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Digital Twin Modeling of Quantum Dynamical Systems: Dissipative Quantum Reservoir Computing

Abhijit Sen, Bikram Keshari Parida, Shital Chauhan, Mahima Arya, Denys I. Bondar

arXiv:2609.36901v1quant-phcs.AI

TL;DR

Driven many-body quantum responses are nonlinear, history dependent, and expensive to simulate, and black-box forecasting can fail beyond the training series. This work uses a fixed dissipative quantum reservoir with a trained readout as a digital twin; a single-qubit reservoir reproduces the response and generalizes across a 20× variation in drive strength.

  • Problem

    Driven many-body quantum dynamics are nonlinear, history dependent, and costly to simulate as system size grows, while black-box models can fail beyond their training series.

    Source: §I · p. 1

  • Method

    DQRC uses a fixed open quantum reservoir with engineered Lindblad dissipation and trains only a classical readout to learn input–output response.

    Source: §I · pp. 1–2

  • Results

    A single fixed reservoir generalizes across a 20× variation in drive strength, with R2 = 0.9997 in the single-amplitude regime and R2 = 0.9709 in the multi-amplitude regime.

    Source: E. Discussion · p. 11

  • Takeaways & Limitations

    The reported results show that a single-qubit dissipative reservoir can reproduce a driven many-body response while keeping the dynamical substrate fixed and training only a lightweight readout.

    Source: §V · p. 11

    At the lowest tested frequencies, the fixed dissipation rate leaves the longest-memory trajectories marginally underrepresented, though this effect is small and tunable.

    Source: B. Intermediate drive regime (F0 = 20) · p. 8

Abstract

from arXiv · show

Modeling the response of driven many-body quantum systems from input--output data is difficult: the dynamics are nonlinear, history dependent, and expensive to simulate as system size grows. A paradigmatic case is High-Harmonic Generation~(HHG), where a strong field drives a medium to emit radiation that is highly sensitive to the drive and encodes long-range temporal correlations. We introduce a dissipative quantum reservoir computing~(DQRC) framework that builds a digital twin of such a system, learning its input--output map directly from data while the reservoir---itself a small open quantum system---stays fixed and only a classical readout is trained. We show that a minimal single-qubit reservoir reproduces the HHG response of a substantially larger Ising spin chain, and on a representative benchmark matches and on several metrics surpasses previously reported temporal convolutional and Kolmogorov--Arnold-network models, while using a simpler, physically realizable system. A single fixed reservoir further generalizes across a broad range of drives, indicating that it learns a shared physical response structure rather than memorizing trajectories. These results establish dissipative quantum reservoirs as compact, physically grounded digital twins for nonlinear, memory-dependent quantum dynamics. Code is available at \href{https://github.com/AI-and-Quantum-Computing/DQuRC}{https://github.com/AI-and-Quantum-Computing/DQuRC}.

I. INTRODUCTION

Driven many-body quantum responses are nonlinear, history dependent, and costly to simulate, while black-box models may fail beyond their training series. The paper proposes dissipative quantum reservoir computing as a fixed open-system surrogate with a trained readout, using engineered dissipation to tune memory.

I. INTRODUCTION

  • DQRC combines machine learning with a fixed quantum reservoir to learn a system's input–output map without solving its full microscopic dynamics.Only a lightweight classical readout is trained.Source: §I · pp. 1–2
  • Driven many-body responses are nonlinear, history dependent, and increasingly expensive to simulate as system size grows.Black-box models can fail when forecasting beyond the training series.Source: §I · p. 1
  • Engineered Lindblad dissipation tunes the reservoir's memory kernel and contraction, shaping a continuous drive-to-observable transducer.The framework uses a Lindblad master equation to ensure completely positive, trace-preserving evolution.Source: §I · p. 2
  • The study isolates driven dissipative dynamics in a single-qubit reservoir, without entanglement or exponential Hilbert-space scaling.Its purpose is not to demonstrate quantum advantage.Source: §I · p. 2

II. QUANTUM RESERVOIR COMPUTING

  • Reservoir computing leaves recurrent dynamics fixed and trains a linear readout, while the reservoir's nonlinear state encodes input history with fading memory.Dissipation draws the reservoir toward an input-driven trajectory and thereby implements fading memory.Source: §II · pp. 2, 4
  • For HHG, fading memory links recent laser-pulse history to the induced dipole, while controlled dissipation can sharpen nonlinear feature mapping.The paper treats HHG as motivation and studies a minimal driven open system rather than modeling microscopic HHG processes directly.Source: §I · p. 2, §II · p. 4

III. PROBLEM FORMULATION

The study uses a hierarchy of driving protocols to separate baseline reservoir behavior from performance under richer inputs.

III. PROBLEM FORMULATION

  • Single-amplitude excitation provides a minimal-input baseline for isolating reservoir memory, sensitivity to initial conditions, and trajectory geometry.The protocol is designed to avoid confounding these properties with structure in the drive.Source: §III · p. 4
  • A subsequent multi-amplitude protocol tests whether reservoir capacity persists when input richness increases.Its performance is read against the single-amplitude baseline.Source: §III · p. 4
  • The protocol treats degradation under richer inputs as evidence of dependence on specific dynamical conditions rather than a general computational resource.This is the study's stated interpretation of the hierarchy's comparison.Source: §III · p. 4

A. Physical system and dataset construction

The benchmark maps a driven Ising-chain control field to a measured observable, contrasting a high-dimensional data-generating system with a minimal reservoir.

A. Physical system and dataset construction

  • The data-generating system defines a continuous-time map from control field h(t) to measured observable Y(t).Datasets are generated from this physical model by varying the driving field.Source: A. Physical system and dataset… · p. 4
  • The data-generating system is a ten-spin Ising chain, while the reservoir operates with a single qubit.The paper uses this contrast to separate predictive capability from Hilbert-space scaling or entanglement.Source: A. Physical system and dataset… · pp. 4–5
  • The chain is driven by monochromatic fields of the form h(t) = F0 sin(ωt).The benchmark varies the driving field.Source: A. Physical system and dataset… · p. 4
  • Each trajectory spans one period of the lowest driving frequency and is discretized into 512 time steps.Driving frequencies cover sub- and super-resonant regimes relative to the system energy gap.Source: A. Physical system and dataset… · p. 5

B. Reservoir

The reservoir is a driven dissipative quantum system, evaluated on the longitudinal-spin expectation generated by the benchmark dynamics.

B. Reservoir

  • The reservoir uses a dissipative quantum system governed by a Lindblad master equation, with the reservoir set to N = 1.The data-generating system is deliberately high-dimensional in contrast.Source: B. Reservoir · pp. 5–6
  • The target observable is the expectation value of the longitudinal spin operator.Trajectories are generated by integrating the master equation from the ground state using QuTiP's mesolve solver.Source: B. Reservoir · p. 6
  • The collapse operator drives the reservoir toward |0⟩, making initial-condition transients minimal in the stated initialization.The resulting input–output trajectories, parameterized by driving amplitude and frequency, form the fidelity benchmark.Source: B. Reservoir · p. 6

C. Preprocessing and data partitioning

Inputs and ground-truth observables are globally min–max normalized to [0,1], preserving amplitude-dependent variation across frequencies while supporting numerical stability.

C. Preprocessing and data partitioning

  • Global min–max normalization to [0,1] preserves physically meaningful amplitude variation across frequencies while maintaining numerical stability.The minimum and maximum are computed over the full dataset, not separately for each trajectory.Source: C. Preprocessing and data partitioning · p. 6

D. Reservoir embedding and readout

Each normalized input drives the dissipative reservoir to produce temporal features, which are compressed with training-set PCA and mapped to targets by a cross-validated KRR readout.

D. Reservoir embedding and readout

  • Each input yields 512 temporal features; for N = 1, the feature tensor has shape (1×512), with reservoir gain fixed at g = 5.0.The density matrix is evolved over T = 512 time steps, recording an expectation value at each step.Source: D. Reservoir embedding and readout · p. 6
  • PCA retains 15 components, fitting the projection on training features only and applying it unchanged to test features to prevent information leakage.The PCA also suppresses noise from rapid transient fluctuations.Source: D. Reservoir embedding and readout · p. 6
  • The PCA-compressed features train an RBF-kernel KRR readout, with regularization and bandwidth selected by five-fold cross-validation before held-out evaluation.The readout maps reservoir-generated dynamical features nonlinearly to the target observable.Source: D. Reservoir embedding and readout · p. 6

E. Evaluation metrics

The weak-drive benchmark reports high reconstruction accuracy from a single-qubit reservoir, while evaluation also examines error, explained variance, and frequency-resolved performance.

E. Evaluation metrics

  • Evaluation uses mean squared error and R2, with per-frequency R2 scores providing a frequency-resolved view, particularly near reservoir resonance (ω ≈ 1).R2 quantifies the fraction of ground-truth variance explained by the model.Source: E. Evaluation metrics · p. 6

IV. RESULTS AND DISCUSSION

  • The reservoir parameters use a fixed random seed for reproducibility, and the broader evaluation proceeds from a single-amplitude case to multi-amplitude inputs.The single-amplitude case isolates reservoir behavior; the multi-amplitude case probes generalization across a parametrized family of inputs.Source: §IV · p. 6

A. Weak-drive regime: F0 = 10

  • With 400 trajectories, mean R2 = 0.9954; 97.5% of test cases exceed R2 > 0.90 and 93.75% exceed R2 > 0.95.With 200 samples, mean R2 = 0.978; the remaining below-threshold cases are predominantly at the lowest driving frequencies, where the output varies slowly and greater temporal context is needed.Source: A. Weak-drive regime: F0 = 10 · p. 7
  • The reported accuracy is achieved with a single-qubit reservoir, whose driven dissipative dynamics provide transduction capacity independently of Hilbert-space scaling.The weak-drive setting corresponds to F0 = 10 and is compared with dataset 2 of Ref. at A = 10.Source: A. Weak-drive regime: F0 = 10 · pp. 6–7

B. Intermediate drive regime (F0 = 20)

In the F0 = 20 regime, a single-qubit reservoir achieves near-perfect reconstruction across the test distribution. Its fixed dissipative dynamics also show a finite-memory boundary at the lowest frequencies, while adding a qubit reduces the training-data requirement.

B. Intermediate drive regime (F0 = 20)

  • MSE = 10^-5, with near-perfect reconstruction of ⟨σ_z⟩(t) across the test distribution using a single reservoir qubit.The result is obtained without entanglement or exponential Hilbert-space scaling.Source: B. Intermediate drive regime (F0 = 20) · p. 8
  • At the lowest probed frequencies, the fixed γ = 5.0 memory window is marginally insufficient for the longest-memory trajectories, though the effect is small and tunable.The sub-resonant accuracy reduction is consistent with increasing temporal-memory requirements as dynamics become drive-dominated.Source: B. Intermediate drive regime (F0 = 20) · p. 8
  • The 512-dimensional reservoir features compress to 15 PCA dimensions with no measurable accuracy loss, indicating a low-dimensional feature manifold.The selected RBF kernel has γrbf = 15.0.Source: B. Intermediate drive regime (F0 = 20) · p. 8
  • Across the frequency band, the readout reproduces representative target trajectories, including both slowly varying envelopes and fine oscillatory structure.The results establish driven-dissipative dynamics as sufficient for near-perfect transduction in this setting.Source: B. Intermediate drive regime (F0 = 20) · p. 8
  • N = 2 reaches 100% at both R2 thresholds from Nω = 1000, while N = 1 reaches 99.59% for R2 > 0.90 and 98.38% for R2 > 0.95 at Nω = 3700.The two-qubit reservoir therefore reaches the reported pass rates with more than threefold fewer training trajectories.Source: Table II · p. 8, B. Intermediate drive regime (F0 = 20) · p. 8

C. Multi-amplitude dataset (F0 ∈[1, 20])

A single QRC readout generalizes over the multi-amplitude range F0 ∈ [1, 20], with lower accuracy than in the single-amplitude regime. Accuracy varies with drive conditions and signal complexity, while the amplitude coordinate is encoded through reservoir dynamics rather than appended to the feature vector.

C. Multi-amplitude dataset (F0 ∈[1, 20])

  • The amplitude coordinate is not appended to the feature vector; the readout resolves it from reservoir dynamics while one model covers the two-dimensional (ω, F0) input space.The amplitude span ranges from weakly driven near-linear dynamics at F0 = 1 to strongly nonlinear behavior at F0 = 20.Source: C. Multi-amplitude dataset (F0 ∈[1, 20]) · p. 9, Table III · p. 9
  • MSE = 3.2 × 10^-4, with five-fold cross-validation selecting α = 5 × 10^-4 and γrbf = 1.0 for the more heterogeneous feature distribution.The broader kernel contrasts with the narrower single-amplitude optimum.Source: C. Multi-amplitude dataset (F0 ∈[1, 20]) · p. 9
  • At fixed frequency, R2 decreases as F0 increases, with the sharpest degradation near resonance at high amplitude; scores are higher at intermediate and super-resonant frequencies.Residual errors in representative predictions are largely confined to the low-frequency, high-amplitude regime.Source: C. Multi-amplitude dataset (F0 ∈[1, 20]) · pp. 9–10
  • R2 = 0.9709 overall, with 85.88% of samples exceeding R2 > 0.95 across F0 ∈ [1, 20].The multi-amplitude result is lower than the single-amplitude R2 = 0.9997.Source: C. Multi-amplitude dataset (F0 ∈[1, 20]) · p. 9, Figure 5 · p. 9, Figure 7 · p. 10
  • Representative predictions trace target envelopes and fine oscillations in most test cases, while dissipation remains fixed at γ = 5.0 across both regimes.Residual phase drift or amplitude mismatch occurs mainly in the low-frequency, high-amplitude regime.Source: C. Multi-amplitude dataset (F0 ∈[1, 20]) · pp. 8, 10

D. Output Complexity Analysis

Amplitude-aware permutation entropy tracks how the target signals’ temporal complexity changes across drive strengths and relates that complexity to prediction accuracy. It rises with drive amplitude, while its variation across frequencies helps distinguish signal difficulty from amplitude alone.

D. Output Complexity Analysis

  • Figure 8 illustrates predicted outputs against targets for four multi-amplitude driving frequencies: 2.92, 1.72, 1.55, and 1.62.The frequencies are highlighted in the R2-versus-ω plot.Source: Figure 8 · p. 10
  • The analysis averages normalized amplitude-aware permutation entropy over frequencies, using embedding dimension m=4 and unit delay τ=1.The amplitude-weighted formulation gives greater weight to high-variance segments and captures ordinal structure together with amplitude information.Source: D. Output Complexity Analysis · p. 10
  • Amplitude-aware permutation entropy increases overall with F0, from nearly linear, low-dimensional responses toward stronger nonlinear mixing and richer temporal structure.Figure 9 reports entropy averaged over driving frequencies for each amplitude.Source: D. Output Complexity Analysis · p. 10, Figure 9 · p. 11
  • R2 is near 1 at F0=20 but 0.9709 across the multi-amplitude space, where entropy is broader and shifted higher.Per-sample R2 co-varies with entropy across frequency and amplitude; at fixed F0=20, the largest deviations occur in the sub-resonant band with highest entropy.Source: D. Output Complexity Analysis · p. 10
  • At fixed amplitude, frequency-dependent entropy corresponds with per-sample R2, supporting its use as a prediction-difficulty measure independent of drive strength.The correspondence is described for the single-amplitude regime, where entropy varies only with frequency.Source: D. Output Complexity Analysis · pp. 10–11

E. Discussion

The discussion reports that a fixed, minimal dissipative reservoir reproduces a larger spin chain’s response across drive strengths, with performance related to signal complexity. The conclusion interprets this compression as an effective quasiparticle description and identifies the high-complexity, high-amplitude near-resonant regime as the main bottleneck.

E. Discussion

  • Reservoir dynamics implicitly encode amplitude without feature augmentation, while contractive Lindblad damping stabilizes predictions in the hardest parameter-space region.Residual errors are associated with target-signal complexity rather than reservoir capacity.Source: E. Discussion · p. 11
  • The main bottleneck is elevated signal complexity at high amplitude near resonance, where entropy is largest and R2 scores are most dispersed.Amplitude-stratified training and nonlinear readout layers are proposed for future comparison.Source: E. Discussion · p. 11

V. CONCLUSION

  • R2=0.9997 in the single-amplitude regime and 0.9709 across a 20× drive-strength range; the single-qubit reservoir matches or surpasses cited models and reconstructs a ten-qubit chain.The cited comparisons are with the physics-informed Kolmogorov–Arnold-network model and temporal convolutional approach.Source: §V · p. 11, E. Discussion · p. 11
  • The authors interpret the compression as a data-driven effective quasiparticle description: a few emergent modes carry the ten-qubit chain’s observable response.In this interpretation, the reservoir represents an effective degree of freedom rather than the chain’s microscopic structure.Source: §V · p. 11
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