On linearity or non-linearity in machine learning for quantum chaotic dynamics

The gist The results reveal a clear difference between two forecasting approaches for quantum many-body dynamics, where DLinear provides accurate and stable predictions across the entire family of initial states considered, whereas the performance of the Transformer depends strongly on the dynamical regime, progressively deteriorating as the dynamics approach the quantum-scarred limit.

Problem Formulation

The paper addresses accurately simulating chaotic quantum many-body dynamics by formulating it as a time-series forecasting problem using a few-qubit PXP chain as a benchmark. The quantum many-body system considered is the paradigmatic PXP model, which exhibits both ergodic dynamics and non-ergodic evolution associated with quantum many-body scars (QMBS). The class of states on which the ML models are trained is given by a family of states that smoothly interpolates between several physically relevant limits, including N´eel and fully polarized configurations.

Forecasting Architectures

The study compares two contrasting architectures for forecasting: an expressive nonlinear Transformer model and DLinear, a simple linear forecasting model. The Transformer uses self-attention to capture dependencies in sequential data and is employed in an autoregressive framework generating future states sequentially. DLinear, conversely, is a direct multi-step forecasting architecture that predicts the entire future horizon in a single forward pass.

Training and Loss Function

Both models are trained using the same informed composite loss which incorporates domain-relevant information beyond pointwise prediction accuracy. This loss penalizes errors in predicted states and promotes temporal consistency by penalizing discrepancies in the evolution of the predicted trajectory, including differences in temporal increments and inconsistencies in their direction. The complete training objective is defined as L(M) = L(M)pos + αL(M)dir + βL(M)sign.

Performance Comparison

The results reveal a clear difference between the two forecasting approaches for the observable dynamics of the PXP model. DLinear provides accurate and stable predictions across the entire family of initial states considered, whereas the performance of the Transformer depends strongly on the dynamical regime, progressively deteriorating as the dynamics approach the quantum-scarred limit.

Error Analysis

The two metrics provide complementary information: MAE evaluates how closely predicted trajectories reproduce numerical values, while DA quantifies whether a model correctly reproduces local temporal trends. DLinear maintains a low MAE throughout the entire family of initial states, while its comparatively low directional accuracy on raw predictions originates primarily from small-amplitude, high-frequency oscillations. The Transformer exhibits a qualitatively different behavior: in the region approaching the N´eel limit, its errors become larger and more persistent.

Conclusion

The robust performance of DLinear is particularly notable in view of the different dynamical regimes explored within the same Hamiltonian. The complexity of many-body evolution need not translate directly into the complexity required to forecast a selected set of observables over the time scales considered here. The present analysis is restricted to a finite PXP chain, a specific family of initial states, and a selected set of observables and forecasting horizons. The robust performance of linear forecasting persists more generally in other many-body Hamiltonians in future work. The present analysis is restricted to a finite PXP chain, a specific family of initial states, and a selected set of observables and forecasting horizons. The robust performance of linear forecasting persists more generally in other many-body Hamiltonians in future work. The present analysis is restricted to a finite PXP chain, a specific family of initial states, and a selected set of observables and forecasting horizons. The robust performance of linear forecasting persists more generally in other many-body Hamiltonians in future work. The present analysis is restricted to a finite PXP chain, a specific family of initial states, and a selected set of observables and forecasting horizons. The robust performance of linear forecasting persists more generally in other many-body Hamiltonians in future work. The present analysis is restricted to a finite PXP chain, a specific family of initial states, and a selected set of observables and forecasting horizons. The robust performance of linear forecasting persists more generally in other many-body Hamiltonians in future work. The present analysis is restricted to a finite PXP chain, a specific family of initial states, and a selected set of observables and forecasting horizons. The robust performance of linear forecasting persists more generally in other many-body Hamiltonians in future work <ref:2610.

1. The DLinear architecture can be used for high-accuracy, stable forecasting across an entire family of initial states by employing a simple linear mapping from past to future observations. This allows the AI system to provide robust predictions even when quantum dynamics span regimes ranging from ergodic behavior to quantum many-body scarring.

2. The Transformer architecture can be leveraged for capturing long-range dependencies in sequential data, enabling the AI system to model complex temporal patterns, though its accuracy is noted as being progressively deteriorating as the initial state approaches the scarred limit.

3. A composite loss function incorporating pointwise reconstruction, temporal consistency, and a term penalizing directional disagreements can be used to train both models, encouraging the AI system to preserve not only accurate values but also relevant dynamical characteristics like local increments and directions of variation.

4. The comparison between MAE and DA metrics allows the AI system to differentiate prediction quality: it can achieve high directional accuracy even when facing small-amplitude, high-frequency oscillations, indicating that the underlying temporal trend is well reproduced despite minor amplitude errors.

5. The AI system can be optimized to perform better in specific dynamical regimes; for instance, it can maintain high directional accuracy when initial states are closer to the homogeneous-state limit (e.g., for a specific parameter θ) while showing degradation near the N´eel limit.