution operators, respectively.

In the recursive ML framework, modeling the time evolution of Eq. (From short-sighted to far-sighted: A comparative study of recursive machine learning approaches for open quantum systems) is formulated as learning a mapping function $\ignorespaces\mathcal{M}$ that maps the input descriptors into predicted RDMs. In general, we have

where $\ignorespaces\{\mathbb{R}^{n\times n}\}^{k^{\prime}}$ is a collection of $\ignorespaces k^{\prime}$ input matrices (of size $\ignorespaces n\times n$ ) that encode physical information such as historical RDM data, initial conditions, and simulation parameters, and $\ignorespaces\{\mathbb{R}^{n\times n}\}^{l}$ is a sequence of $\ignorespaces l$ predicted RDMs corresponding to different time steps. In our study, we consider four distinct approaches for predicting the time evolution of RDM:

The procedure is applied iteratively: after predicting $\ignorespaces\rho_{\rm S}(t_{k+1})$ , this new RDM is appended to the history while the oldest entry is removed, keeping the memory size constant at $\ignorespaces k^{\prime}$ .

As with the standard SR-PINN, the process is applied recursively with a fixed history length.

with

In this case, the model outputs $N_{f}$ future RDMs simultaneously, thus providing a multi-step prediction without requiring iterative updating.

By integrating the simulation parameters $\ignorespaces\mathbf{p}$ , the model can adjust its predictions to account for different physical conditions while forecasting multiple future time steps concurrently.

Each of these approaches leverages the past history of RDMs (and optionally simulation parameters) to predict the future dynamics of the system, differing primarily in whether they predict a single RDM or multiple RDMs in one go.

To evaluate the proposed methods, we analyze the relaxation dynamics of the SB model and the EET process in the FMO complex (see Methods section for details). The models are implemented using a hybrid deep learning architecture that integrates convolutional neural networks (CNNs) with long short-term memory (LSTM) layers, followed by fully connected dense layers (CNN-LSTM). Following the approach outlined in Ref. 39, training is optimized using a composite loss function, expressed as:

where each loss term is defined as follows.

The first term, $\ignorespaces\mathcal{L}_{1}$ , represents the mean squared error (MSE) between the predicted elements of the RDM, $\ignorespaces\mathbf{\rho}_{\text{S}}$ , and the reference values, $\ignorespaces\tilde{\mathbf{\rho}}_{\text{S}}$ :

Here, $\ignorespaces N_{t}$ denotes the number of time steps.

To ensure trace conservation of the density matrix, the second loss term, $\ignorespaces\mathcal{L}_{2}$ , penalizes deviations of the trace from unity:

The third term, $\ignorespaces\mathcal{L}_{3}$ , enforces positive semi-definiteness by penalizing negative eigenvalues $\ignorespaces\mu_{i}(t)$ of the density matrix:

Additionally, $\ignorespaces\mathcal{L}_{4}$ ensures that all eigenvalues remain within the valid range $\ignorespaces[0,1]$ , enforcing a key physical constraint of the RDM:

The clipping function used here is defined as:

The weighting coefficients $\ignorespaces\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ control the relative contributions of these loss terms. In our case, we set them all to unity ( $\ignorespaces\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=1.0$ ). Collectively, these loss components ensure that the predicted RDM satisfies key physical properties: accuracy ( $\ignorespaces\mathcal{L}_{1}$ ), trace conservation ( $\ignorespaces\mathcal{L}_{2}$ ), positive semi-definiteness ( $\ignorespaces\mathcal{L}_{3}$ ), and eigenvalue constraints ( $\ignorespaces\mathcal{L}_{4}$ ).

For demonstration, we use data from the publicly available QD3SET-1 database[62] for both the SB model and the FMO complex. The models are trained on 80% of the simulations, with the remaining 20% reserved for testing. Further details on the dataset and training process can be found in the Methods section.

Figure 1 presents the predictive performance of all the four models for the time evolution of RDM elements in both symmetric and asymmetric SB models. Each model is provided with an initial short-time seed ( $\ignorespaces 4/\Delta$ ) and tasked with recursively forecasting the system’s future evolution.

The results highlight the limitations of SR-PINN, which exhibits significant errors in both diagonal and off-diagonal terms (population and coherence), leading to a rapid divergence from the expected dynamics. PSR-PINN, despite incorporating simulation parameters, further degrades accuracy, indicating that the past history window of the model remains insufficient for stable recursive predictions. In contrast, MR-PINN, which leverages a longer forecasting horizon, effectively mitigates error accumulation and successfully captures both population and coherence dynamics across the prediction window. PMR-PINN performs similarly to MR-PINN, suggesting that the inclusion of simulation parameters does not provide additional benefits in this setting.

To test a larger system, in Figure 2 we showcase the predicted evolution of RDM elements for the FMO complex under initial excitation on site-1. The models are trained with an initial short-time seed (0.2 ps) and recursively predict the system’s future dynamics.

SR-PINN exhibits considerable inaccuracies, particularly in long-term dynamics, leading to deviations from the expected population transfer trends. PSR-PINN, despite integrating simulation parameters, fails to improve performance and even amplifies errors, especially in diagonal elements. As in the SB model, MR-PINN achieves significantly enhanced accuracy, demonstrating robust predictions of both energy transfer and coherence decay. PMR-PINN yields results comparable to MR-PINN, reinforcing the observation that the longer forecasting window is the primary factor driving predictive stability.

A quantitative analysis of model performance, summarized in the accompanying table (Table 1), further substantiates these findings. As in the SB model, SR-PINN struggles to maintain accuracy, particularly for coherence elements, and PSR-PINN further aggravates errors. MR-PINN consistently outperforms both single-time-step approaches, achieving the lowest mean absolute errors (MAE) across all RDM elements. The inclusion of simulation parameters in PMR-PINN does not lead to meaningful improvements over MR-PINN.

Notably, the errors are higher for the asymmetric SB model compared to the symmetric case, indicating that prediction accuracy degrades as system complexity increases. This trend suggests that more intricate dynamical behaviors impose additional challenges for PINN-based models, particularly when using single-RDM-predicting training strategies.

For the FMO complex, the disparity between methods remains evident. SR-PINN and PSR-PINN perform poorly, with PSR-PINN producing the highest errors for both population and coherence terms, particularly when the initial excitation occurs at site-6 (as shown in the table). MR-PINN and PMR-PINN provide a marked improvement, although predictive errors remain higher compared to the SB model, reflecting the increased complexity of the system. The trend observed in the SB model, where errors increase with system complexity, is also evident in the FMO complex. The lack of significant gains from PMR-PINN over MR-PINN suggests that the longer prediction window is the dominant factor in improving accuracy, while the inclusion of simulation parameters has a limited effect.

In summary, this work investigated four PINN-based architectures for predicting the time evolution of the RDM in selected open quantum systems: single-RDM-predicting models (SR-PINN, PSR-PINN) and multi-RDMs-predicting models (MR-PINN, PMR-PINN). These models use historical RDM data to predict future dynamics, with some incorporating environment-specific parameters such as temperature and system-bath coupling.

Our findings reveal the limitations of single-RDM-predicting models (SR-PINN and PSR-PINN) in capturing long-term quantum dynamics. These models, constrained by a narrow history window, struggle to predict long-term quantum evolution accurately, as they fail to capture the full complexity of system evolution. However, as demonstrated in previous works, [40, 43, 44, 60] when trained on simpler observables—such as the population difference in the spin-boson model—they yield reasonable predictions, suggesting that single-variable evolution is easier to propagate recursively.

In contrast, multi-RDMs-predicting models (MR-PINN and PMR-PINN) consistently provide stable and accurate long-term predictions across various scenarios. By predicting multiple RDMs in one step, these models mitigate cumulative errors and better capture long-range temporal correlations, improving their ability to generalize to unseen conditions. This emphasizes that extending the forecast horizon is more effective than merely increasing the historical input length, as explicitly forecasting future states stabilizes predictions more effectively than relying solely on past dynamics.

Surprisingly, explicitly incorporating simulation parameters—such as reorganization energy, characteristic frequency, and temperature (in PSR-PINN and PMR-PINN)—did not consistently improve predictive accuracy and, in some cases, slightly reduced performance. This suggests that the effects of these parameters are already implicitly captured in the RDM evolution, rendering their explicit inclusion redundant and potentially introducing unnecessary complexity.

Overall, this work underscores the limitations of short-sighted, single-step recursive models in complex quantum systems and reinforces the advantages of far-sighted, multi-step approaches for robust, long-term predictions. Our findings highlight that incorporating a longer predictive horizon is key to improving prediction stability, capturing complex dynamics, and reducing the impact of short-term fluctuations in open quantum systems.