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On Dequantization of Supervised Quantum Machine Learning via Random Fourier Features

Mehrad Sahebi, Alice Barthe, Yudai Suzuki, Zoë Holmes, Michele Grossi

TL;DR

This work analyzes when classical Random Fourier Features can dequantize supervised quantum ML models, including QNNs, quantum kernels, and SVMs. It derives sufficient conditions—alignment between the RFF frequency distribution and the quantum model’s Fourier spectrum, and concentration of the distribution—under which RFF can closely match the true risk of quantum models for regression and classification. The authors extend dequantization results from QNN regression to QK regression and QSVM/QNN-SVM, providing theoretical bounds and practical prescriptions, and validate these findings with numerical experiments on particle-collision data. Overall, the paper clarifies when quantum advantages may or may not arise in practical learning tasks and guides the design of classical surrogates via RFF.

Abstract

In the quest for quantum advantage, a central question is under what conditions can classical algorithms achieve a performance comparable to quantum algorithms--a concept known as dequantization. Random Fourier features (RFFs) have demonstrated potential for dequantizing certain quantum neural networks (QNNs) applied to regression tasks, but their applicability to other learning problems and architectures remained unexplored. In this work, we derive bounds on the true risk gap between classical RFF models and quantum models for regression and classification tasks with both QNN and quantum kernel architectures. Furthermore, we provide sufficient conditions under which this gap is small and thus the quantum system can be dequantized via the RFF method. We support our findings with numerical experiments that illustrate the practical dequantization of existing quantum kernel-based methods. Our findings not only broaden the applicability of RFF-dequantization but also enhance the understanding of potential quantum advantages in practical machine-learning tasks.

On Dequantization of Supervised Quantum Machine Learning via Random Fourier Features

TL;DR

This work analyzes when classical Random Fourier Features can dequantize supervised quantum ML models, including QNNs, quantum kernels, and SVMs. It derives sufficient conditions—alignment between the RFF frequency distribution and the quantum model’s Fourier spectrum, and concentration of the distribution—under which RFF can closely match the true risk of quantum models for regression and classification. The authors extend dequantization results from QNN regression to QK regression and QSVM/QNN-SVM, providing theoretical bounds and practical prescriptions, and validate these findings with numerical experiments on particle-collision data. Overall, the paper clarifies when quantum advantages may or may not arise in practical learning tasks and guides the design of classical surrogates via RFF.

Abstract

In the quest for quantum advantage, a central question is under what conditions can classical algorithms achieve a performance comparable to quantum algorithms--a concept known as dequantization. Random Fourier features (RFFs) have demonstrated potential for dequantizing certain quantum neural networks (QNNs) applied to regression tasks, but their applicability to other learning problems and architectures remained unexplored. In this work, we derive bounds on the true risk gap between classical RFF models and quantum models for regression and classification tasks with both QNN and quantum kernel architectures. Furthermore, we provide sufficient conditions under which this gap is small and thus the quantum system can be dequantized via the RFF method. We support our findings with numerical experiments that illustrate the practical dequantization of existing quantum kernel-based methods. Our findings not only broaden the applicability of RFF-dequantization but also enhance the understanding of potential quantum advantages in practical machine-learning tasks.

Paper Structure

This paper contains 40 sections, 45 theorems, 163 equations, 8 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical Framework
  4. Quantum Framework
  5. RFF Dequantization of QML models
  6. Dequantization for Regression
  7. QNN Regression by Ref. Lg
  8. QK Regression
  9. Dequantization for SVM Classification
  10. Quantum SVM
  11. QNN-SVM
  12. Numerical experiments
  13. Conclusion
  14. Appendix
  15. Related Work
  16. ...and 25 more sections

Key Result

Lemma 3.1

A QK $k_Q$ as in Eq. eq:DefKernel using a Hamiltonian encoding (Assumption ass:Hencoding) can be written as where $F$ is a positive semi-definite, unit-trace ($\sum_{\bm{\omega}} F_{\bm{\omega}\bm{\omega}} = 1$) matrix and $F_{-\bm{\omega}, -\bm{\nu}} = F^*_{\bm{\omega} \bm{\nu}}$, and $\Omega$ is a set of vector frequencies determined by the eigenvalues of encoding Hamiltonians.

Figures (8)

  • Figure 1: Alignment and concentration conditions. For QK methods, given a distribution $q_{\boldsymbol{\omega}}$ corresponding to the diagonal of the Fourier transform $F$ of the target QML model (Black), the model is dequantizable if the probability distribution $p_{\boldsymbol{\omega}}$ used for RFF is aligned to the target and concentrated (Green). On the other hand, when $p_{\boldsymbol{\omega}}$ is not aligned (Red) or concentrated (Purple), it might not be dequantizable.
  • Figure 2: Bounded RKHS norm condition. The RKHS norm describes the model complexity and is thus associated with the generalization performance. In regression, a well-behaved output $f_1$ (green) has the small RKHS norm, whereas $f_2$ that is more complex has a large norm. In SVM classification, the inverse of the norm is equal to the margin.
  • Figure 3: Comparison of Sampling Strategies. For a high-energy physics dataset, the true risk of a RFF-SVM is plotted against the number of frequency samples $D$ for the RFF, for three different distributions. The solid lines are the average performance over 60 runs of the RFF algorithm, and the shaded regions represent the standard deviation. More details can be found in App. \ref{['app:dataset']}.
  • Figure 4: Convolutional and Truncated Sampling Strategies used in Numerical Experiments (Section \ref{['subsec:numerics']}) Convolutional sampling (Green) inspired by the distribution of Fourier coefficients in QNNs barthe2024gradients comes from the rows of Pascal triangle. Truncated sampling (Golden) is sets the probability of the higher half of the frequencies to zero and for the lower half is proportional to the convolutional distribution.
  • Figure 5: RFF-SVM vs QSVM. The true risk of SVM against the number of frequency samples $D$ for the RFF-SVM using a truncated-convolutional sampling distribution. The dashed lines are the performance of a 5-layer 16-qubit QK, with different levels of shot noise. More details can be found in App. \ref{['app:dataset']}.
  • ...and 3 more figures

Theorems & Definitions (93)

  • Definition 2.1: Shift-invariant Kernel
  • Definition 2.3: RFF Dequantization
  • Lemma 3.1
  • Definition 3.2: Kernel distribution
  • Proposition 3.3: Sufficient conditions for RFF dequantization of QK regression
  • Proposition 3.4: Sufficient Conditions for RFF dequantization of QSVM
  • Proposition 3.5: Sufficient Conditions for RFF dequantization of QNN-SVM
  • Definition B.1: Kernel Function
  • Definition B.2: Reproducing Kernel Hilbert Space RKHS
  • Definition B.3: Kernel Integral Operator
  • ...and 83 more