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Quantum Machine Intelligence

Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks

Patrick Holzer, Ivica Turkalj

Published Mar 16, 2026• 1 citation
Publisher

Abstract

Quantum Neural Networks (QNNs) are a popular approach in Quantum Machine Learning. We analyze the frequency spectrum using Minkowski sums, which yields a compact algebraic description and permits explicit computation. Using this description, we prove several maximality results for broad classes of QNN architectures. Under some mild technical conditions we establish a bijection between classes of models with the same area $$A:=R\cdot L$$ that preserves the frequency spectrum, where R denotes the number of qubits and L the number of layers, which we consequently call spectral invariance under area-preserving transformations. With this we explain the symmetry in R and L in the results often observed in the literature and show that the maximal frequency spectrum depends only on the area $$A=RL$$ and not on the individual values of R and L. Moreover, we collect and extend existing results and specify the maximum possible frequency spectrum of a QNN with an arbitrary number of layers as a function of the spectrum of its generators. In the case of arbitrary dimensional generators, where our two introduced notions of maximality differ, we extend existing Golomb ruler based results and introduce a second novel approach based on a variation of the turnpike problem, which we call the relaxed turnpike problem. We clarify comprehensively how the generators of a QNN must be chosen in order to obtain a maximal frequency spectrum for a given area A, thereby contributing to a deeper theoretical understanding. However, our numerical experiments show that trainability depends not only on $$A = RL$$ , but also on the choice of (R, L), so that knowledge of the maximum frequency spectrum alone is not sufficient to ensure good trainability. While the maximality analysis limits the space of attainable frequency spectra and characterizing these spectra helps to understand the effect of the ansatz choice, it does not by itself provide sufficient criteria for ansatz selection, and further considerations are required.

DOI
10.1007/s42484-026-00372-x

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Quantum Neural Networks (QNNs) are a popular approach in Quantum Machine Learning. We analyze the frequency spectrum using Minkowski sums, which yields a compact algebraic description and permits explicit computation. Using this description, we prove several maximality results for broad classes of QNN architectures. Under some mild technical conditions we establish a bijection between classes of models with the same area A:=R⋅LA:=R\cdot LA:=R⋅L that preserves the frequency spectrum, where R denotes the number of qubits and L the number of layers, which we consequently call spectral invariance under area-preserving transformations. With this we explain the symmetry in R and L in the results often observed in the literature and show that the maximal frequency spectrum depends only on the area A=RLA=RLA=RL and not on the individual values of R and L. Moreover, we collect and extend existing results and specify the maximum possible frequency spectrum of a QNN with an arbitrary number of layers as a function of the spectrum of its generators. In the case of arbitrary dimensional generators, where our two introduced notions of maximality differ, we extend existing Golomb ruler based results and introduce a second novel approach based on a variation of the turnpike problem, which we call the relaxed turnpike problem. We clarify comprehensively how the generators of a QNN must be chosen in order to obtain a maximal frequency spectrum for a given area A, thereby contributing to a deeper theoretical understanding. However, our numerical experiments show that trainability depends not only on A=RLA = RLA=RL , but also on the choice of (R, L), so that knowledge of the maximum frequency spectrum alone is not sufficient to ensure good trainability. While the maximality analysis limits the space of attainable frequency spectra and characterizing these spectra helps to understand the effect of the ansatz choice, it does not by itself provide sufficient criteria for ansatz selection, and further considerations are required.
Quantum Neural Networks (QNNs) are a popular approach in Quantum Machine Learning. We analyze the frequency spectrum using Minkowski sums, which yields a compact algebraic description and permits explicit computation. Using this description, we prove several maximality results for broad classes of QNN architectures. Under some mild technical conditions we establish a bijection between classes of models with the same area that preserves the frequency spectrum, where R denotes the number of qubits and L the number of layers, which we consequently call spectral invariance under area-preserving transformations. With this we explain the symmetry in R and L in the results often observed in the literature and show that the maximal frequency spectrum depends only on the area and not on the individual values of R and L. Moreover, we collect and extend existing results and specify the maximum possible frequency spectrum of a QNN with an arbitrary number of layers as a function of the spectrum of its generators. In the case of arbitrary dimensional generators, where our two introduced notions of maximality differ, we extend existing Golomb ruler based results and introduce a second novel approach based on a variation of the turnpike problem, which we call the relaxed turnpike problem. We clarify comprehensively how the generators of a QNN must be chosen in order to obtain a maximal frequency spectrum for a given area A, thereby contributing to a deeper theoretical understanding. However, our numerical experiments show that trainability depends not only on, but also on the choice of (R, L), so that knowledge of the maximum frequency spectrum alone is not sufficient to ensure good trainability. While the maximality analysis limits the space of attainable frequency spectra and characterizing these spectra helps to understand the effect of the ansatz choice, it does not by itself provide sufficient criteria for ansatz selection, and further considerations are required.
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.
Quantum neural networks constitute a key class of near-term quantum learning models, yet their training dynamics remain not fully understood. Here, we present a unified theoretical framework for the frequency principle (F-principle) that characterizes the training dynamics of both classical and quantum neural networks. Within this framework, we prove that quantum neural networks exhibit a spectral bias toward learning low-frequency components of target functions, mirroring the behavior observed in classical deep networks. We further analyze the impact of noise and show that, when single-qubit noise is applied after encoding-layer rotations and modeled as a Pauli channel aligned with the rotation axis, the Fourier component labeled by ω\boldsymbolωω is suppressed by a factor (1−2γ)∥ω∥1(1-2γ)^{\|\boldsymbolω\|_1}(1−2γ)∥ω∥1​. This leads to exponential attenuation of high-frequency terms while preserving the learnability of low-frequency structure. In the same setting, we establish that the resulting noisy circuits admit efficient classical simulation up to average-case error. Numerical experiments corroborate our theoretical predictions: Quantum neural networks primarily learn low-frequency features during early optimization and maintain robustness against dephasing and depolarizing noise acting on the encoding layer. Our results provide a frequency-domain lens that unifies classical and quantum learning dynamics, clarifies the role of noise in shaping trainability, and guides the design of noise-resilient quantum neural networks.
Abstract 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.
Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.
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.