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 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.
This thesis investigates the integration of quantum computing into machine learning, specifically focusing on reinforcement learning. The study focuses on the building of a code library to enable the exploration of the use of variational quantum circuits as function approximators within reinforcement learning algorithms, specifically policy gradient methods. A comprehensive framework is developed to analyze the performance and trainability of variational quantum circuit-based reinforcement learning models, supported by this library, to understand how different quantum circuit designs and data re-uploading techniques might affect the performance and trainability. To demonstrate the library’s capabilities, experiments were conducted in classic control environments like CartPole-v1 and Acrobot-v1, evaluating the performance (measured as average return) and trainability of three quantum models. These experiments examined the effects of data re-uploading, classical postprocessing, and inverse temperature parameter scheduling on both metrics. The empirical results show that variational quantum circuits are effective function approximators, with data re-uploading significantly enhancing model performance. Additionally, the globality of the measured observables plays a critical role, with higher globality values leading to better performance. Despite increasing circuit expressiveness, the gradient magnitudes and variances remain substantial throughout training. Furthermore, the tested inverse temperature schedules, used to adjust the policy’s level of exploration, did not show any significant advantage, indicating the challenge in finding optimal settings for these schedules. The study also introduces the universal quantum classifier adapted for reinforcement learning. Contrary to the barren plateau phenomenon, it has been empirically demonstrated that although increasing the number of qubits leads to performance drops in the Acrobot environment, it does not lead to vanishing gradients during the agents’ training process. Furthermore, under specific sets of hyperparameters, all three quantum models demonstrate strong and similar performance levels in both environments.
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.
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.
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.
Fixed encoding data re-uploading quantum circuits provide a striking example of universality emerging from a highly constrained architecture. However, universality alone is insufficient for assessing the theoretical and practical value of fixed and tunable upload circuits. The resource cost of removing tunability remains poorly understood. In this work, we establish quantitative depth-error scaling for approximating tunable upload circuits with fixed upload circuits. We show that a tunable upload circuit can be approximated by a fixed upload circuit using depth D=Oσ[(log(1/ε))σ] for every σ>1, with a target dependent constant overhead, thereby improving the previously known polynomial dependence on 1/ε with the same overhead. Our proof is based on an auxiliary extension approximation mechanism that combines Gevrey class construction, Jackson's theorem and generalized quantum signal processing theorem. Thus, the expressive power lost by removing tunability can be recovered using only polylogarithmic growth in circuit depth with a target dependent constant overhead. We further identify a periodic mismatch obstruction intrinsic to fixed upload approximations and use Turán-Nazarov inequalities to prove logarithmic lower bounds D=Ω(log(1/ε)) for the approximation of mismatch class target tunable upload circuits. Conceptually, our analysis reveals two structural mechanisms underlying approximation in fixed upload architectures: auxiliary extensions and mismatch obstructions. These results provide a quantitative understanding of how expressivity is transferred from tunable frequencies into circuit depth, and suggest a broader framework for studying approximation complexity in quantum signal processing and related quantum learning models.
Near-term quantum machine learning must balance expressivity, optimization, and hardware constraints. We study quantum re-uploading units (QRUs) as compact circuits and compare them, at matched parameter count, to a standard mono-encoded variational quantum circuit (VQC) baseline. On a three-feature calorimetry classification task, we train a single-qubit QRU that outputs a scalar in [−1,1] and map it to three classes via fixed thresholds. In this setting, QRUs obtain higher accuracy than the mono-encoded baseline. A controlled ablation over depth, input scaling, circuit template, optimizer, and gradient accumulation indicates that most gains occur at small depths, with diminishing returns as depth increases while training cost grows approximately linearly. To interpret these observations, we analyze reachable Fourier components and find that repeated data re-encoding expands the per-coordinate harmonic support relative to mono-encoding, consistent with a spectral activation study over random initializations. Finally, we report an end-to-end proof-of-execution of the trained model on a superconducting QPU via a cloud workflow, illustrating practical deployability under current constraints.