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.
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.
Quantum learnability is an algebraic invariant. While optimization and statistical generalization have long been analyzed through disconnected mathematical formalisms in quantum machine learning, both phenomena emerge as dual geometric projections of the dynamical Lie algebra (DLA) dimension g = Lie{-i H_k} ⊆ su(2^n); across any parameterized circuit family, polynomial trainability (Var[∇C] ≥ Ω(1/poly(n))) and polynomial generalizability (M*(ε) ≤ O(poly(n)/ε^2)) are strictly equivalent and hold if and only if dim(g_n) ∈ O(poly(n)). For any observable with adjoint orbit O_O, the operational product satisfies Var[∇C] · M*(ε) = Θ(dim(O_O) / (dim(g_n) ε^2)), which collapses to the universal, dimension-free invariant Θ(1/ε^2) for all generic, non-degenerate observables, identically independent of system size n and algebraic dimension dim(g_n). Unconstrained expressivity (dim(g_n) ~ 4^n) triggers a simultaneous catastrophe: exponential gradient vanishing accompanied by exponential sample complexity explosion. Subalgebra confinement cures both failures. By restricting generators to permutation-equivariant or symmetry-protected subalgebras, the circuit enforces dim(g_n) ≤ O(poly(n)) and secures non-vanishing gradient flow alongside statistical sample efficiency. Exact statevector simulations in PennyLane across n ∈ {2, 3, 4, 5, 6} corroborate this duality: hardware-efficient ansätze exhibit a 17.22× variance collapse, whereas symmetry-protected models sustain Θ(1) gradients and achieve twofold lower test error.
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 Fx 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 Fx frozen for the entire run, while escaping circuits migrate their frequency content (direct training: rpb=−0.48; curriculum: d=1.34; both p<0.001). The replicated signature is this spectral mobility, not any endpoint value of Fx, and trapped circuits retain a fully non-degenerate parameter-space QFIM (rpb≈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→3.0) convexifies the early loss landscape; escaping circuits raise Fx 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.
Orbit dimension, not parameter count, bounds quantum sample complexity. For continuous parameterized quantum circuits on n qubits, the minimax estimation risk obeys R(deff,M)≥Ω(Mdeff), where deff=dim(OO) is the real dimension of the observable's adjoint orbit under the dynamical Lie group. For balanced Pauli observables O=Z⊗n, quotienting the dynamical Lie algebra by the stabilizer subgroup yields the complex Grassmannian Gr(2n−1,2n) of real dimension deff=22n−1=214n≈21dim(g), which matches upper bounds of Caro et al.~(2022) and Ohno~(2025) up to a universal constant and resolves the exact tightness rate Θ(dim(g)/M). For rank-1 projective observables, the orbit contracts instead to complex projective space CP2n−1 with dimension deff=2(2n−1). Sample requirements therefore explode exponentially with qubit count. The scaling identifies Wang \& Wu's (2025) dimension-independent O(1/M) uniform convergence bound as an artifact of uniform stability rather than true estimation power. PennyLane benchmarks across n=1,2,3,4 qubits confirm this exponential threshold. Barren plateaus and statistical sample bottlenecks stem from an identical geometric origin: volume growth in SU(2n) simultaneously suppresses gradient variances as ∼1/dim(g) and drives sample complexity up to ∼dim(g)/M.
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, R-reupload architecture (QDR). (i) We make explicit that the accessible frequencies form the set KR=S(R)−S(R), which equals the R-fold sumset T(R) of the difference set T=S−S of the encoding spectrum, and we characterise it exactly: KR={k∈L:ℓT(k)≤R}, where L=spanZ(T) is an integer lattice and ℓT is word length in T. (ii) We show that the \emph{directions} of the spectrum stabilise already at R=1 (the real span of KR equals that of L), whereas the spectrum itself never saturates: the maximal frequency grows linearly in R and the number of frequencies grows polynomially, as Θ(Rm) with m=rankL. (iii) Using the spectrum size, we give a Rademacher-complexity bound on the capacity of QDRs of order ∣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≥4), where it is highly sensitive to the input-encoding scale; it beats the MLP only on equations with d≤3 inputs.
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.
Quantum machine learning is a promising paradigm for learning from limited data, a central bottleneck in domains such as medical imaging, clinical trials, and rare diseases. Quantum convolutional neural networks (QCNNs) are particularly attractive in this setting, combining a hierarchical architecture with strong inductive bias and a parameter count that grows only logarithmically with system size. Their appeal rests on the generalization bounds of Caro et al. (2022), which show that the generalization error of a quantum model scales with the number of trainable parameters rather than with the Hilbert-space dimension, placing QCNNs in a potentially sample-efficient regime. We develop a hardware-compatible QCNN with mid-circuit measurement and classical feed-forward, and show on a binary handwritten-digit task that strong test performance is achievable from as few as 10 training samples, with the generalization error decreasing as the training set grows. At a matched 45-parameter budget the QCNN learns where an equally small classical convolutional network stays at chance, although an unconstrained classical baseline with roughly 25,000 parameters remains strongest when data are plentiful. Transpiling amplitude and angle encoded circuits across image resolutions from 2x2 to 512x512 pixels then exposes the dominant scaling bottleneck: amplitude encoding stays qubit-efficient but grows extremely deep, whereas angle encoding stays shallow but becomes qubit-prohibitive. On the medically motivated BreastMNIST benchmark the QCNN does not surpass the unconstrained classical network, yet it learns consistently above chance using orders of magnitude fewer parameters. Our results indicate that for QCNNs, learning from few samples is attainable in practice, whereas scaling to realistic image data is constrained less by optimization than by data encoding and hardware execution.