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Classically Approximating Variational Quantum Machine Learning with Random Fourier Features

Jonas Landman, Slimane Thabet, Constantin Dalyac, Hela Mhiri, Elham Kashefi

TL;DR

The paper reframes variational quantum circuits as shift-invariant kernels whose Fourier spectrum Ω is determined by encoding Hamiltonians, and proposes classical Random Fourier Feature (RFF) surrogates to approximate VQCs without running on quantum hardware. Three RFF strategies—Distinct, Tree, and Grid sampling—are developed to sample frequencies from Ω and construct scalable classical models via kernel ridge regression, with theoretical bounds showing linear scaling in input dimensionality and logarithmic dependence on spectrum size. Empirical results across Pauli and more complex encodings, including real datasets like Fashion-MNIST and California Housing, demonstrate that RFF surrogates can match or even outperform VQCs using only a small fraction of Ω, particularly when frequency redundancy is high. These findings challenge the assumed quantum advantage of VQCs in many practical settings and emphasize the role of encoding structure and spectrum properties in determining learnability, while outlining conditions under which quantum advantage might still emerge.

Abstract

Many applications of quantum computing in the near term rely on variational quantum circuits (VQCs). They have been showcased as a promising model for reaching a quantum advantage in machine learning with current noisy intermediate scale quantum computers (NISQ). It is often believed that the power of VQCs relies on their exponentially large feature space, and extensive works have explored the expressiveness and trainability of VQCs in that regard. In our work, we propose a classical sampling method that may closely approximate a VQC with Hamiltonian encoding, given only the description of its architecture. It uses the seminal proposal of Random Fourier Features (RFF) and the fact that VQCs can be seen as large Fourier series. We provide general theoretical bounds for classically approximating models built from exponentially large quantum feature space by sampling a few frequencies to build an equivalent low dimensional kernel, and we show experimentally that this approximation is efficient for several encoding strategies. Precisely, we show that the number of required samples grows favorably with the size of the quantum spectrum. This tool therefore questions the hope for quantum advantage from VQCs in many cases, but conversely helps to narrow the conditions for their potential success. We expect VQCs with various and complex encoding Hamiltonians, or with large input dimension, to become more robust to classical approximations.

Classically Approximating Variational Quantum Machine Learning with Random Fourier Features

TL;DR

The paper reframes variational quantum circuits as shift-invariant kernels whose Fourier spectrum Ω is determined by encoding Hamiltonians, and proposes classical Random Fourier Feature (RFF) surrogates to approximate VQCs without running on quantum hardware. Three RFF strategies—Distinct, Tree, and Grid sampling—are developed to sample frequencies from Ω and construct scalable classical models via kernel ridge regression, with theoretical bounds showing linear scaling in input dimensionality and logarithmic dependence on spectrum size. Empirical results across Pauli and more complex encodings, including real datasets like Fashion-MNIST and California Housing, demonstrate that RFF surrogates can match or even outperform VQCs using only a small fraction of Ω, particularly when frequency redundancy is high. These findings challenge the assumed quantum advantage of VQCs in many practical settings and emphasize the role of encoding structure and spectrum properties in determining learnability, while outlining conditions under which quantum advantage might still emerge.

Abstract

Many applications of quantum computing in the near term rely on variational quantum circuits (VQCs). They have been showcased as a promising model for reaching a quantum advantage in machine learning with current noisy intermediate scale quantum computers (NISQ). It is often believed that the power of VQCs relies on their exponentially large feature space, and extensive works have explored the expressiveness and trainability of VQCs in that regard. In our work, we propose a classical sampling method that may closely approximate a VQC with Hamiltonian encoding, given only the description of its architecture. It uses the seminal proposal of Random Fourier Features (RFF) and the fact that VQCs can be seen as large Fourier series. We provide general theoretical bounds for classically approximating models built from exponentially large quantum feature space by sampling a few frequencies to build an equivalent low dimensional kernel, and we show experimentally that this approximation is efficient for several encoding strategies. Precisely, we show that the number of required samples grows favorably with the size of the quantum spectrum. This tool therefore questions the hope for quantum advantage from VQCs in many cases, but conversely helps to narrow the conditions for their potential success. We expect VQCs with various and complex encoding Hamiltonians, or with large input dimension, to become more robust to classical approximations.
Paper Structure (31 sections, 5 theorems, 35 equations, 11 figures, 3 algorithms)

This paper contains 31 sections, 5 theorems, 35 equations, 11 figures, 3 algorithms.

Table of Contents

  1. Spectral Properties of Variational Quantum Circuits
  2. Definitions
  3. Quantum Models are Large Fourier Series
  4. Quantum models are shift-invariant kernel methods
  5. Random Fourier Features approximates high-dimensional kernels
  6. RFF Methods for Approximating VQCs
  7. Related Work
  8. RFF Sampling strategies
  9. RFF with Distinct sampling
  10. RFF with Tree sampling
  11. RFF with Grid sampling
  12. Number of Samples and Approximation Error
  13. Pauli encoding
  14. Grid sampling
  15. Limitations of RFF for Approximating VQCs
  16. ...and 16 more sections

Key Result

Theorem 1

Let $\mathcal{X}$ be a compact set of $\mathbb{R}^d$, and $\epsilon > 0$. We consider a training set $\{(x_i, y_i)\}_{i=1}^M$. Let $f$ be a VQC model with $L$ encoding Pauli gates on each of the $d$ dimensions and full freedom on the associated frequency coefficients, trained with a regularization $ with $C_1$, $C_2$ being constants depending on $\sigma_y$, $|\mathcal{X}|$. We recall that in Eq.eq

Figures (11)

  • Figure 1: Random Fourier Features as a classical approximator of quantum models. Instead of training a Variational Quantum Circuit by using a quantum computer, we propose to train a classical kernel built by sampling a few frequencies of the quantum model. These frequencies can be derived from the quantum circuit architecture, in particular from the encoding gates. Using Random Fourier Features, one can build a classical model which performs as good as the quantum model with a bounded error and a number of samples that grows nicely.
  • Figure 2: From encoding Hamiltonians to Frequencies. The frequencies composing the VQC model (on one dimensional input) come from all the combinations of eigenvalues from each encoding Hamiltonians. This can be seen as a tree, with $L=3$ Hamiltonians in this figure. We also see potential redundancy in the leaves.
  • Figure 3: Variational Quantum Circuits give rise to Fourier Series. In a quantum machine learning task, classical data is encoded in a subset of variational gates of a quantum circuit (green), while blue gates are trainable.
  • Figure 4: Random instance of a VQC. In this example, three encoding Hamiltonians $\{H_1, H_2, H_3 \}$ are randomly assigned over four qubits, and load a 1-dimensional vector $x$. Following each encoding gate $H_i$, an ansatz with trainable parameters and a ladder of CNOTs is applied, $l_i$ times in a row.
  • Figure 5: RFF performance for $L=5, d=4$, to approximate random VQCs with Pauli encoding.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof