Quantum Random Features: A Spectral Framework for Quantum Machine Learning

TL;DR

Quantum Random Features (QRF) and Quantum Dynamical Random Features (QDRF) address the challenge that quantum models often require deep, parameterized circuits to capture high-frequency components, limiting near-term scalability. By embedding data through layered $R_z$ rotations and spectral scrambling via fixed permutations or Ising dynamics, these models reproduce the spectral statistics of classical Random Fourier Features (RFF) while achieving an $N_f=2^N$ feature map with preprocessing cost $O(\,log(N_f))$. Empirical results on Fashion-MNIST reach up to $89.3\%$ accuracy with modest qubit counts, and QDRF matches or surpasses QRF_Rff, demonstrating hardware-amenable expressivity without variational optimization. The work establishes a principled, spectral-theoretic route to scalable QML on NISQ devices and opens avenues for spectral engineering in time-series, generative tasks, and reinforcement learning.

Abstract

Quantum machine learning (QML) models often require deep, parameterized circuits to capture complex frequency components, limiting their scalability and near-term implementation. We introduce \textit{Quantum Random Features} (QRF) and \textit{Quantum Dynamical Random Features} (QDRF), lightweight quantum reservoir models inspired by classical random Fourier features (RFF) that generate high-dimensional spectral representations without variational optimization. Using $Z$-rotation encoding combined with random permutations or Hamiltonian dynamics, these models achieve $N_f$-dimensional feature maps at preprocessing cost $O(\log(N_f))$. Spectral analysis shows that QRF and QDRF reproduce the behavior of RFF, while simulations on Fashion-MNIST reach up to 89.3\% accuracy-matching or surpassing classical baselines with scalable qubit requirements. By linking spectral theory with experimentally feasible quantum dynamics, this work provides a compact and hardware-compatible route to scalable quantum learning.

Figures (4)

• Figure 1: Schematic diagram of the QRF model. (a) the architecture consists of three key parameters: $d$ (input dimension), $N$ (number of qubits), and $L$ (number of encoding layers). (b) the circuit sequentially encodes classical data through $R_z$ rotations and applies layerwise permutations or dynamical evolutions to generate a high-dimensional quantum feature map.
• Figure 2: Test performance of (a) RFF-like Quantum Random Features ($\mathrm{QRF}_\mathrm{Rff}$), (b) Quantum Random Features (QRF), and (c) Quantum Dynamical Random Features (QDRF). In (a) comparison between classical RFF and the extreme quantum model as a function of $1/\sigma$ for $N = 10\sim 13$. In panel (b), QRF performance converges toward that of the extreme case as the number of layers $L$ increases, with $L \approx 20$–$30$ sufficient for all tested qubit counts and kernel widths $\sigma$. This result shows that QRF attains exponential feature expressivity with only linear preprocessing cost. In panel (c), QDRF employs Ising-model dynamics to achieve spectral scrambling and reaches accuracy comparable to the extreme case, demonstrating that natural Hamiltonian evolution offers an effective and experimentally compatible alternative to random permutations. In this simulation, we set $g/J=1.0$, $Jt=3.5$ and $\alpha=1.5$, where $t$ is a duration time per layers.
• Figure 3: Number of shots required. (a) shows the test accuracy obtained from the theoretical probability distribution and from empirical distributions computed with different shot numbers ranging from $2^8$ to $2^{20}$. The dashed lines represent the reference accuracy based on the theoretical distribution. (b) plots the absolute difference between theoretical and empirical accuracies as a function of the scaling variables $\sqrt{2^N/N_s}$ and $\sqrt{N^2/N_s}$. Here, $\mathrm{acc}_\mathrm{TH}$ and $\mathrm{acc}_\mathrm{EN}$ represent the testing accuracy using theoretical and empirical probability distributions, respectively. The parameters are set to $1/\sigma = 15$ and $L = 30$.
• Figure 4: Comparison of theoretical and numerical results for row vector correlations. The solid gray line represents the theoretical values calculated from the Bernoulli distribution, while the colored dots show numerical results obtained using permutation matrices. For each trial, 100 random pairs of indices $(i, j)$ are selected to compute $\langle \boldsymbol{V}_{i, \cdot}, \boldsymbol{V}_{j, \cdot} \rangle$. The gray dashed line indicates the asymptotic value $\sigma^2 / \sqrt{d}$. Each data point is averaged over $10^4$ realizations to estimate the standard deviation $\mathbb{V}\!\left[ \langle \boldsymbol{V}_{i, \cdot}, \boldsymbol{V}_{j, \cdot} \rangle \right]$. Different marker shapes correspond to different numbers of qubits. In this comparison, we set $\sigma = 1/10$.