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arXiv:2610.01874

Fourier Symmetrization for Geometric Quantum Machine Learning

Letao Wang, Abdel Lisser, Sreejith Sreekumar, Zeno Toffano

quant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

Geometric quantum machine learning incorporates symmetry into quantum models, but how symmetry shapes their expressivity and guides effective model design remains insufficiently understood. We address this question through the Fourier representation of quantum Fourier models (QFMs). Symmetry organizes the frequency spectrum into orbits and sums the Fourier coefficients within each orbit into a symmetrized coefficient. When QFM trainable layers form independent exact 2-designs, the variance of each symmetrized coefficient equals the sum of the coefficient variances in its orbit. For single-layer QFMs with $\varepsilon$-approximate 2-design trainable layers, we bound the deviation from this identity. The resulting bound for individual Fourier coefficients can be exponentially tighter than an existing bound. The hyperoctahedral group provides an example of orbit growth that can mitigate vanishing expressivity of the symmetrized coefficients. Symmetrization of QFMs over an elementary abelian 2-group also yields pure multivariate Chebyshev polynomial basis functions. We introduce randomized encoding, which implements invariant models without ancilla qubits or the additional circuit depth for quantum twirling. We evaluate the models as quantum physics-informed neural networks (QPINNs) on two-dimensional screened Poisson and stationary viscous Hamilton-Jacobi equations. Under hard boundary constraints, QPINNs using exact symmetrization and randomized encoding achieve the lowest mean errors in the two benchmarks, respectively.

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47 pages, 9 figures

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Geometric quantum machine learning incorporates symmetry into quantum models, but how symmetry shapes their expressivity and guides effective model design remains insufficiently understood. We address this question through the Fourier representation of quantum Fourier models (QFMs). Symmetry organizes the frequency spectrum into orbits and sums the Fourier coefficients within each orbit into a symmetrized coefficient. When QFM trainable layers form independent exact 2-designs, the variance of each symmetrized coefficient equals the sum of the coefficient variances in its orbit. For single-layer QFMs with ε\varepsilonε-approximate 2-design trainable layers, we bound the deviation from this identity. The resulting bound for individual Fourier coefficients can be exponentially tighter than an existing bound. The hyperoctahedral group provides an example of orbit growth that can mitigate vanishing expressivity of the symmetrized coefficients. Symmetrization of QFMs over an elementary abelian 2-group also yields pure multivariate Chebyshev polynomial basis functions. We introduce randomized encoding, which implements invariant models without ancilla qubits or the additional circuit depth for quantum twirling. We evaluate the models as quantum physics-informed neural networks (QPINNs) on two-dimensional screened Poisson and stationary viscous Hamilton-Jacobi equations. Under hard boundary constraints, QPINNs using exact symmetrization and randomized encoding achieve the lowest mean errors in the two benchmarks, respectively.
Geometric quantum machine learning incorporates symmetry into quantum models, but how symmetry shapes their expressivity and guides effective model design remains insufficiently understood. We address this question through the Fourier representation of quantum Fourier models (QFMs). Symmetry organizes the frequency spectrum into orbits and sums the Fourier coefficients within each orbit into a symmetrized coefficient. When QFM trainable layers form independent exact 2-designs, the variance of each symmetrized coefficient equals the sum of the coefficient variances in its orbit. For single-layer QFMs with ε-approximate 2-design trainable layers, we bound the deviation from this identity. The resulting bound for individual Fourier coefficients can be exponentially tighter than an existing bound. The hyperoctahedral group provides an example of orbit growth that can mitigate vanishing expressivity of the symmetrized coefficients. Symmetrization of QFMs over an elementary abelian 2-group also yields pure multivariate Chebyshev polynomial basis functions. We introduce randomized encoding, which implements invariant models without ancilla qubits or the additional circuit depth for quantum twirling. We evaluate the models as quantum physicsinformed neural networks (QPINNs) on two-dimensional screened Poisson and stationary viscous Hamilton-Jacobi equations. Under hard boundary constraints, QPINNs using exact symmetrization and randomized encoding achieve the lowest mean errors in the two benchmarks, respectively.
Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.
The redundancy distribution of the frequency spectrum has been shown in the literature to impact the expressivity and trainability of Quantum Fourier Models (QFMs). In this work, we address the question of how this redundancy spectrum is shaped by the choice of eigenvalues of the data re-uploading Hamiltonians, using a simple mathematical formalism based on generating functions. We derive exact and asymptotic redundancy profiles for several structured eigenvalue choices identical for every layer, including arithmetic progressions and single-qubit Pauli encodings, and show that both approach a Gaussian profile as the number of layers grows. We then show that this Gaussian limit is not specific to these constructions but is a generic feature of QFMs built from integer eigenvalues that are identical in every layer. These results can be tied to the central limit theorem for random walks. These results clarify why generic or unstructured encoding choices give rise to a redundancy bias that favours low frequencies, highlighting the importance of well-thought-out encoding strategies when constructing a QFM.
Wai-Hong Tam, Reza Safari and Hiromichi Matsuyama (2026) introduce Geometric Quantum Physics-Informed Neural Networks (GQPINNs) as a symmetry-aware extension of quantum physics-informed neural networks (QPINNs). Their framework embeds finite-group (Klein four-group (K_4)) and compact Lie-group (SO(2) rotational) symmetries directly into the parametrized quantum circuit ansatz using twirling-based construction of equivariant generator sets. This builds on geometric quantum machine learning (GQML) to enforce solution-level invariance/equivariance by construction, rather than through loss penalties or post-hoc constraints. The paper benchmarks GQPINNs on representative linear and nonlinear PDEs (2D Poisson, 2D diffusion, 1D acoustic wave, 1D viscous Burgers), demonstrating superior mean absolute error (MAE) and parameter efficiency compared to baseline QPINNs and symmetry-adapted classical PINNs. Also available on Academia.edu: https://www.academia.edu/attachments/132740816/download_file?s=portfolio
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.
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.