Parameterized quantum circuits are widely investigated in quantum machine learning, yet how their architecture can provide task-aligned inductive biases remains poorly understood. We study data reuploading quantum circuits with two distinct inputs: a signal that varies between samples and a fixed, task-defined spectral query that specifies how the signal should be probed. We show that encoding the query can change the circuit's hypothesis class over the signal by modulating how its existing signal frequencies are combined, without introducing new sample-dependent information. This mechanism matches tasks built on nonuniform Fourier transforms, whose evaluation frequencies specify the query and whose sampling positions guide its encoding. We examine this connection in a classification task inspired by synthetic aperture radar, where acquisition geometry determines a query for classifying unfocused spectral patches before image formation. Across small-scale spectral and radar-like tasks, aligned queries improve learning from limited data over parameter-matched circuits with mismatched or absent queries. These results provide a route from known spectral task structure to the design and validation of inductive biases in parameterized quantum circuits.
Parameterized quantum circuits are widely investigated in quantum machine learning, yet how their architecture can provide task-aligned inductive biases remains poorly understood. We study data reuploading quantum circuits with two distinct inputs: a signal that varies between samples and a fixed, task-defined spectral query that specifies how the signal should be probed. We show that encoding the query can change the circuit's hypothesis class over the signal by modulating how its existing signal frequencies are combined, without introducing new sample-dependent information. This mechanism matches tasks built on nonuniform Fourier transforms, whose evaluation frequencies specify the query and whose sampling positions guide its encoding. We examine this connection in a classification task inspired by synthetic aperture radar, where acquisition geometry determines a query for classifying unfocused spectral patches before image formation. Across small-scale spectral and radar-like tasks, aligned queries improve learning from limited data over parameter-matched circuits with mismatched or absent queries. These results provide a route from known spectral task structure to the design and validation of inductive biases in parameterized quantum circuits.
A central design principle in modern machine learning and artificial intelligence is to align a model's inductive bias with the structure of its input data. For matrix-valued inputs, relevant matrix-level relationships can be characterised through spectral values and spectral subspaces; however, common coordinate-wise rotation-gate data-encoding unitaries used in most quantum machine learning models do not explicitly construct such a matrix-level representation. We introduce Quantum Spectral Models (QSMs), in which we construct the generator of the data-encoding unitary directly from each input matrix. We study three QSM variants based on symmetric, global block, and non-overlapping patch-local block Hamiltonians. Their outputs admit truncated Fourier representations in which input-dependent spectral gaps supply candidate phase carriers, while spectral subspaces help determine their coefficients. We evaluate the QSMs and comparison quantum models on two matrix representations of Pendigits and two controlled synthetic tasks defined by spectral statistics. At the largest evaluated circuit depth, QSM variants lead the tested quantum models in mean test accuracy across all four benchmarks. The patch-local QSM leads on Pendigits, whereas the global block-Hamiltonian QSM leads on the controlled spectral tasks. Ablations show a task-dependent reversal: subspace-preserving controls perform better on Pendigits, whereas spectral-value-only controls lead among the tested ablations on the synthetic tasks. Together, these results shed new light on quantum machine-learning model design by showing how input-conditioned spectral representations can provide an analysable inductive bias, while offering a broader perspective on structure-aware model design in machine learning and artificial intelligence.
Parametrised quantum circuits learn by adjusting gate parameters, while their design shapes the functions they can represent and how readily they learn them. We introduce the circuit harmonic matrix, a fixed matrix organising Fourier expansions over inputs and parameters. It makes the effects of encoding, gates, initial state and observable explicit in coefficient variance, covariance and the quantum neural tangent kernel, linking variation across parameter space to local sensitivity. The construction also constrains representable functions and lower-bounds fitting error. For fixed Clifford gates and independently parametrised Pauli or controlled rotations, we develop a two-copy propagation method that computes covariance and the averaged tangent kernel analytically for uniform parameters, without constructing the full matrix or sampling parameters. Across 400 configurations spanning eight circuit families, we compare these quantities with actual learning. Larger coefficient variance is strongly associated with faster learning and lower error for the corresponding target frequency, revealing spectral bias towards lower frequencies. Increasing depth and qubit count generally suppresses total variance, yet deeper circuits usually learn faster and achieve lower error, while adding qubits slows early learning and has a frequency-dependent effect on final error. Matched targets show no robust overall advantage from covariance alignment. For one circuit setting, initial tangent kernels closely forecast the modest loss reductions reached in small-step gradient descent. These results connect circuit choices to representational constraints and learning performance, providing a basis for selecting designs suited to particular learning tasks.
Existing theory of variational quantum learning characterises circuits through three complementary perspectives: representation, optimization, and statistical capacity, but does not ask whether a chosen data encoding is compatible with the learning task at hand. We develop a task-centric theory of learnability to answer this question. Building on the Fourier characterization of parameterized quantum circuits, we introduce spectral task-circuit alignment, a quantitative measure of the overlap between the hypothesis space generated by a data encoding and the spectral structure of a target function. We prove that this quantity determines an irreducible approximation floor that cannot be reduced by deeper ansätze, additional trainable parameters, improved optimization, or more training data. Combining this result with the Effective Dimension framework gives a two-stage characterization of learnability that separates approximation barriers from statistical barriers. The resulting theory further leads to a principled criterion for encoding design, showing that alignment is maximized by encodings whose accessible frequency sets capture the dominant Fourier components of the target. We illustrate the framework through an analytical study of the feature-map construction of Abbas et al. on the checkerboard classification target, demonstrating that the empirical advantage of the hard feature map follows directly from its superior task-circuit alignment. More broadly, this work introduces task compatibility as a fourth theoretical component of quantum machine learning, complementing expressibility, trainability, and statistical capacity. The resulting framework provides a unified foundation for analysing approximation, learnability, and encoding design before any circuit is trained.
This thesis investigates how principles from physics can serve as inductive biases that make learning and optimization more tractable in machine learning and quantum computing. Physical structure - encoded in Hamiltonians, physics-inspired dynamics, circuit architectures, and measurement protocols - constrains the hypothesis class, enabling learning under limited computational and data budgets in high-dimensional settings. The thesis presents four case studies showing how these physics-inspired inductive biases can be incorporated into both classical and quantum models. First, quantum diffusion models are introduced as generative models for quantum states. A Markovian noising process, motivated by statistical physics, defines the forward dynamics, while parameterized quantum circuits implement the reverse dynamics. Second, a Potts-model Hamiltonian is used to formulate a graph-coloring objective, leading to a physics-informed neural architecture that generalizes beyond the training distribution. Third, parameterized quantum circuits and hybrid classical-quantum models are evaluated for an anomaly detection task in highly imbalanced classical transaction data, using circuit structure as an additional inductive bias. Fourth, a data-driven classical-shadow method is developed, in which a neural network maps shadow snapshots to observable expectation values. This work fits into the broader effort of quantum device certification. Overall, the results show that physics-inspired inductive biases are a robust lever for resource-efficient learning, while NISQ-era quantum resources provide modest task- and metric-dependent benefits rather than a consistent advantage over classical baselines.
Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high-frequency or non-dominant ones; a phenomenon we term the quantum Fourier parameterization bias. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single-stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.
Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high‑frequency or non-dominant ones - a phenomenon we term the \textit{quantum Fourier parameterization bias}. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single‑stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.