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.
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.
Variational quantum algorithms (VQAs) have shown advantages over classical approaches in solving many problems, yet their quantum neural networks (QNNs) models still perform poorly in complex tasks like multi-class classification and high-frequency time-series prediction, raising questions about their feasibility. While existing studies have characterized their performance from several perspectives, but research on their learning performance and visualization remains scarce. This work develops the universal approximation theory of variational quantum algorithms for multivariate function fitting, supported by a verification in visual training analysis on two-input binary datasets. We first visualize how a single-qubit VQA with single-encoding process four types of binary data, revealing inherent limitations, and then formulate the measurement expectation of VQA as a partial Fourier series, establishing its universal approximation capacity and showing that the Fourier spectrum is determined by the data encoding strategy: multiple encoding layers broaden the spectrum and enhance learning performance. Guided by this theory, we adjust the circuit structure and achieve nearly 100% accuracy across all four binary classification tasks.
When parametrized quantum circuits are viewed as models that translate data inputs into predictions, quantum computers can be utilized for supervised learning. Many crucial theoretical aspects of these models remain unknown, even though considerable work has been done to understand the practical consequences of this method. Quantum models can access richer and richer frequency spectra by repeatedly repeating straightforward data-encoding gates. Since simulation time increases exponentially, simulation-based training cannot be used to big QML models. Variational quantum algorithms and quantum GAN are expected to give the best results if applied for different health care applications in microseconds. The present work explains how QML can be applied for variational quantum eigen solving. This paper also explains quantum Fourier transform analysis.
Fourier analysis has become a crucial tool for understanding the expressivity of Variational Quantum Circuit (VQC) models, as well as an important indicator of barren plateaus (BP). While existing literature has only studied angle-embedded VQCs in a noiseless environment, here we develop the Fourier analysis of VQCs with non-linear data embedding, with particular focus on amplitude embedding, which provides a naturally compact encoding scheme. We first investigate a subtle difference in the domain of input features within amplitude embedding that leads to a distinct expressivity of the zero-frequency Fourier coefficient. By assuming that the ensemble of unitaries generated from the parameter space forms at least a 2-design with respect to the unitary group, we derive, via Weingarten calculus, that the mean of the Fourier coefficients is concentrated at zero, and the variance scales at an exponentially decaying order with respect to the multi-dimensional frequency magnitude. When a noise channel with unitary Kraus operators and probabilities {pk} is taken into account, the variance is further suppressed by a factor (∑kpk2)Q<1, where Q is the number of channel instances applied. Furthermore, we demonstrate and validate the analytical results through simulations, both noiseless and noisy, including a case where target functions are decomposed into non-integer frequencies, highlighting the practical utility of the approach. Our results establish a rigorous Fourier framework for amplitude-encoded VQCs, offering both theoretical guarantees on expressivity, hence trainability scaling in the frequency domain, as well as practical simulations for deployment on noisy quantum devices.
Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data-encoding gates in the circuit. By repeating simple data-encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.
Introduction: A variational quantum classifier realises a function whose Fourier spectrum is fixed by the circuit and whose coefficients must be found by an optimiser. We study a readout borrowed from quantum phase estimation (QPE): a data-dependent operator acts on a target register through controlled powers, and the class label is read from an estimation register of m qubits after an inverse quantum Fourier transform (QFT). Materials and methods: We derive the exact readout distribution—a Fejér-weighted expansion in the moments of the operator—and characterise its Fourier spectrum for two single-qubit blocks: H ⋅ P h ⋅ R Y (family A) and a universal U 2 block (family B). All analytical statements are verified against a state-vector simulator, including attainment of the predicted spectrum. Three ablation controls alter one component each at an unchanged parameter count: the encoding QFT is removed, the exponential ladder of controlled powers is replaced by single powers, and the inverse QFT is replaced by independent Hadamard tests. The benchmark comprises 7680 training runs of 24 circuits on three tasks, with one generated dataset or one cross-validation fold as the unit of inference (twenty per sample size), exact signed-rank tests, and the corrected resampled t -test for the overlapping folds. Results: The support of every readout probability lies in parity-locked lattice cubes of maximal harmonic order 2 m + 1 . The maximal order is attained at every register size, in every feature and every outcome, except for family A with one target qubit, where the two highest orders cancel identically; the full support is attained at every register size used here ( t ≤ 4 , m ≤ 3 ), and the band is tapered towards its edge. Removing the encoding QFT costs family B 0.144 and 0.082 in test accuracy on the two-feature tasks, and replacing the power ladder by single powers costs 0.086 – 0.225 wherever m ≥ 2 ; replacing the inverse QFT by Hadamard tests has no detectable effect. The architecture leads the independently parameterised reference circuits only on the three-class task ( + 0.119 , p < 0.0001 ), where a parameter-tied re-uploading circuit with 12 rather than 20 parameters is not resolved from it ( + 0.023 , 95% confidence interval (CI)
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.