Quantum Random Features: A Spectral Framework for Quantum Machine Learning

Quantum machine learning (QML) models often require deep, parameterized circuits to capture complex frequency components, limiting their scalability and near-term implementation. We introduce $\mathit{\text{Quantum}}\mathit{\text{Random}}\mathit{\text{Features}}$ (QRF) and $\mathit{\text{Quantum}}\mathit{\text{Dynamical}}\mathit{\text{Random}}\mathit{\text{Features}}$ (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.