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arXiv:2607.11013

Overcoming Fourier Locking in Quantum Data Re-uploading Classifiers via Spectral Homotopy

Spencer Topel

quant-phcs.LGSubmitted Jul 13, 2026• Announced Jul 14, 2026• 0 citations
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Abstract

Data re-uploading parameterized quantum circuits (DRU-PQCs) are universal function approximators, yet their expressivity produces oscillatory, non-convex loss landscapes that resist gradient-based optimization. We show that the primary optimization bottleneck in DRU-PQCs is not insufficient capacity but a structural failure mode we term Fourier locking (FL): because encoding weights and entangling layers are nonlinearly coupled, random initialization on high-frequency targets collapses the encoding parameters into spurious local minima. Two Fisher diagnostics characterize FL. The input-space quantum Fisher information $F_x$ measures the effective frequency content of the encoded state; the Fisher discriminant ratio of the measured features measures their alignment with the class labels. In two independent 50-seed experiments, the locking is literal: trapped circuits hold $F_x$ frozen for the entire run, while escaping circuits migrate their frequency content (direct training: $r_{pb} = -0.48$; curriculum: $d = 1.34$; both $p < 0.001$). The replicated signature is this spectral mobility, not any endpoint value of $F_x$, and trapped circuits retain a fully non-degenerate parameter-space QFIM ($r_{pb} \approx 0$): the failure is spectral misalignment of a responsive state, not a loss of geometric sensitivity. A frequency-staged homotopy protocol that paces the target frequency ($f: 1.0 \to 3.0$) convexifies the early loss landscape; escaping circuits raise $F_x$ in step with the curriculum, and the escape rate triples (18% vs. 6%). Fourier locking is a frequency-alignment problem, and its remedy is frequency pacing.

Comments
14 pages, 5 figures, 1 table. Submitted to PRX Quantum. Code and data: https://github.com/moth-quantum/fourier_locking_study. ORCID: 0009-0000-0081-9786

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Data re-uploading parameterized quantum circuits (DRU-PQCs) are universal function approximators, yet their expressivity produces oscillatory, non-convex loss landscapes that resist gradient-based optimization. We show that the primary optimization bottleneck in DRU-PQCs is not insufficient capacity but a structural failure mode we term Fourier locking (FL): because encoding weights and entangling layers are nonlinearly coupled, random initialization on high-frequency targets collapses the encoding parameters into spurious local minima. Two Fisher diagnostics characterize FL. The input-space quantum Fisher information FxF_xFx​ measures the effective frequency content of the encoded state; the Fisher discriminant ratio of the measured features measures their alignment with the class labels. In two independent 50-seed experiments, the locking is literal: trapped circuits hold FxF_xFx​ frozen for the entire run, while escaping circuits migrate their frequency content (direct training: rpb=−0.48r_{pb} = -0.48rpb​=−0.48; curriculum: d=1.34d = 1.34d=1.34; both p<0.001p < 0.001p<0.001). The replicated signature is this spectral mobility, not any endpoint value of FxF_xFx​, and trapped circuits retain a fully non-degenerate parameter-space QFIM (rpb≈0r_{pb} \approx 0rpb​≈0): the failure is spectral misalignment of a responsive state, not a loss of geometric sensitivity. A frequency-staged homotopy protocol that paces the target frequency (f:1.0→3.0f: 1.0 \to 3.0f:1.0→3.0) convexifies the early loss landscape; escaping circuits raise FxF_xFx​ in step with the curriculum, and the escape rate triples (18% vs. 6%). Fourier locking is a frequency-alignment problem, and its remedy is frequency pacing.
Angle-encoded variational quantum circuits implement truncated Fourier series. For one-dimensional inputs, a single-qubit re-uploading (serial) circuit represents every function that a multi-qubit circuit with the same encoding budget EEE can; we find that this exchange of width for depth is not free in first-order training. Because its state lives on the Bloch sphere, all parameters of the serial circuit move the state, at each input, along one complex direction, which we call \emph{phase-locking}. We prove that this costs no Fourier direction but leaves no direction to spare: however many parameters it has, the serial circuit has at most 2E+22E+22E+2 tangent directions, one more than its 2E+12E+12E+1 Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least 6E+66E+66E+6. The cost in training is measured against this control, as the number of runs out of 100 that reach the optimum within a fixed step budget. On an exact loss with flat-spectrum targets the gap grows with EEE; at E=32E=32E=32 the serial circuit converges in 3, 53 and 83 runs under gradient descent, Adam and Levenberg--Marquardt, the control in 99, 100 and 100. On three real-world series, at matched parameter counts, the serial circuit reaches the least-squares optimum in at most 9 runs under Adam and in 32 to 93 under Levenberg--Marquardt, the control in at least 98 and in all 100; its held-out error is up to about 1.51.51.5 times that of the control under Adam, but no larger in the typical run under Levenberg--Marquardt.
In this work, we investigate the phenomenon of spectral bias in quantum machine learning, where, in classical settings, models tend to fit low-frequency components of a target function earlier during training than high-frequency ones, demonstrating a frequency-dependent rate of convergence. We study this effect specifically in parameterised quantum circuits (PQCs). Leveraging the established formulation of PQCs as Fourier series, we prove that spectral bias in this setting arises from the ``redundancy'' of the Fourier coefficients, which denotes the number of terms in the analytical form of the model contributing to the same frequency component. The choice of data encoding scheme dictates the degree of redundancy for a Fourier coefficient. We find that the magnitude of the Fourier coefficients' gradients during training strongly correlates with the coefficients' redundancy. We then further demonstrate this empirically with three different encoding schemes. Additionally, we demonstrate that PQCs with greater redundancy exhibit increased robustness to random perturbations in their parameters at the corresponding frequencies. We investigate how design choices affect the ability of PQCs to learn Fourier sums, focusing on parameter initialization scale and entanglement structure, finding large initializations and low-entanglement schemes tend to slow convergence.
Commutation failure controls quantum generalization. When data encoding gates fail to commute with trainable parameter gates across consecutive re-uploading layers, the effective dynamical Lie algebra becomes data-dependent; this dependency invalidates the parameter-isolation assumption underpinning existing Lie-algebraic frameworks (Caro et al., 2022; Ohno, 2025). We resolve this structural gap by proving that the statistical generalization capacity of L-layer data re-uploading circuits is tightly bounded by the effective hypothesis dimension: d_eff(L) = min(L * p_layer, |Ω_L|) <= min(L * p_layer, (2L + 1)^d), where d is the input data dimension, p_layer the number of trainable parameters per layer, and |Ω_L| the cardinality of accessible Fourier frequencies generated through non-commuting Lie brackets in the joint dynamical Lie algebra g_joint = Lie_R{i H_j, i G_k}. Harmonic analysis combined with metric entropy chaining establishes matching upper and lower bounds; together, they settle the exact minimax estimation rate R*(F_L, M) = \tilde{\Theta}(\sqrt{d_eff(L)/M}). Two distinct orbit concepts govern learning. The point-wise adjoint orbit dimension dim(O_O) <= dim(g_joint) at fixed x quantifies single-point state variation, whereas the global function-space Fourier capacity |Ω_L| governs sample complexity across the full data domain; conflating these quantities yields vacuous bounds. An initial basis rotation unlocks harmonic modes up to k_max = L; without this rotation, eigenstate initialization constrains the cutoff to k_max = L - 1. Bandwidth dictates depth. For target functions with harmonic bandlimit k_target, the network underfits when L < k_target and achieves full expressivity convergence once L >= k_target. Permutation-equivariant data encoding compresses the joint DLA to an su(2) subalgebra of dimension 3. This algebraic confinement suppresses the 4^n - 1 dimensional explosion across arbitrary qubit counts and shields the circuit from the curse of dimensionality. Numerical benchmarks in PennyLane across L in {1, 2, 4, 8} and sample sizes M in [10, 200] corroborate this harmonic-algebraic scaling and its associated uniform convergence rates.
Data reuploading---repeatedly encoding classical inputs at multiple layers of a parameterized quantum circuit---is a central mechanism for enhancing the expressivity of quantum neural networks (QNNs). Building on the Fourier-theoretic framework of Schuld et al.~\cite{schuld2021fourier}, we study the frequency spectrum of the general multivariate, RRR-reupload architecture (QDR). (i) We make explicit that the accessible frequencies form the set KR=S(R)−S(R)K_R=S^{(R)}-S^{(R)}KR​=S(R)−S(R), which equals the RRR-fold sumset T(R)T^{(R)}T(R) of the difference set T=S−ST=S-ST=S−S of the encoding spectrum, and we characterise it exactly: KR={k∈L: ℓT(k)≤R}K_R=\{k\in\mathcal L:\ \ell_T(k)\le R\}KR​={k∈L: ℓT​(k)≤R}, where L=spanZ(T)\mathcal L=\mathrm{span}_{\mathbb Z}(T)L=spanZ​(T) is an integer lattice and ℓT\ell_TℓT​ is word length in TTT. (ii) We show that the \emph{directions} of the spectrum stabilise already at R=1R=1R=1 (the real span of KRK_RKR​ equals that of L\mathcal LL), whereas the spectrum itself never saturates: the maximal frequency grows linearly in RRR and the number of frequencies grows polynomially, as Θ(Rm)Θ(R^{m})Θ(Rm) with m=rank Lm=\mathrm{rank}\,\mathcal Lm=rankL. (iii) Using the spectrum size, we give a Rademacher-complexity bound on the capacity of QDRs of order ∣KR∣/n\sqrt{|K_R|/n}∣KR​∣/n​. (iv) We report experiments on function approximation and classification with train/test splits, several seeds and parameter-matched classical baselines. The simulated spectra match the theory exactly. A single-qubit QDR beats a matched MLP on several smooth or periodic univariate targets and on the spiral task, but it is clearly worse on multivariate regression (d≥4d\ge4d≥4), where it is highly sensitive to the input-encoding scale; it beats the MLP only on equations with d≤3d\le3d≤3 inputs.
Near-term quantum machine learning must balance expressivity, optimization, and hardware constraints. We study quantum re-uploading units (QRUs) as compact circuits and compare them, at matched parameter count, to a standard mono-encoded variational quantum circuit (VQC) baseline. On a three-feature calorimetry classification task, we train a single-qubit QRU that outputs a scalar in [−1,1][-1,1][−1,1] and map it to three classes via fixed thresholds. In this setting, QRUs obtain higher accuracy than the mono-encoded baseline. A controlled ablation over depth, input scaling, circuit template, optimizer, and gradient accumulation indicates that most gains occur at small depths, with diminishing returns as depth increases while training cost grows approximately linearly. To interpret these observations, we analyze reachable Fourier components and find that repeated data re-encoding expands the per-coordinate harmonic support relative to mono-encoding, consistent with a spectral activation study over random initializations. Finally, we report an end-to-end proof-of-execution of the trained model on a superconducting QPU via a cloud workflow, illustrating practical deployability under current constraints.
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.