Quantum Machine Learning (QML) models have shown great promise in solving complex problems, but their expressivity and trainability is often limited by the choice of architecture. In the Fourier framework, layer-dependent data re-uploading has been proposed to increase expressivity, but applying this scheme often results in a trade-off between generalizability and trainability. By ensuring that the redundancies of some frequencies in the Fourier spectrum are high when selecting an encoding, the flexibility of the corresponding coefficients is increased, which in turn influences the trainability of the model. In this work we empirically investigate how a data-tailored re-uploading scheme affects trainability of QML models and find an indication for better performing models.
Quantum Machine Learning (QML) models have shown great promise in solving complex problems, but their expressivity and trainability is often limited by the choice of architecture. In the Fourier framework, layer-dependent data re-uploading has been proposed to increase expressivity, but applying this scheme often results in a trade-off between generalizability and trainability. By ensuring that the redundancies of some frequencies in the Fourier spectrum are high when selecting an encoding, the flexibility of the corresponding coefficients is increased, which in turn influences the trainability of the model. In this work we empirically investigate how a data-tailored re-uploading scheme affects trainability of QML models and find an indication for better performing models.
Quantum machine learning models incorporating data re-uploading circuits have garnered significant attention due to their exceptional expressivity and trainability. However, their ability to generate accurate predictions on unseen data, referred to as the predictive performance, remains insufficiently investigated. This study reveals a fundamental limitation in predictive performance when deep encoding layers are employed within the data re-uploading model. Concretely, we theoretically demonstrate that when processing high-dimensional data with limited-qubit data re-uploading models, their predictive performance progressively degenerates to near random-guessing levels as the number of encoding layers increases. In this context, the repeated data uploading cannot mitigate the performance degradation. These findings are validated through experiments on both synthetic linearly separable datasets and real-world datasets. Our results demonstrate that when processing high-dimensional data, the quantum data re-uploading models should be designed with wider circuit architectures rather than deeper and narrower ones.
The redundancy distribution of the frequency spectrum has been shown in the literature to impact the expressivity and trainability of Quantum Fourier Models (QFMs). In this work, we address the question of how this redundancy spectrum is shaped by the choice of eigenvalues of the data re-uploading Hamiltonians, using a simple mathematical formalism based on generating functions. We derive exact and asymptotic redundancy profiles for several structured eigenvalue choices identical for every layer, including arithmetic progressions and single-qubit Pauli encodings, and show that both approach a Gaussian profile as the number of layers grows. We then show that this Gaussian limit is not specific to these constructions but is a generic feature of QFMs built from integer eigenvalues that are identical in every layer. These results can be tied to the central limit theorem for random walks. These results clarify why generic or unstructured encoding choices give rise to a redundancy bias that favours low frequencies, highlighting the importance of well-thought-out encoding strategies when constructing a QFM.
Quantum re-uploading models have been extensively investigated as a form of machine learning within the context of variational quantum algorithms. Their trainability and expressivity are not yet fully understood and are critical to their performance. In this work, we address trainability through the lens of the magnitude of the gradients of the cost function. We prove bounds for the differences between gradients of the better-studied data-less parameterized quantum circuits and re-uploading models. We coin the concept of absorption witness to quantify such difference. For the expressivity, we prove that quantum re-uploading models output functions with vanishing high-frequency components and upper-bounded derivatives with respect to data. As a consequence, such functions present limited sensitivity to fine details, which protects against overfitting. We performed numerical experiments extending the theoretical results to more relaxed and realistic conditions. Overall, future designs of quantum re-uploading models will benefit from the strengthened knowledge delivered by the uncovering of absorption witnesses and vanishing high frequencies.
Quantum machine learning (QML) is a new field in its infancy, promising performance enhancements over many classical machine learning (ML) algorithms. Data reuploading is a QML algorithm with a focus on utilizing the power of a singular qubit as an individually capable classifier. Recently, there have been studies set out to explore the concept of data re-uploading in a classification setting, however, important aspects are often not considered in experiments, which may hinder our understanding of the methodology’s performance. In this work, we conduct an analysis of the single-qubit data re-uploading methodology, in relation to the effect that system depth has on classification and robustness performances against the influence of environmental noise during training. This is aimed towards bridging together previous works in order to solidify the concepts of the methodology, and provide reasonable insight into how transferable the methodology is when applied to non-synthetic data. To further demonstrate the findings, we also analyse the results of a case study using a subset of MNIST data. From this work, our experimental results support that an increase in system depth can lead to higher classification performances, as well as improved stability during training in noisy environments, with the sharpest performance improvements seemingly occurring between 1–3 uploading layer repetitions. Leading on from our experimental results, we suggest areas for further exploration, to ensure we can maximize classification performance when using the data re-uploading methodology.
The encoding of classical data into quantum states constitutes the primary performance bottleneck in Quantum Machine Learning (qml) on Noisy Intermediate-Scale Quantum (nisq) devices. No existing framework jointly characterises resource cost, expressivity, and noise robustness, nor provides actionable selection guidelines for practitioners. This survey addresses that gap through a systematic review of 66 primary works (2017-2026) assembled via a PRISMA-adapted protocol across five academic databases. Four principal contributions are made. First, a three-axis cost-expressivity-robustness taxonomy classifies all major encoding families - basis, angle, dense-angle, amplitude, data re-uploading, and IQP - along independently measurable axes. Second, closed-form depth-fidelity bounds under nisq decoherence channels identify the critical gate-error rate p* ~ 10^-3 below which amplitude encoding is viable. Third, a unified treatment of Fourier expressivity, barren-plateau onset, and quantum kernel concentration as functions of the encoding circuit provides the first joint trainability analysis. Fourth, a five-regime decision framework maps (D, n, p, tau) - feature dimension, qubit budget, error rate, and task type - to a hardware-grounded encoding recommendation. The central finding is that for p >= 10^-3, shallow angle-based encodings consistently outperform amplitude encoding in practice, despite the latter's exponential qubit advantage.
In this work, we investigate the phenomenon of spectral bias in quantum machine learning, where, in classical settings, models tend to fit low-frequency components of a target function earlier during training than high-frequency ones, demonstrating a frequency-dependent rate of convergence. We study this effect specifically in parameterised quantum circuits (PQCs). Leveraging the established formulation of PQCs as Fourier series, we prove that spectral bias in this setting arises from the ``redundancy'' of the Fourier coefficients, which denotes the number of terms in the analytical form of the model contributing to the same frequency component. The choice of data encoding scheme dictates the degree of redundancy for a Fourier coefficient. We find that the magnitude of the Fourier coefficients' gradients during training strongly correlates with the coefficients' redundancy. We then further demonstrate this empirically with three different encoding schemes. Additionally, we demonstrate that PQCs with greater redundancy exhibit increased robustness to random perturbations in their parameters at the corresponding frequencies. We investigate how design choices affect the ability of PQCs to learn Fourier sums, focusing on parameter initialization scale and entanglement structure, finding large initializations and low-entanglement schemes tend to slow convergence.