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arXiv:2602.23409

Long Range Frequency Tuning for QML

Michael Poppel, Markus Baumann, Sebastian Wölckert, Claudia Linnhoff-Popien, Jonas Stein

cs.LGcs.AIcs.ETquant-phSubmitted Feb 26, 2026• Announced Feb 27, 2026• 0 citations• Revised Jul 20, 2026 (v3)
v1 · Feb 26, 2026v2 · May 15, 2026v3 · Jul 20, 2026
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Abstract

Angle-encoded variational quantum circuits admit a truncated Fourier series representation of their output, but approximating functions with maximum frequency $ω_{\max}$ using fixed unary encoding requires $\mathcal{O}(ω_{\max})$ encoding gates. Trainable-frequency (TF) circuits promise a reduction by learning the data-encoding prefactors alongside the ansatz parameters, adapting the accessible frequency spectrum to the target during training. We identify a practical barrier that prevents this promise from being realized: the prefactor gradient is suppressed by the spectral gap between the circuit's accessible frequencies and the target spectrum, independently of the ansatz parameters, confining gradient-driven prefactor movement to a narrow neighborhood of initialization. We propose \emph{ternary grid initialization} -- setting prefactors to $\{1, 3, 9, \ldots, 3^{k-1}\}$ -- which ensures every target frequency within $[-ω_{\max}, ω_{\max}]$ lies within $\tfrac{1}{2}$ unit of the accessible spectrum at initialization, so that the spectral-gap bound no longer constrains the target-driven gradient to be small. This is a necessary condition for reliable convergence, whose sufficiency we establish empirically. On a synthetic benchmark with target frequencies shifted well beyond the standard initialization range, ternary initialization achieves median $R^2 = 0.997$ versus $0.18$ for unary initialization, with $100\%$ of runs achieving $R^2 > 0.95$ against $0\%$. CMA-ES with $20\times$ the evaluation budget reaches only $25\%$ success, confirming the limitation is a property of the optimization landscape rather than of gradient-based optimization specifically. Real-world validation on two benchmark datasets demonstrates consistent advantages over both fixed and trainable unary baselines.

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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.
Parameterized quantum circuits as machine learning models are typically well described by their representation as a partial Fourier series of the input features, with frequencies uniquely determined by the feature map's generator Hamiltonians. Ordinarily, these data-encoding generators are chosen in advance, fixing the space of functions that can be represented. In this work we consider a generalization of quantum models to include a set of trainable parameters in the generator, leading to a trainable-frequency (TF) quantum model. We numerically demonstrate how TF models can learn generators with desirable properties for solving the task at hand, including nonregularly spaced frequencies in their spectra and flexible spectral richness. Finally, we showcase the real-world effectiveness of our approach, demonstrating an improved accuracy in solving the Navier-Stokes equations using a TF model with only a single parameter added to each encoding operation. Since TF models encompass conventional fixed-frequency models, they may offer a sensible default choice for variational quantum machine learning.
In the domain of variational quantum algorithms, quantum Fourier models (QFMs) provide a mathematically well defined structure for quantum machine learning (QML). There has been a substantial amount of work on the scalability and trainability of such models showcasing the potential but also the limitations for the prospective application of QFMs. However, much less is known in the context of pulse-level quantum computing, where the microwave parameters that implement unitary operations on the hardware are used to perform computations directly instead of through the interface of quantum circuits. In this work, we evaluate QFMs through the lens of pulse parameters and link metrics such as expressibility and Fourier coefficient correlation (FCC) to this extended set of variational parameters. We show that while control over pulse shapes does not significantly alter the global expressibility or structural correlations of the Ansatz, it fundamentally alters the local optimisation landscape. For composite gates, independent pulse scalings replace a single logical angle by multiple independently tunable sub-angles. This relaxes the rigid monomial couplings induced by the gate-level parameterisation, and provides gradient descent with higher-dimensional escape routes, decoupling local parameter constraints and significantly boosting performance during training. Following an analytical proof, we show numerical results validating our theory on training a QFM with an exponential (ternary) feature map on a Fourier series with the same frequencies.
Quantum Neural Networks (QNNs) offer a promising framework for integrating quantum computing principles into machine learning, yet their practical capabilities and limitations remain insufficiently studied. In this work, we systematically investigate the trainability and approximation properties of QNNs by benchmarking diverse circuit architectures and encoding strategies across synthetic and real-world datasets. We analyze several ansätze, including Hamming, binary, exponential, ternary, turnpike and Golomb, by evaluating their ability to learn synthetic data modeled as random finite Fourier series. To assess real-world applicability, we further evaluate QNNs on two time-series classification tasks: a Fischertechnik pneumatic leak detection dataset and the publicly available NASA bearing fault dataset. Our experiments show that while broader frequency spectra can theoretically enhance expressivity, practical trainability is strongly influenced by architectural factors such as qubit count and circuit depth. Notably, we find that QNNs perform best when the frequency spectrum is tailored to the target function's complexity but remains as compact as possible. Moreover, architectures with identical frequency spectra can differ in trainability, with configurations using more qubits and fewer layers generally performing better, except in the single-layer case. These findings provide guidelines for selecting QNN ansätze and offer new insights into the interplay between expressivity and trainability in quantum machine learning.