Re-uploading quantum data: a universal function approximator for quantum inputs

Amplitude embedding lets a variational quantum model read many classical features with few qubits, but the model output is then only a quadratic form in the normalized input. Data re-uploading, which encodes the input repeatedly between trainable blocks, can be used to enrich such a model. Because every additional upload of an amplitude embedding repeats a costly state preparation, the gain that an upload buys has to be known before the gates are spent. This tutorial thus addresses the problem of explaining what data re-uploading adds to an amplitude-embedded model and how large the gain is. We express sequential re-uploading and parallel re-uploading with two copies in one notation and show that each upload raises the reachable polynomial order by two. Under one trainable circuit family and one budget of trainable parameters, simulation results show that re-uploading raises the test coefficient of determination from 0.689 to between 0.883 and 0.975 on targets of order four and from 0.337 to between 0.642 and 0.783 on targets of order six, on all ten seeds. On a quadratic target and on handwritten digits, one upload remains the best choice, and every additional upload lowers the gate error that the classifier tolerates. These results help designers match the number of uploads to the polynomial order of the task at hand.

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.

Autoencoders transformed classical machine learning by solving the curse of dimensionality, enabling principled weight initialization and learning compact, structured representations. In this work, we extend this paradigm to quantum machine learning by introducing a variational autoencoder framework that learns task-specific quantum embeddings of classical data. We demonstrate that high-dimensional datasets, including ImageNet, can be compressed into a 13-qubit quantum representation while remaining reconstructable through a learned decoder. On MNIST (3 vs 5), our approach achieves 98.5% validation accuracy using a circuit-centric quantum classifier, within 1.2 percentage points of a classical neural network baseline (99.7%) and more than 30 percentage points above a naive amplitude-embedding approach. Unlike amplitude embeddings, which require full quantum state tomography for recovery, or angle embeddings, which generally rely on circuit inversion under restrictive assumptions, the proposed framework reconstructs the original data from only a polynomial number of measurements. The framework was further validated on IBM quantum hardware, confirming that the learned embeddings remain stable and reconstructable under real device noise.