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Neural Fourier Surrogates for Data Reuploading Quantum Neural Networks

Oliver Knitter, Jonathan Mei, Sang Hyub Kim, Chi Chen, Masako Yamada, Martin Roetteler

arXiv:2610.00841v1quant-phcs.LG

TL;DR

Comparisons between QNNs and classical models with different function classes do not readily clarify the boundary between their performance. This paper introduces Neural Fourier Surrogates, a classical model of the finite Fourier-series space used by data-reuploading QNNs, and finds comparable QNN performance and broad competitiveness with classical baselines on tabular data. Its results support NFS as a classical baseline for evaluating QNN performance.

  • Problem

    The boundary between classical and quantum advantage remains unclear, and comparisons are difficult to interpret when QNNs and classical baselines represent different function classes.

    Source: §1 · p. 1

  • Method

    NFS is a stochastic classical neural network that learns coefficients over the finite Fourier-series space modeled by data-reuploading QNNs, using an autoregressive network to learn a distribution over frequencies.

    Source: §1 · pp. 1–2

  • Results

    On small-to-medium tabular data, NFS is competitive with standard classical baselines and qualitatively matches QNN performance, with synthetic-data analysis showing its use for examining QNN Fourier spectra.

    Source: §1 · p. 2, §5 · p. 8

  • Takeaways & Limitations

    NFS is a strong candidate classical baseline for evaluating QNN performance and can provide insight into the Fourier spectra learned by QNNs.

    Source: §1 · p. 2

    The compared QNN uses a fixed measurement observable that constrains its expressivity beyond the finite Fourier-series structure; more complicated or tunable observables could close the performance gap.

    Source: §4.2 · pp. 6–7

Abstract

from arXiv · show

For quantum machine learning, the exact boundary between classical and quantum advantage is still poorly understood. Direct comparison between quantum neural networks (QNNs) and existing classical models, which encompass fundamentally different function classes, often fails to provide broader insight into the difference between the two. Inspired by the techniques of Neural Quantum States and Random Fourier Features, this work introduces Neural Fourier Surrogates (NFS), a stochastic classical neural network architecture for efficiently learning coefficients over the same finite Fourier series support as quantum neural networks. Testing on a selection of tabular benchmark datasets, we find that NFS is an effective classifier architecture broadly competitive with established classical baselines, including a comparable Random Fourier Features model, and possessing comparable performance to data-reuploading QNNs; combined with additional analysis comparing the learned Fourier spectra of QNNs and NFS on synthetic data, these results establish NFS as a natural classical baseline for evaluating QNN performance.

1. Introduction

The paper addresses the difficulty of interpreting QNN performance against classical models with different function classes. It introduces NFS as a classical model over the finite Fourier-series space of data-reuploading QNNs and evaluates it against classical baselines and QNNs.

  • QNN performance is difficult to interpret against classical baselines because their function classes differ, and many assessments use small datasets, simulators, and simple baselines.The paper notes concerns that such baselines may be substandard representations of classical performance.Source: §1 · p. 1
  • NFS models the same finite Fourier-series space as a data-reuploading QNN with angle encoding, while not inheriting the fixed-measurement restriction typical of QNNs.Its autoregressive network learns a distribution over relevant frequencies; a separate phase network supplies sample values, and global scaling makes their sample mean approximate the series.Source: §1 · pp. 1–2
  • Across small-to-medium tabular data, NFS is competitive with standard classical baselines, qualitatively matches QNN performance, and was tested against six classical baselines on 15 datasets.The paper also uses synthetic data to examine the Fourier spectra learned by NFS and QNNs.Source: §1 · p. 2
  • The experiments compare NFS and a data-reuploading QNN across training-set sizes and model widths, and use synthetic data to assess NFS’s ability to learn a QNN Fourier spectrum.Source: §1 · p. 2

2. Related Work

This section situates NFS alongside classical surrogates, Random Fourier Features, and Neural Quantum States. It distinguishes NFS by learning a distribution over Fourier frequencies to model an entire finite Fourier series.

  • Unlike Neural Quantum States, which model a quantum state and its Born distribution, NFS models the behavior of an entire finite Fourier series.Source: §2 · p. 2
  • A prior classical surrogate learns a trained QNN’s Fourier coefficients by enumerating the frequency space, whereas NFS learns a frequency distribution and Random Fourier Features samples from a fixed model-defined distribution.Source: §2 · p. 2
  • NFS provides a tangible classical surrogate example relevant to research on QNN expressivity constraints and boundaries where quantum models or classical surrogates outperform each other.Source: §2 · p. 2

3. Methodology

NFS represents QNN-like finite Fourier series with neural networks for coefficient magnitudes and phases, then estimates their outputs by sampling frequencies. Its architecture supports efficient sampling, real-valued outputs, multiple outputs, and a Gaussian prior over frequencies.

3.1. Neural Representation of Fourier Series

  • NFS models a QNN-style finite Fourier series by learning its frequency coefficients’ magnitudes and phases, then approximates the series through random frequency sampling.The frequency support is determined by the input dimension and number of encoding re-uploads.Source: §3.1 · p. 3
  • Using shared frequency samples to estimate outputs and gradients enables automatic differentiation but introduces covariance-derived bias inversely proportional to sample batch size.The authors state that the autoregressive sampler permits batch sizes for which this bias becomes negligible.Source: §3.1 · p. 3

3.2. Neural Network Architecture

  • NFS enforces conjugate-symmetric coefficients through an even modulus and odd phase; an input-free scaling network restores the coefficient 1-norm.Masking restricts sampling to frequencies with nonzero first entries, and importance sampling gradient optimization reuses samples across updates without bias.Source: §3.2 · pp. 4–5
  • A transformer modulus network produces nested conditional probabilities for autoregressive frequency sampling, while a shallow feedforward network models phase.The conditional probabilities multiply to form a normalized joint frequency distribution, enabling efficient, parallelizable sampling.Source: §3.2 · p. 4
  • For multidimensional outputs, NFS represents a separate Fourier series per output with shared support and shared modulus and phase networks.A final-layer projection head determines each output, and the study trains classification logits with mean squared error.Source: §3.2 · p. 5

3.3. Gaussian Prior Regularization

  • NFS regularizes its learned frequency distribution toward a Gaussian to approximate the QNN prior that higher-frequency coefficient magnitudes tend to be smaller.The regularizer uses the frequency distribution’s entropy and the Gaussian normalization constant.Source: §3.3 · p. 5

4. Experiments

Across tabular benchmarks, NFS is broadly competitive with classical models and shows performance similar to QNNs in some comparisons, though QNNs outperform it in a qubit-dimension sweep. Synthetic tests show that NFS capacity interacts with data complexity, and its learned spectra do not always match those of the QNNs it imitates.

4.1. Full Comparison on Tabular Benchmark Data

  • NFS is broadly competitive across 15 tabular benchmarks, without superior performance, but generally degrades on higher-dimensional datasets at d = 10.The authors suggest the fixed dimension may help explain the degradation.Source: §4.1 · p. 6

4.2. Learning Efficiency Relative to QNNs

  • At encoding depth 4, NFS and QNNs show consistently similar performance across six datasets and training-set sizes, with RFF qualitatively similar on most datasets.Source: §4.2 · p. 6
  • In the qubit-dimension sweep, QNNs generally outperform NFS, while both models are fairly insensitive to dimension changes; the authors identify dimension reduction as the primary bottleneck.The sweep is the closest feasible analog to a parameter-matched comparison, which the architectures do not permit.Source: Figure 3 · p. 6, §4.2 · p. 7

4.3. Fourier Capacity Analysis of NFS

  • On synthetic targets generated by QNNs, increasing NFS encoding depth generally improves output replication, but depth 6 appears to need more data on lower-complexity targets, likely due to overfitting.Source: §4.3 · pp. 7–8
  • With 500 training samples, increasing target Fourier complexity reduces test accuracy while increasing encoding depth improves it, with only marginal capacity gains at sufficiently high complexity.Together with the training-size results, this suggests higher-capacity models require more data to train effectively.Source: Figure 4 · p. 7, §4.3 · p. 8
  • When trained to match QNN outputs, NFS frequency-distribution correlations are bimodal, clustering near 1 and 0: it either recovers the QNN spectrum or finds a different suitable coefficient set.The authors attribute this to NFS being more expressive than the fixed-observable QNN used.Source: Figure 5 · p. 7, §4.3 · p. 8

4.4. Hyperparameter Tuning

  • NFS training costs more computation than the classical baselines, and the authors frame it primarily as an interpretive surrogate rather than a competitor to classical methods.Source: §4.4 · p. 8
  • Hyperparameter tests emphasize sufficiently high learning rates and Gaussian-prior regularization; lower model capacity performs better with smaller training sets, while other reasonable settings are generally robust.Source: §4.4 · p. 8

5. Conclusion

NFS learns coefficients over a Fourier-series function class related to data-reuploading QNNs and performs similarly to QNNs while remaining competitive with classical baselines on small-to-medium tabular datasets. The authors position it as a baseline for assessing QNN performance and identify hardware comparisons and lower-cost scaling as future directions.

5. Conclusion

  • NFS performs similarly to QNNs and is competitive, though not superior, to classical baselines on small-to-medium tabular datasets.Source: §5 · p. 8
  • NFS spans coefficient space theoretically achievable by the entire QNN function class, but slightly exceeds what typical QNN architectures can reach.Source: §5 · p. 8
  • Future work includes comparisons with hardware-executed QNNs and testing recurrent networks as a way to improve NFS scaling cost without sacrificing accuracy.Source: §5 · p. 8

AI Use Statement

  • The authors report using Claude for project and experiment ideation, benchmark-code generation, outlining, and proofreading, but not for derivations, manuscript writing, theory, or data generation.Source: AI Use Statement · p. 8

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