Introduction: A variational quantum classifier realises a function whose Fourier spectrum is fixed by the circuit and whose coefficients must be found by an optimiser. We study a readout borrowed from quantum phase estimation (QPE): a data-dependent operator acts on a target register through controlled powers, and the class label is read from an estimation register of m qubits after an inverse quantum Fourier transform (QFT). Materials and methods: We derive the exact readout distribution—a Fejér-weighted expansion in the moments of the operator—and characterise its Fourier spectrum for two single-qubit blocks: H ⋅ P h ⋅ R Y (family A) and a universal U 2 block (family B). All analytical statements are verified against a state-vector simulator, including attainment of the predicted spectrum. Three ablation controls alter one component each at an unchanged parameter count: the encoding QFT is removed, the exponential ladder of controlled powers is replaced by single powers, and the inverse QFT is replaced by independent Hadamard tests. The benchmark comprises 7680 training runs of 24 circuits on three tasks, with one generated dataset or one cross-validation fold as the unit of inference (twenty per sample size), exact signed-rank tests, and the corrected resampled t -test for the overlapping folds. Results: The support of every readout probability lies in parity-locked lattice cubes of maximal harmonic order 2 m + 1 . The maximal order is attained at every register size, in every feature and every outcome, except for family A with one target qubit, where the two highest orders cancel identically; the full support is attained at every register size used here ( t ≤ 4 , m ≤ 3 ), and the band is tapered towards its edge. Removing the encoding QFT costs family B 0.144 and 0.082 in test accuracy on the two-feature tasks, and replacing the power ladder by single powers costs 0.086 – 0.225 wherever m ≥ 2 ; replacing the inverse QFT by Hadamard tests has no detectable effect. The architecture leads the independently parameterised reference circuits only on the three-class task ( + 0.119 , p < 0.0001 ), where a parameter-tied re-uploading circuit with 12 rather than 20 parameters is not resolved from it ( + 0.023 , 95% confidence interval (CI)
Introduction: A variational quantum classifier realises a function whose Fourier spectrum is fixed by the circuit and whose coefficients must be found by an optimiser. We study a readout borrowed from quantum phase estimation (QPE): a data-dependent operator acts on a target register through controlled powers, and the class label is read from an estimation register of m qubits after an inverse quantum Fourier transform (QFT). Materials and methods: We derive the exact readout distribution—a Fejér-weighted expansion in the moments of the operator—and characterise its Fourier spectrum for two single-qubit blocks: H ⋅ P h ⋅ R Y (family A) and a universal U 2 block (family B). All analytical statements are verified against a state-vector simulator, including attainment of the predicted spectrum. Three ablation controls alter one component each at an unchanged parameter count: the encoding QFT is removed, the exponential ladder of controlled powers is replaced by single powers, and the inverse QFT is replaced by independent Hadamard tests. The benchmark comprises 7680 training runs of 24 circuits on three tasks, with one generated dataset or one cross-validation fold as the unit of inference (twenty per sample size), exact signed-rank tests, and the corrected resampled t -test for the overlapping folds. Results: The support of every readout probability lies in parity-locked lattice cubes of maximal harmonic order 2 m + 1 . The maximal order is attained at every register size, in every feature and every outcome, except for family A with one target qubit, where the two highest orders cancel identically; the full support is attained at every register size used here ( t ≤ 4 , m ≤ 3 ), and the band is tapered towards its edge. Removing the encoding QFT costs family B 0.144 and 0.082 in test accuracy on the two-feature tasks, and replacing the power ladder by single powers costs 0.086 – 0.225 wherever m ≥ 2 ; replacing the inverse QFT by Hadamard tests has no detectable effect. The architecture leads the independently parameterised reference circuits only on the three-class task ( + 0.119 , p < 0.0001 ), where a parameter-tied re-uploading circuit with 12 rather than 20 parameters is not resolved from it ( + 0.023 , 95% confidence interval (CI)
Variational quantum classifiers (VQCs) are often compared with classical models under unequal data partitions, tuning budgets, or statistical treatment. We benchmark a 4-qubit VQC against logistic regression, an RBF-SVM, random forest, a quantum-kernel SVM (QSVM), and a majority-class control on Breast Cancer Wisconsin (n=569), Coimbra (n=116), and Golub leukemia (n=72), using five stratified folds across five seeds (25 paired blocks per dataset), identical training-fold-fitted preprocessing, and Nadeau–Bengio corrected paired tests with Holm correction. A first arm with a frozen configuration and the common [0, π] angle encoding left the VQC at 0.563, 0.515, and 0.576 accuracy (majority 0.617, 0.552, 0.652). In a second arm, in which every model was tuned by the same inner cross-validation and the encoding scale was a tuned hyperparameter, VQC accuracy rose to 0.864, 0.616, and 0.747. On Wisconsin the VQC stayed significantly below every matched classical model (differences 0.079–0.097, Holm p≤0.006) but above the majority class; on Golub it was significantly below only the SVM (Holm p=0.016); on Coimbra no comparison was significant. A QSVM with the same feature map exceeded the VQC numerically on every dataset, and three optimizers reached the same training loss in pilot diagnostics, pointing to the encoding and the variational head rather than the optimizer. The Wisconsin arms also differ in sample size (120 versus 569). Results apply to this low-qubit, PCA-compressed, exactly simulated setting and do not indicate a quantum advantage.
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 E 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+2 tangent directions, one more than its 2E+1 Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least 6E+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 E; at E=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 1.5 times that of the control under Adam, but no larger in the typical run under Levenberg--Marquardt.
Abstract Variational quantum algorithms are among the most widely studied approaches for applying near-term quantum devices to learning tasks. In this work we investigate a supervised quantum-autoencoder-style classifier that encodes images by amplitude encoding and predicts labels from a designated readout sub-register. Building on the quantum-autoencoder (QAE)-style classifier of Asaoka and Kudo, we make three contributions. First, we give a rigorous equivalence result showing that, for computational-basis label states, the original SWAP-test-based training objective can be estimated without ancilla and reference registers by directly measuring the readout subsystem. This simplification reduces circuit resources while preserving the same training signal. Second, we propose a lightweight hybrid extension: instead of using only the probability of the correct readout bitstring, we estimate the full readout distribution and apply a learnable temperature-and-bias post-processing followed by a softmax normalization. Across MNIST, KMNIST, and Fashion-MNIST (restricted to eight classes), this classical post-processing yields substantial and statistically significant improvements in evaluation accuracy, consistently across cross-entropy and mean-squared-error objectives. Finally, to interpret these gains we analyze Fisher-geometry summaries based on effective dimension and complementary Fisher-spectrum diagnostics for both the quantum models and the classical baselines. Empirically, the hybrid quantum model is associated with a more spectrally concentrated and anisotropic Fisher geometry, with Fisher mass concentrated in fewer dominant directions and with a substantially larger local Fisher scale after training.
Introduction: Variational quantum algorithms can show unstable trainability and uneven generalisation that are not fully explained by circuit depth or raw parameter count alone. We study the data quantum Fisher information matrix (DQFIM) as a data-dependent effective-capacity diagnostic for controlled supervised unitary-learning tasks. Materials and methods: We used exact-state, classically simulated, 4-qubit, supervised unitary-learning benchmarks with two ansatz families: a hardware-efficient ansatz (HEA) for the baseline trainability study and a symmetry-preserving ansatz for symmetry-controlled comparisons. The analysis covered four linked experiments: E0 trainability phase boundaries, E1 symmetry-controlled data-regime comparisons, E2 trainability versus generalisation, and E3 pre-training prediction of optimisation success. For the main analyses we used the support-basis DQFIM, 30 random seeds per main configuration, grouped cross-validation in the predictive benchmark, and additional diagnostics for threshold sensitivity, phase indeterminacy, Hamming-sector generalisation, and test set size sensitivity. Results: In E0, the empirical trainability boundary increased from M c = 32 at L = 1 to M c = 192 at L = 8 , and the support-basis DQFIM-predicted boundary matched the empirical boundary on the resolved scanned grid under the main analysis setting. In E1, the sector-preserving and sector-broken conditions showed no resolved empirical boundary shift on the scanned grid, with only mild low-L asymmetry in the DQFIM-predicted boundary. In E2, trainability and generalisation separated clearly: some regimes generalised well, while others reached near-zero training loss but retained high test loss. The phase diagnostic supported relative phase indeterminacy as a mechanism for failure on in-sector superpositions after basis-state training. In E3, parameter count alone was a weak predictor of optimisation success, with receiver operating characteristic area under the curve (ROC AUC) = 0.648 ; the structural baseline was stronger, with ROC AUC = 0.958 ; and the DQFIM-enhanced model performed best, with ROC AUC = 0.989 . Conclusions: In these small, idealised, classically simulated matched-family tasks, the support-basis DQFIM provides a useful data-dependent pre-training diagnostic of effective capacity on the retained data support. It tracks resolved trainability boundaries and contributes predictive information beyond raw parameter count and structural metadata. The conclusions remain scoped to controlled small-system simulations; larger-qubit, noisy, hardware-executed, and less-matched settings require further validation before claiming practical scalability.
Variational quantum algorithms (VQAs) have shown advantages over classical approaches in solving many problems, yet their quantum neural networks (QNNs) models still perform poorly in complex tasks like multi-class classification and high-frequency time-series prediction, raising questions about their feasibility. While existing studies have characterized their performance from several perspectives, but research on their learning performance and visualization remains scarce. This work develops the universal approximation theory of variational quantum algorithms for multivariate function fitting, supported by a verification in visual training analysis on two-input binary datasets. We first visualize how a single-qubit VQA with single-encoding process four types of binary data, revealing inherent limitations, and then formulate the measurement expectation of VQA as a partial Fourier series, establishing its universal approximation capacity and showing that the Fourier spectrum is determined by the data encoding strategy: multiple encoding layers broaden the spectrum and enhance learning performance. Guided by this theory, we adjust the circuit structure and achieve nearly 100% accuracy across all four binary classification tasks.
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] 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.