PaperlayerView exact revision on arXiv

Source-linked AI summary

Evaluating Hybrid Quantum-Classical Models for Reduced-Order Brain Deformation Dynamics

Tao Liu, Ge He, Dongyu Liang, Wujie Wen

arXiv:2610.00554v1cs.LGcs.CE

TL;DR

High-dimensional, time-dependent physical fields remain a practical challenge for QML models constrained to low-dimensional inputs. This study uses POD to compare classical, minimal quantum, and enhanced hybrid models on brain-deformation regression and forecasting; classical networks are strongest in the reported setting.

  • Problem

    Realistic spatiotemporal field prediction remains a largely unexplored QML application, while current variational quantum models are largely constrained to low-dimensional inputs.

    Source: §I · p. 1

  • Method

    The study projects brain displacement fields into a compact POD latent space and compares classical, minimal quantum, and enhanced hybrid models on static regression and temporal forecasting.

    Source: §I · p. 1

  • Results

    Compact classical models remain the strongest and most robust performers in the present setting, although hybrid enhancements substantially improve weak quantum baselines.

    Source: §I · p. 1, §V · p. 6

  • Takeaways & Limitations

    Reduced-order physical field learning provides a rigorous testbed for near-term QML, with practical guidance emphasizing physical-space reconstruction, robustness, and temporal context.

    Source: §I · p. 1, §V · p. 6

Abstract

from arXiv · show

We evaluate hybrid quantum-classical machine learning for the reduced-order prediction of spatiotemporal brain deformation fields. To mitigate the computational intractability of high-dimensional displacement fields, we employ Proper Orthogonal Decomposition (POD) to project the data into a compact latent space. Within this framework, we formulate two distinct learning objectives: static temporal-to-latent regression and autoregressive latent state forecasting. We systematically benchmark compact classical baselines against both minimal and enhanced hybrid quantum architectures. Our results demonstrate that classical networks provide the strongest baselines in the present setting. For static regression, a classical POD-MLP outperforms all evaluated quantum variants, although an enhanced Variational Quantum Circuit (VQC) substantially improves upon a minimal VQC baseline. For temporal forecasting, a classical POD-LSTM delivers superior predictive accuracy and statistical robustness compared to an enhanced Quantum LSTM (QLSTM) across varying history windows and random initializations. Overall, this study establishes reduced-order physical field learning as a rigorous testbed for near-term QML, highlighting that while hybrid enhancements successfully recover expressivity in weak quantum circuits, classical architectures retain a definitive advantage in both fidelity and stability.

I. INTRODUCTION

The study addresses the largely unexplored intersection of QML and realistic spatiotemporal field prediction by using brain deformation as a reduced-order testbed. It compares classical, minimal quantum, and enhanced hybrid models across static regression and temporal forecasting.

I. INTRODUCTION

  • QML for realistic spatiotemporal field prediction remains largely unexplored, despite the use of data-driven ROMs in biomechanics,,,.The introduction identifies current variational quantum models' low-dimensional input constraints as a practical gap for high-dimensional, spatially distributed, time-dependent data.Source: §I · p. 1
  • The framework projects full displacement fields into a compact POD latent space, making two learning tasks tractable while retaining evaluation in reconstructed physical space.The tasks are static timestep-to-latent regression and forecasting future states from historical windows.Source: §I · p. 1
  • The comparison spans compact classical baselines, minimal quantum baselines, and enhanced hybrid quantum models for both static regression and temporal forecasting.It asks whether quantum enhancement benefits smooth reduced-order regression and whether a hybrid quantum bottleneck can capture latent dynamics with improved trainability and robustness.Source: §I · p. 1
  • Compact classical models remain the strongest and most robust performers in the present setting, while hybrid enhancements substantially improve weak quantum baselines.The framework also emphasizes physical-space reconstruction, robustness, and sensitivity to temporal context when applying QML to reduced-order field prediction.Source: §I · p. 1

II. METHODOLOGY: REDUCED-ORDER FORMULATION AND HYBRID QUANTUM ARCHITECTURES

The methodology frames brain deformation as a time-evolving displacement field on a decimated mesh and uses POD to make its prediction tractable. It compares learning the reduced representation with forecasting its temporal evolution under a shared reduced-order formulation.

II. METHODOLOGY: REDUCED-ORDER FORMULATION AND HYBRID QUANTUM ARCHITECTURES

  • Brain deformation is represented as a time-evolving spatial displacement field, and direct learning in the original field space is high-dimensional and computationally demanding for near-term hybrid models.The mesh depiction contrasts initial and final timesteps, with node color indicating displacement magnitude in millimeters.Source: §II · p. 1, Figure 1 · p. 2
  • POD projects the displacement field into a compact latent space, where the timestep-t coefficients z_t represent the reduced deformation state,.The reduced representation reformulates the original high-dimensional problem as learning in a compact space.Source: §II · pp. 1–2
  • The study distinguishes static reduced-order regression from temporal reduced-order forecasting, using model families as a shared comparison ladder rather than an exhaustive architecture survey.The comparison includes compact classical, minimal quantum, and stronger hybrid quantum variants, examining direct regression and latent dynamics forecasting.Source: §II · p. 2

A. Static Model Families

The static model family isolates approximation of the reduced deformation manifold by removing recurrent memory and autoregressive rollout. It compares a classical POD-MLP with a hybrid quantum-feature variant.

A. Static Model Families

  • The static setting removes recurrent memory and autoregressive rollout to isolate each model's ability to approximate the reduced deformation manifold.This control setting helps distinguish limited representational power from a failure to learn dynamics when interpreting weak temporal performance.Source: A. Static Model Families · p. 2

1) Standard POD-MLP:

  • The Standard POD-MLP maps normalized timestep directly to the k-dimensional latent vector through two tanh hidden layers and a linear output layer.It serves as the primary classical baseline for smooth reduced-order regression.Source: 1) Standard POD-MLP · p. 2

2) Quantum-Enhanced POD-MLP:

  • The Quantum-Enhanced POD-MLP adds a quantum feature transformation between classical input and output layers to test its value on a classically approachable regression task.The timestep is lifted to a bounded feature vector, encoded with angle embedding, processed by trainable entangling layers, and mapped back through a classical output layer.Source: 2) Quantum-Enhanced POD-MLP · p. 2

3) Standard POD-VQC:

The model comparison proceeds from a minimal memoryless VQC to an enhanced VQC, then turns to recurrent models for latent dynamics forecasting. The temporal task adds autoregressive rollout and possible error accumulation.

3) Standard POD-VQC:

  • The Standard POD-VQC is a minimal memoryless regressor that broadcasts the scalar timestep to all qubits, encodes it once, and measures a shallow variational circuit for latent predictions.It serves as the absolute minimal quantum baseline.Source: 3) Standard POD-VQC · p. 2

4) Enhanced POD-VQC:

  • The Enhanced POD-VQC adds classical input and output dressing and repeated input embedding across circuit depth to test recovery of expressivity lost in the shallow minimal design.The repeated embedding uses the data re-uploading paradigm.Source: 4) Enhanced POD-VQC · p. 2

B. Temporal Model Families

  • Temporal models predict latent dynamics by inferring the reduced-state update law, with forecasting quality affected by autoregressive rollout and error accumulation.The comparison includes one classical recurrent baseline and two hybrid recurrent variants.Source: B. Temporal Model Families · p. 2

1) Classical POD-LSTM:

The classical POD-LSTM encodes a latent history window and predicts the next latent vector through a classical output head. The hybrid variants replace or deepen the quantum interface between sequence encoding and latent prediction.

1) Classical POD-LSTM:

  • The Classical POD-LSTM processes a latent history window with an LSTM and maps its final hidden state to the next latent vector through a classical output head.Source: 1) Classical POD-LSTM · p. 2
  • The Standard POD-QLSTM replaces the classical output head with a shallow quantum bottleneck that maps the encoded hidden state to the next latent prediction.Source: 3) Enhanced POD-QLSTM · p. 2

3) Enhanced POD-QLSTM:

  • The Enhanced POD-QLSTM adds a classical pre-quantum bridge, deeper VQC, and output dressing network to strengthen the interface between recurrent encoding and latent prediction.Source: 3) Enhanced POD-QLSTM · p. 2
  • All latent predictions are assessed after inverse POD reconstruction and inverse scaling in the physical displacement field.Source: 3) Enhanced POD-QLSTM · p. 2

A. Dataset Preparation

The study uses a single brain-deformation trajectory represented as flattened displacement fields over time. Training-only scaling and POD produce a compact latent representation that is reconstructed to assess physical-space fidelity.

A. Dataset Preparation

  • The dataset contains 51 snapshots of a decimated brain mesh with 1,940 nodes, and each displacement state concatenates the three node-wise components into a full-field vector.Source: A. Dataset Preparation · p. 3
  • Training uses the first 46 snapshots and testing holds out the final five, creating a temporal extrapolation test rather than interpolation among observed snapshots.Source: A. Dataset Preparation · p. 3

B. Reduced-Order Preprocessing

  • Displacement features are scaled to [−1, 1] using a MinMaxScaler fitted only on training snapshots before applying the same transformation to test data.Source: B. Reduced-Order Preprocessing · p. 3
  • POD fitted to scaled training data projects snapshots onto leading modes, yielding a compact latent representation whose leading modes capture most cumulative variance.Source: B. Reduced-Order Preprocessing · p. 3
  • The benchmark fixes latent dimension k = 8 as a practical near-term QML size, not the intrinsic optimal rank, and evaluates fidelity after inverse POD and scaling.Source: B. Reduced-Order Preprocessing · p. 3

C. Training and Evaluation Protocol

The evaluation uses autoregressive rollout for temporal forecasting and common training settings for the models. It also measures latent and reconstructed physical-space accuracy, with a multi-window, multi-seed sweep for temporal robustness.

A. Static Reduced-Order Regression

Across static regression and temporal forecasting, compact classical models outperform the evaluated quantum variants in this setting. Hybrid enhancements improve weak quantum baselines, but do not overturn the classical performance hierarchy.

A. Static Reduced-Order Regression

  • R2 = 0.7812 in latent space and 0.9728, 0.9794, and 0.9452 in physical Ux, Uy, and Uz make Standard POD-MLP the strongest static model, exceeding Quantum-Enhanced POD-MLP and enhanced VQC.Source: A. Static Reduced-Order Regression · pp. 3–4

B. Temporal Reduced-Order Forecasting

A. Static Reduced-Order Regression

B. Temporal Reduced-Order Forecasting

C. Lessons for Reduced-Order Spatiotemporal QML

The results show no clear QML advantage overall: compact classical models lead on smooth static regression, while hybrid enhancement mainly recovers expressivity in weaker quantum circuits. Temporal forecasting is more fragile, and evaluation should include reconstructed physical fidelity and robustness across initializations.

C. Lessons for Reduced-Order Spatiotemporal QML

  • POD projection compresses the high-dimensional displacement field into a tractable lower-dimensional task while retaining physical-fidelity evaluation.The authors describe this reduced-order bridge as foundational for applying current QML models to the field.Source: C. Lessons for Reduced-Order… · p. 6
  • The Standard POD-MLP achieved the strongest overall static performance, outperforming both minimal and enhanced quantum variants.For smooth, low-dimensional static mappings, the lightweight feedforward model provides a difficult-to-beat baseline.Source: C. Lessons for Reduced-Order… · p. 6
  • Deep enhancements, including data re-uploading and classical dressing, improved VQC and QLSTM minimal baselines but did not establish quantum supremacy.Source: C. Lessons for Reduced-Order… · p. 6
  • QLSTM variants showed sensitivity to history context and significant variance, while stronger gradient signals in the Enhanced POD-QLSTM did not guarantee higher predictive accuracy.The authors identify accumulated rollout errors as a stability challenge in recurrent prediction.Source: C. Lessons for Reduced-Order… · p. 6
  • Static latent errors could coexist with strong physical reconstructions, whereas small temporal latent discrepancies compounded into severe physical-space degradation.The authors therefore emphasize reconstructed physical fidelity and robustness across initializations over latent fit alone.Source: C. Lessons for Reduced-Order… · p. 6
  • The study found no clear QML advantage overall, and recommends comparisons against strong classical baselines using physically meaningful metrics.Source: C. Lessons for Reduced-Order… · p. 6

V. CONCLUSION

The study uses a POD latent bridge to evaluate classical and hybrid quantum models on static and temporal brain-deformation prediction. Classical POD-MLP and POD-LSTM models outperform the quantum variants, while hybridization improves weak quantum baselines without surpassing classical models in this low-rank setting.

V. CONCLUSION

  • A POD latent bridge compresses high-dimensional brain displacement fields into tractable static regression and temporal forecasting tasks.Source: §V · p. 6
  • Classical POD-MLP and POD-LSTM models consistently outperformed all quantum variants in static and temporal prediction, respectively.Source: §V · p. 6
  • Deep hybridization recovered expressivity in minimal quantum baselines but did not surpass classical models in this low-rank regime.The conclusion identifies multi-trajectory loading and non-smooth transient dynamics as scenarios for future study.Source: §V · p. 6
Loading 2610.00554v1…