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arXiv:2610.21778

Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits

Michael Poppel, David Bucher, Maximilian Zorn, Markus Baumann, Sebastian Wölckert, Claudia Linnhoff-Popien, Philipp Altmann, Jonas Stein

quant-phcs.LGSubmitted Oct 15, 2026• Announced Oct 16, 2026• 0 citations
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Abstract

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.

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Fourier spectra and benchmarking of phase-estimation variational quantum classifiers

Victor Sineglazov, Petro Chynnyk

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)

Academia quantum.•Oct 29, 2026•0 citations

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

Spencer Topel

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.

quant-pharXiv:2607.11013•Jul 14, 2026•0 citations

Quantum Re-Uploading for Calorimetry: Optimized Architectures with Extended Expressivity

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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.

arXiv:2412.12397•Dec 16, 2024•0 citations

Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency

Michael Poppel, David Bucher, Maximilian Zorn +5 more

Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging $N$ qubits with $L$ encoding layers each -- sharing encoding budget $E = NL$ -- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape $(N,L)$ at fixed $E$. We identify structural rank deficiency of the coefficient matching Jacobian $J$ as the mechanism responsible. For serial single-qubit architectures, we prove $\mathrm{rank}(J) \leq 2L+1$ regardless of parameter count $P$, with $\dim(\ker J) \geq P-(2L+1)$ growing without bound -- a phenomenon we term \emph{structural gradient starvation}: a growing fraction of parameters become structurally decoupled from the loss as $P$ increases at fixed $L$. Parallel architectures avoid this via independent phase trajectories, ensuring $σ_{\min}(J^{(\mathrm{par})}) > 0$ generically for $P \leq 2E+1$, so no parameter lies in $\ker J$. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves $R^2 \geq 0.95$ with $1.6$--$2.2\times$ fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.

quant-pharXiv:2605.05942•May 8, 2026•0 citations

A Tutorial on Quantum Amplitude Embedding with Data Re-Uploading

Hyojun Ahn, Joongheon Kim

Amplitude embedding lets a variational quantum model read many classical features with few qubits, but the model output is then only a quadratic form in the normalized input. Data re-uploading, which encodes the input repeatedly between trainable blocks, can be used to enrich such a model. Because every additional upload of an amplitude embedding repeats a costly state preparation, the gain that an upload buys has to be known before the gates are spent. This tutorial thus addresses the problem of explaining what data re-uploading adds to an amplitude-embedded model and how large the gain is. We express sequential re-uploading and parallel re-uploading with two copies in one notation and show that each upload raises the reachable polynomial order by two. Under one trainable circuit family and one budget of trainable parameters, simulation results show that re-uploading raises the test coefficient of determination from 0.689 to between 0.883 and 0.975 on targets of order four and from 0.337 to between 0.642 and 0.783 on targets of order six, on all ten seeds. On a quadratic target and on handwritten digits, one upload remains the best choice, and every additional upload lowers the gate error that the classifier tolerates. These results help designers match the number of uploads to the polynomial order of the task at hand.

quant-pharXiv:2610.17836•Oct 14, 2026•0 citations

The Shot Gap: Bloch-Sphere Geometry and the Hidden Measurement Budget of Single-Qubit Quantum Classifiers

Alfredo Merlet

Abstract Single-qubit data re-uploading classifiers map every input to a point on the Bloch sphere and are routinely evaluated by thresholding an expectation value estimated from many repetitions of the circuit (shots). A single measurement, however, can deliver at most the Helstrom accuracy of the two class-conditional mixtures, which for one qubit reduces to the length of the difference between the class-mean Bloch vectors. We call the difference between thresholded accuracy and single-shot accuracy the shot gap and measure it in a computational experiment with known ground truth: four synthetic tasks with exact Bayes accuracy, five circuit depths, three training losses, two label-noise levels and ten seeds (1,200 runs), simulated exactly and validated against a state-vector simulation, finite-difference gradients and a brute-force search over measurements. The Helstrom bound held in every run. For circuits of depth two or more, models whose thresholded accuracy averaged 0.857 delivered 0.696 per single measurement, a mean shot gap of 0.161; at depth six, models with 0.93–0.98 thresholded accuracy on the linear, circle and XOR tasks delivered 0.67–0.86 per shot. The embedding, not the choice of measurement, accounted for 96.3% of the single-shot shortfall. Training with the single-shot log-likelihood narrowed the shot gap relative to a smoothed hinge loss in 89% of paired configurations (median reduction 0.085) at a median cost of 0.011 in thresholded accuracy, and eleven shots recovered 74% of the gap against 41% under the hinge loss. Two preregistered hypotheses were not supported: the gap did not grow with boundary nonlinearity, and training loss predicted it better than class-mean Bloch-vector length. Accuracy reports for quantum classifiers should state the number of shots per prediction, as they state training data and compute.

•Oct 21, 2026•0 citations
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.
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)
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.
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.
Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging NNN qubits with LLL encoding layers each -- sharing encoding budget E=NLE = NLE=NL -- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape (N,L)(N,L)(N,L) at fixed EEE. We identify structural rank deficiency of the coefficient matching Jacobian JJJ as the mechanism responsible. For serial single-qubit architectures, we prove rank(J)≤2L+1\mathrm{rank}(J) \leq 2L+1rank(J)≤2L+1 regardless of parameter count PPP, with dim⁡(ker⁡J)≥P−(2L+1)\dim(\ker J) \geq P-(2L+1)dim(kerJ)≥P−(2L+1) growing without bound -- a phenomenon we term \emph{structural gradient starvation}: a growing fraction of parameters become structurally decoupled from the loss as PPP increases at fixed LLL. Parallel architectures avoid this via independent phase trajectories, ensuring σmin⁡(J(par))>0σ_{\min}(J^{(\mathrm{par})}) > 0σmin​(J(par))>0 generically for P≤2E+1P \leq 2E+1P≤2E+1, so no parameter lies in ker⁡J\ker JkerJ. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves R2≥0.95R^2 \geq 0.95R2≥0.95 with 1.61.61.6--2.2×2.2\times2.2× fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.
Amplitude embedding lets a variational quantum model read many classical features with few qubits, but the model output is then only a quadratic form in the normalized input. Data re-uploading, which encodes the input repeatedly between trainable blocks, can be used to enrich such a model. Because every additional upload of an amplitude embedding repeats a costly state preparation, the gain that an upload buys has to be known before the gates are spent. This tutorial thus addresses the problem of explaining what data re-uploading adds to an amplitude-embedded model and how large the gain is. We express sequential re-uploading and parallel re-uploading with two copies in one notation and show that each upload raises the reachable polynomial order by two. Under one trainable circuit family and one budget of trainable parameters, simulation results show that re-uploading raises the test coefficient of determination from 0.689 to between 0.883 and 0.975 on targets of order four and from 0.337 to between 0.642 and 0.783 on targets of order six, on all ten seeds. On a quadratic target and on handwritten digits, one upload remains the best choice, and every additional upload lowers the gate error that the classifier tolerates. These results help designers match the number of uploads to the polynomial order of the task at hand.
Abstract Single-qubit data re-uploading classifiers map every input to a point on the Bloch sphere and are routinely evaluated by thresholding an expectation value estimated from many repetitions of the circuit (shots). A single measurement, however, can deliver at most the Helstrom accuracy of the two class-conditional mixtures, which for one qubit reduces to the length of the difference between the class-mean Bloch vectors. We call the difference between thresholded accuracy and single-shot accuracy the shot gap and measure it in a computational experiment with known ground truth: four synthetic tasks with exact Bayes accuracy, five circuit depths, three training losses, two label-noise levels and ten seeds (1,200 runs), simulated exactly and validated against a state-vector simulation, finite-difference gradients and a brute-force search over measurements. The Helstrom bound held in every run. For circuits of depth two or more, models whose thresholded accuracy averaged 0.857 delivered 0.696 per single measurement, a mean shot gap of 0.161; at depth six, models with 0.93–0.98 thresholded accuracy on the linear, circle and XOR tasks delivered 0.67–0.86 per shot. The embedding, not the choice of measurement, accounted for 96.3% of the single-shot shortfall. Training with the single-shot log-likelihood narrowed the shot gap relative to a smoothed hinge loss in 89% of paired configurations (median reduction 0.085) at a median cost of 0.011 in thresholded accuracy, and eleven shots recovered 74% of the gap against 41% under the hinge loss. Two preregistered hypotheses were not supported: the gap did not grow with boundary nonlinearity, and training loss predicted it better than class-mean Bloch-vector length. Accuracy reports for quantum classifiers should state the number of shots per prediction, as they state training data and compute.