Linear DLinear forecasts PXP-chain observables stably across the whole initial-state family, while the nonlinear Transformer degrades progressively toward the Néel (scarred) limit

With the PXP Hamiltonian held fixed, the authors compare a direct multi-step linear DLinear against an autoregressive nonlinear Transformer on the same forecasting task, using a family of initial states randomly sampled in parameter space that continuously connects the ergodic regime with the Néel limit. Prior machine-learning studies of quantum dynamics often focus on a single architecture or a single dynamical region; here linear and nonlinear architectures are contrasted under the same initial-state family, input window, forecasting horizon, and composite loss, so that architectural differences and dynamical-regime differences can be observed separately. Training and test data are split at the trajectory level so that different time steps of the same physical trajectory never appear in both sets; preprocessing is fitted only on training trajectories; the two models share preprocessing, input window, forecasting horizon, loss weights, and evaluation protocol, with model-specific optimization settings chosen independently.

DLinear keeps a low and approximately uniform MAE across the initial-state parameter space, and its comparatively low directional accuracy on raw predictions originates mainly from small-amplitude high-frequency oscillations; after Savitzky–Golay smoothing, directional accuracy rises above 0.9 over a broad region of parameter space, indicating that the underlying temporal trend is nevertheless well reproduced. This suggests that a drop in raw directional accuracy need not mean the trend forecast has failed, and that amplitude error and directional error should be read separately; smoothing is used only as a post-processing analysis tool and does not affect training or the raw model predictions. The conclusion rests on two complementary metrics (MAE and directional accuracy), on feature- and time-resolved absolute-error heatmaps, and on error-versus-forecast-time curves; DLinear's error settles into a bounded, approximately stationary regime after an initial transient, without a clear systematic increase.

The Transformer's performance depends strongly on the initial-state parameter: it is comparable to DLinear closer to the homogeneous-state limit but shows larger and more persistent errors as initial states approach the Néel limit, and smoothing does not remove this degradation, indicating that it increasingly fails to reproduce the temporal trend rather than merely superimposing high-frequency fluctuations. This provides a quantum many-body counterpoint to the assumption that greater expressive power in a nonlinear architecture is necessarily advantageous, echoing observations in classical time-series forecasting where simple linear models compete with, and in some cases outperform, more sophisticated nonlinear architectures. Feature-resolved error distributions show the Transformer's mean lying systematically above its median, a right-skewed tail driven by a subset of initial conditions; its error peaks grow progressively larger at later prediction times, giving substantially stronger error growth than DLinear.

The authors stress that forecasting observable time series with high accuracy through a simple linear mapping does not imply that the underlying quantum dynamics are themselves linear; rather, the complexity of the many-body evolution need not translate directly into the complexity of the mapping required to forecast a selected set of observables over the time scales considered. This separates the complexity of quantum dynamics from the complexity needed to forecast its observable signatures, and offers a framework for testing that relation in other Hamiltonians, digital quantum circuits, and Markovian versus non-Markovian dynamics. The judgment is grounded in results for a finite PXP chain, a specific initial-state family, a selected set of observables, and the forecasting horizons considered here; the authors frame it as an interpretation of this setting rather than a claim about quantum dynamics in general.

The result applies to forecasting selected observables from a finite observation history: the input is a time window of local magnetizations and connected two-point correlators, the output is their subsequent evolution, and training and test data are generated by numerically exact evolution of a few-qubit PXP chain. The authors state explicitly that the framework is not tied to a specific Hamiltonian and can be extended to other quantum many-body systems and to digital quantum circuits whenever representative time-series data are available; they propose testing whether the robust performance of linear forecasting holds more generally in other many-body Hamiltonians with qualitatively different ergodic and non-ergodic regimes, in digital quantum circuits, and in Markovian versus non-Markovian dynamics. For a reader, this means that when the task is extrapolating observables from past observations, a linear baseline deserves to be the first comparison, and any benefit from a nonlinear architecture should be assessed within the specific dynamical regime.

The authors restrict their conclusions to a finite PXP chain, a specific family of initial states, a selected set of observables, and the forecasting horizons considered, so whether the observed behavior extends to larger many-body systems, other Hamiltonians, or longer horizons remains an open question. Directional accuracy is sensitive to small-amplitude high-frequency oscillations, while an excessively large smoothing window can also suppress genuine dynamical features, so smoothed results must be read jointly with MAE; the authors also note that high directional accuracy is a necessary but not sufficient condition for accurate reconstruction. In addition, the Transformer's larger mean error is driven by a subset of initial conditions, with a right-skewed error distribution, so trajectory-averaged metrics can mask differences between regions. The original presents specific numerical values in equations and figures, so some quantitative details (such as exact error values per region and hyperparameter values) cannot be fully verified from the prose alone.