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.
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.
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.
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.
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.
The quantum kernel method is one of the key approaches to quantum machine learning, which has the advantage of not requiring optimization and its theoretical simplicity. By virtue of these properties, several experimental demonstrations and discussions of the potential advantages have been developed so far. However, as is the case in classical machine learning, not all quantum machine learning models could be regarded as kernel methods. In this work, we explore a quantum machine learning model with a deep parametrized quantum circuit and aim to go beyond the conventional quantum kernel method. In this case, the expressive power and performance are expected to be enhanced, while the training process might be a bottleneck because of the barren plateaus. Moreover, the high computational cost of gradient-based optimization and the large search space of the gradient-free optimization directly make the training intractable. However, we find that parameters of a deep enough quantum circuit do not move much from their initial values during training, allowing for a first-order expansion with respect to the parameters. This behavior is similar to that of the neural tangent kernel in classical machine learning, and such a quantum machine learning with deep variational quantum circuits can be described by another emergent kernel, the . We show that the proposed quantum tangent kernel has the potential to outperform the conventional quantum kernel method by performing a classification task on an ansatz-generated dataset. This work provides a different direction beyond the conventional quantum kernel method and explores the potential power of quantum machine learning with deep parametrized quantum circuits. Published by the American Physical Society 2024
Variational quantum circuits (VQCs) can be understood through the lens of Fourier analysis. The function space represented by any circuit architecture can be described through a truncated Fourier sum. We show that the spectrum available to that truncated Fourier sum is not entirely determined by the encoding gates of the circuit, since the variational part of the circuit can constrain certain coefficients to zero, effectively removing that frequency from the spectrum. To the best of our knowledge, we give the first description of the functional dependence of the Fourier coefficients on the variational parameters as trigonometric polynomials. This allows us to provide an algorithm that computes the exact spectrum of any given circuit and the corresponding Fourier coefficients. Finally, we demonstrate that by comparing the Fourier transform of the dataset to the available spectra, it is possible to predict which VQC of a given list of choices will best fit the data.
The output of a parameterized quantum circuit (PQC) can be expressed as a finite Fourier series whose accessible frequencies are fixed by the data-encoding gates. While the encoder determines which frequencies can appear, the corresponding Fourier coefficients depend on how the trainable and entangling gates are arranged. Although existing studies provide metrics for characterizing how gate structure affects Fourier coefficients, they do not translate these analyses into an explicit design criterion specifying how gates should be arranged to make a target coefficient reachable. In this paper, we provide such a criterion. Using the adjoint action of the encoding generator, we decompose operator space into two-dimensional invariant planes indexed by frequency and show that each Fourier coefficient is exactly a sum of bilinear projections of the effective state and observable onto the planes at that frequency. Because, for Pauli encodings, high-frequency planes are spanned by mixed multi-qubit Pauli strings, a target coefficient can contribute to the output only when local rotations and entangling layers are arranged so that both the effective state and observable acquire support on one of its planes. For Pauli readouts and commuting two-qubit entanglers, this yields a circuit design rule that specifies, for a given interaction graph, a placement of local rotations and entangling layers that makes a target Fourier coefficient reachable. Using a single encoding layer, we validate the proposed design through a placement ablation, regression tasks from PDEBench and physics-informed Maxwell field modeling, and further demonstrate its robustness to moderate simulated gate noise and damping.