The effect of data encoding on the expressive power of variational quantum machine learning models
Using Fourier series analysis, this paper reveals how data encoding strategies influence the expressive power of variational quantum models, demonstrating universality with rich frequency spectra.
Key Findings
Methodology
This work employs Fourier series formalism to represent the output of parameterized quantum circuits as a sum over frequencies determined by the data encoding Hamiltonians. By analyzing single and repeated encoding schemes, it uncovers how the frequency spectrum can be systematically expanded. Eigenvalue decomposition of the encoding Hamiltonian isolates the frequency set Ω, while circuit parameters control Fourier coefficients. Theoretical proofs establish that with sufficiently rich frequency spectra, quantum models can approximate any function in L² space. Numerical experiments with randomly initialized circuits validate the relationship between frequency spectrum expansion and function approximation capacity.
Key Results
- Repeated encoding strategies—such as multiple layers or parallel encodings—linearly extend the frequency spectrum, with the maximum supported frequency being r, the number of repetitions. For example, a single encoding supports frequencies {−1, 0, 1}, while r repetitions support {−r, ..., r}, significantly enhancing the model’s expressive power.
- The frequency set Ω is derived from the eigenvalue differences of the encoding Hamiltonian. When eigenvalues are integers, the spectrum becomes a subset of integers, allowing the model to represent any finite Fourier series. Experiments with Pauli-X rotations confirm that increasing repetitions broadens the spectrum and improves approximation accuracy.
- Theoretically, if the frequency spectrum is dense enough and Fourier coefficients are controllable, the quantum model can approximate any square-integrable function arbitrarily well, demonstrating universality. This is grounded in Fourier series completeness and the spectral richness condition.
Significance
This research offers a fundamental understanding of how data encoding shapes the function class that quantum models can learn. By framing the problem within Fourier analysis, it provides a rigorous foundation for designing quantum circuits with desired expressive capabilities. The insights bridge the gap between quantum circuit architecture and classical function approximation theory, enabling systematic improvements in quantum machine learning models. Practically, it guides the development of more powerful, generalizable quantum algorithms for supervised learning tasks, addressing key limitations of current approaches.
Technical Contribution
The paper introduces a formalism that maps parameterized quantum circuits onto partial Fourier series, explicitly linking frequency spectra to the eigenvalues of data encoding Hamiltonians. It demonstrates how repeated encoding schemes extend the frequency support, thus increasing the model’s capacity. The authors prove that with sufficiently rich spectra and controllable coefficients, quantum models are universal function approximators. This work combines linear algebra, Fourier analysis, and quantum information theory to establish a new theoretical framework for quantum model expressivity, providing quantitative bounds and universality proofs.
Novelty
This work is the first systematic application of Fourier series analysis to quantify the expressive power of variational quantum circuits in supervised learning. It uniquely connects data encoding strategies with spectral properties, showing how repeated encodings expand the frequency support and enable universality. Unlike prior studies focusing solely on quantum information processing capabilities, this research emphasizes the function approximation perspective, offering a new theoretical lens that unifies quantum circuit design with classical approximation theory.
Limitations
- The analysis assumes idealized conditions where the eigenvalues of encoding Hamiltonians are precisely controllable and noise-free. In practical quantum hardware, decoherence and gate errors may distort the spectral properties, reducing the effectiveness of frequency spectrum control.
- The controllability of Fourier coefficients depends on the complexity of the trainable circuit blocks. Shallow or limited-parameter circuits may not achieve arbitrary coefficient control, constraining the approximation power.
- The theoretical universality relies on the assumption of infinitely rich spectra and perfect coefficient control, which are challenging to realize in near-term quantum devices with limited coherence times and hardware constraints.
Future Work
Future research will explore robustness of spectral control under realistic noise conditions, develop hardware-efficient encoding schemes, and extend the framework to multi-variable inputs and higher-dimensional data. Additionally, integrating this spectral analysis with learning theory could establish explicit bounds on generalization and sample complexity, guiding the design of scalable quantum neural networks. Investigating adaptive encoding strategies and their impact on spectral richness also remains a promising direction.
AI Executive Summary
Quantum machine learning has emerged as a promising paradigm, leveraging the unique capabilities of quantum computers to process complex data. However, understanding the fundamental limits of these models—particularly their expressive power—remains a challenge. Traditional approaches often focus on the quantum circuit’s ability to perform universal quantum computation, but this does not directly translate to the ability to learn arbitrary functions from data.
This paper introduces a novel perspective by applying Fourier series analysis to parameterized quantum circuits. The core idea is that the output of a quantum model can be expressed as a sum over frequencies, with the set of accessible frequencies determined by the data encoding Hamiltonians. By analyzing how these frequencies can be expanded through repeated encoding strategies—either in parallel or sequentially—the authors demonstrate that the spectral support of the model can be systematically increased.
The mathematical framework hinges on the eigenvalue decomposition of the encoding Hamiltonian, which reveals the frequency set Ω as differences of eigenvalues. When these eigenvalues are integers, the frequency set becomes a subset of integers, enabling the model to represent any finite Fourier series. The authors rigorously prove that with sufficiently rich frequency spectra and controllable Fourier coefficients, quantum models can approximate any square-integrable function, thus establishing their universality.
Numerical experiments with randomly initialized circuits validate the theoretical predictions. Repeating the data encoding multiple times effectively broadens the frequency spectrum, allowing the quantum model to fit increasingly complex target functions. This insight provides a clear pathway for designing more expressive quantum neural networks by tuning the encoding strategy.
The broader significance of this work lies in its potential to guide the development of quantum algorithms with strong generalization capabilities. By linking spectral properties to function approximation, it offers a principled approach to circuit design, balancing expressivity and hardware constraints. Although practical challenges such as noise and limited parameter control remain, this spectral framework lays a solid foundation for future advances in quantum machine learning.
Looking ahead, research will focus on robustness under realistic noise conditions, multi-variable data encoding, and hardware-efficient implementations. The integration of Fourier analysis with learning theory could also yield explicit bounds on the sample complexity and generalization performance of quantum models, accelerating their transition from theoretical constructs to practical tools in AI and data science.
Deep Dive
Abstract
Quantum computers can be used for supervised learning by treating parametrised quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate practical implications of this approach, many important theoretical properties of these models remain unknown. Here we investigate how the strategy with which data is encoded into the model influences the expressive power of parametrised 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 realise all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.
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