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On linearity or non-linearity in machine learning for quantum chaotic dynamics
Francesco Perciavalle, Agostino Gallo, Francesco Plastina, Gianluigi Greco, Nicola Lo Gullo, Carlo Adornetto
TL;DR
Classical simulation of chaotic quantum many-body dynamics is challenging, prompting the question of whether forecasting observable dynamics requires equally complex models. The paper compares DLinear and a nonlinear Transformer on a PXP chain across initial states spanning ergodic and scarred regimes. DLinear remains accurate across that family while Transformer performance worsens toward the scarred limit, showing that a linear mapping can forecast the observables considered.
Problem
Classical simulation is challenged by entanglement growth, and it remains an open question whether the complexity of quantum evolution translates into complexity required to forecast its observables.
Source: §1 · pp. 1–2Method
The paper compares DLinear and a nonlinear Transformer on PXP-chain observable forecasts across initial states spanning ergodic and scarred regimes.
Source: §1 · pp. 2–3Results
DLinear remains accurate across the initial-state family, while Transformer performance deteriorates toward the scarred limit; DLinear’s main deviations are small high-frequency oscillations.
Source: §1 · p. 3, §5 · p. 11Takeaways & Limitations
The observables studied can be forecast accurately by a linear mapping from finite past observations, without implying that the underlying quantum dynamics are linear.
Source: §5 · p. 11The analysis is limited to a finite PXP chain, a specific initial-state family, selected observables, and selected forecasting horizons.
Source: §5 · p. 11
Abstract
from arXiv · showhide
Accurately simulating chaotic quantum many-body dynamics remains a major computational challenge for classical methods, due to the rapid buildup and spatial spreading of entanglement during the evolution. This raises the question of whether machine learning can provide an effective alternative for predicting quantum dynamics. We address this question by formulating quantum dynamics as a time-series forecasting problem, using a few-qubit PXP chain, realizable with Rydberg-atom arrays, as a benchmark. By varying the initial state, the system spans dynamical regimes ranging from ergodic behavior to quantum many-body scarring, providing a controlled setting for testing forecasting models across qualitatively different dynamics. We compare two contrasting architectures: an expressive nonlinear Transformer and DLinear, a simple linear forecasting model. The Transformer accurately predicts dynamics in the more ergodic regime, but its performance progressively deteriorates as the initial state approaches the scarred limit. In contrast, DLinear remains accurate across the entire family of initial states, with its main deviations consisting of small high-frequency oscillations that have little effect on the overall prediction error. Remarkably, these results show that observables generated by complex quantum many-body dynamics can be forecast with high accuracy through a simple linear mapping from past to future observations. This reveals that the complexity of the underlying quantum evolution need not translate into an equally complex forecasting problem.
1 Introduction
The paper asks whether quantum-many-body forecast complexity must match the complexity of the underlying dynamics, given the difficulty of classical simulation as entanglement spreads. It compares linear and nonlinear forecasting on PXP-chain observables across initial states spanning different dynamical regimes.
1 Introduction
- Classical simulation is challenged by exponentially growing Hilbert spaces and entanglement spreading, which increases the resources needed for accurate MPS representations and limits simulation time.Exact diagonalization is limited by system size, while tensor-network methods are limited by accessible evolution time.Source: §1 · p. 1
- The study asks whether complex quantum evolution requires an equally complex model to forecast its observable dynamics.The forecasting formulation uses finite histories of observations to predict subsequent evolution without explicitly propagating the full many-body state.Source: §1 · pp. 1–2
- The comparison uses a few-qubit PXP chain and initial states spanning ergodic to scarred regimes, contrasting a nonlinear Transformer with DLinear.The PXP model describes interacting Rydberg atoms in the blockade regime, and the initial-state family varies the dynamics while keeping the Hamiltonian fixed.Source: §1 · p. 2
- DLinear predicts accurately across the initial-state family, while Transformer performance progressively deteriorates toward the quantum-scarred limit.For the observables considered, the result shows accurate forecasting without an expressive nonlinear architecture.Source: §1 · pp. 2–3
2 The Physical Problem
The PXP chain exhibits distinct dynamical regimes depending on its initial state, including ergodic behavior and scar-associated coherent revivals. The study constructs an initial-state family and observable time series to examine these regimes.
2 The Physical Problem
- The PXP model combines predominantly chaotic dynamics with a small nonthermal quantum many-body scar subspace; the fully polarized state is ergodic, while the Néel state exhibits long-lived revivals.The model is an effective description of interacting Rydberg atoms in the blockade regime [26, 27, 28, 29, 30, 31, 32]; the revivals are associated with overlap with the scar subspace.Source: §2 · p. 3
- The initial-state family smoothly interpolates between Néel and fully polarized configurations, with intermediate λ values introducing entangled initial states.The staggered product states are combined using parameters λ and θ, with specific parameter choices recovering the Néel and fully polarized limits.Source: §2 · p. 3
- Initial states are sampled in parameter space and evolved under the PXP Hamiltonian to generate observable trajectories for learning and predicting unseen dynamics.The framework aims to predict previously unseen trajectories across behaviors ranging from chaotic to quantum-scarred regimes.Source: §2 · pp. 3–4
- The observables are local magnetizations and connected two-point correlations, giving F = L(L + 1)/2 features; for L = 10 qubits, F = 55.The feature vector concatenates single-site magnetizations and connected pairwise correlators.Source: §2 · p. 4
3 Methods
The methods cast prediction of quantum observables as forecasting from a finite history, comparing direct linear and autoregressive models on held-out trajectories. Both use a temporally informed training objective and are evaluated with complementary measures of value error and temporal direction.
3.1 Formal Definitions
- Each trajectory is split into an input history of ρ time steps and a target horizon of τ subsequent steps, and a learned mapping predicts the future from the past.Training uses some trajectories, with the remaining unseen trajectories reserved for evaluation.Source: §3.1 · p. 4
3.2 Models
- DLinear predicts the whole future horizon in one pass, while the Transformer generates future states autoregressively; both are trained with a composite objective that also encourages temporal consistency.The objective penalizes discrepancies in predicted states, increments, and their directions.Source: §3.2 · p. 4, §3.3 · p. 5
3.2.1 DLinear
- DLinear decomposes the input into trend and seasonal components, applies separate linear layers, and combines their outputs to forecast the full horizon directly.Its direct multi-step design avoids recursive prediction and the associated error accumulation and exposure bias.Source: §3.2.1 · p. 4
3.2.2 Transformer
- The Transformer is trained to predict the next step and then feeds its own predictions back through a sliding window to generate the forecast.This autoregressive procedure can accumulate errors and exposes a training–inference mismatch.Source: §3.2.2 · p. 5
3.3 Numerical Experiments
- The benchmark uses 500 trajectories of 800 time steps and 55 observables, with 400 trajectories for training and validation and 100 held out for testing.Models use the first 500 steps as input to forecast the next 300.Source: §3.3 · p. 5
3.3.1 Training & Instantiations
- Both architectures use the same temporally informed objective and evaluation protocol, while model-specific optimization settings are selected independently.The objective combines reconstruction with penalties on temporal increments and directional disagreement, without explicitly imposing the equations of motion.Source: §3.3.1 · pp. 5–6
3.3.2 Evaluation Metrics
- Mean absolute error measures average absolute deviation, while directional accuracy measures whether predicted and true observable increments have matching signs.Directional accuracy is reported as a percentage; 100% means every feature and time step has the correct direction.Source: §3.3.2 · p. 6
- Small spurious high-frequency oscillations can lower directional accuracy despite a low mean absolute error.The two metrics can therefore give different assessments of a prediction’s quality.Source: §3.3.2 · pp. 6–7
4 Results
DLinear maintains low error across the initial-state family, while Transformer accuracy worsens toward the quantum-scarred regime. The error analyses show that DLinear’s main deviations are small and rapidly varying, whereas Transformer errors can be larger and grow over time.
Trajectory-resolved forecasting performance
- For DLinear, directional accuracy exceeds 90% across much of parameter space at intermediate smoothing windows w ∈[20, 60], while excessive smoothing suppresses relevant dynamics.The Transformer remains less sensitive to smoothing, and smoothing does not remove its loss of trend accuracy in the unfavorable regime.Source: Trajectory-resolved forecasting… · p. 8
- DLinear maintains low MAE across the initial-state family, while Transformer performance progressively deteriorates toward the quantum-scarred regime.The raw DLinear directional-accuracy loss largely reflects small, high-frequency oscillations; smoothing recovers the underlying trend, whereas Transformer trend accuracy remains poor in the unfavorable regime.Source: Trajectory-resolved forecasting… · p. 8
Feature- and time-resolved prediction errors
- DLinear errors remain comparatively small in the representative trajectories, with rapid fluctuations that can lower raw directional accuracy despite close magnitude agreement.The heatmaps show absolute error over time for the first ten local-magnetization features, with columns comparing the Transformer and DLinear on a shared scale.Source: Figure 3 · p. 8, Feature- and time-resolved prediction… · p. 9
- In the worst representative trajectory, Transformer absolute error locally approaches 0.7, while DLinear errors remain substantially smaller over most of the prediction window.The representative rows are selected using a joint score based on both models’ MAE.Source: Figure 3 · p. 8, Feature- and time-resolved prediction… · p. 9
Error propagation over the forecast horizon
- DLinear MAE settles into a bounded, nearly stationary regime after an initial increase, while Transformer errors oscillate over longer timescales and grow more strongly across the forecast horizon.Larger Transformer errors near θ ≳π/4 contribute substantially to its trajectory-averaged MAE.Source: Error propagation over the forecast… · pp. 9–10, Figure 4 · p. 9
Feature-resolved prediction errors
- DLinear errors and their dispersion are smaller near the chain center and increase toward the boundaries, opposite to the Transformer’s broader distributions around central sites.The Transformer median is approximately uniform along the chain, but its upper quartiles and whiskers broaden around central sites.Source: Feature-resolved prediction errors · p. 10
- DLinear has lower mean feature-resolved MAE at every feature and generally narrower trajectory distributions, while Transformer errors are right-skewed and driven upward by a subset of initial conditions.The feature-level comparison covers the first ten local magnetizations; the boxplots show distributions across test trajectories.Source: Feature-resolved prediction errors · p. 10, Figure 5 · p. 10
5 Discussion and conclusions
DLinear predicts the PXP model’s observable dynamics accurately and comparatively uniformly across the initial-state family, while Transformer performance deteriorates toward the scarred regime. The findings show that complex many-body evolution can still admit accurate forecasting through a simple linear mapping of selected observables.
5 Discussion and conclusions
- DLinear remains accurate across the initial-state family, while Transformer performance deteriorates toward the N´eel limit; both models depend only weakly on λ.Transformer performance is comparable to DLinear nearer the homogeneous-state limit, but worsens as dynamics become more influenced by quantum many-body scarring.Source: §5 · pp. 10–11
- DLinear’s low directional accuracy on raw predictions mainly reflects small, high-frequency oscillations, while Transformer errors near the N´eel limit persist after smoothing.After filtering short-timescale components, DLinear’s directional accuracy increases substantially; feature-resolved errors also show a larger-error tail for Transformer trajectories, unlike DLinear’s more uniform errors.Source: §5 · p. 11
- Selected observable time series can be forecast accurately by mapping a finite window of past observations to future evolution, without implying that the underlying quantum dynamics are linear.The stated scope is the observables and time scales considered in this work.Source: §5 · p. 11
- Transformer expressiveness provides no systematic advantage here, and the relative performance of linear and nonlinear models can depend on the dynamical regime.Transformer degradation toward the quantum-scarred regime remains visible after smoothing, despite coherent and recurrent temporal structures there.Source: §5 · p. 11
- The analysis is limited to a finite PXP chain, one family of initial states, selected observables, and forecasting horizons, leaving broader generality open.The authors identify other Hamiltonians, digital quantum circuits, and Markovian versus non-Markovian dynamics as directions for further study.Source: §5 · p. 11