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arXiv:2609.14424

Certification cost of quantum models: measurement correlation, not parameter count

Pavel Sulimov, Claude Lehmann

quant-phcs.LGSubmitted Sep 13, 2026• Announced Sep 15, 2026• 0 citations
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Abstract

Reporting the Fisher geometry of a trained variational quantum model is routine; quoting the shot budget that would establish it is not. Certifying an empirical Fisher matrix to relative Frobenius error $\varepsilon$ under coordinate-wise parameter shift costs $Θ(B p^{2} V/(\varepsilon^{2} G))$ circuit executions, where $V$ is the measured readout variance and $G$ the measured squared gradient norm, with uniform allocation optimal in that class. One constant reproduces the cost of two circuit families whose exponents differ by a full power of $p$. The exponent is an identity in how $nV$ and $nG$ scale with the register, holding family by family to $0.001$ across 624 matrix-product-state cells once the finite-$p$ prefactor is removed. The cubic cost is therefore a finite-size window, set by whether the readout light cone grows with the register. A product family to 256 qubits gives $1.966$ (95% CI $1.934$--$1.997$); a brickwork entangler falls from $2.853$ below ten qubits to $1.715$ beyond sixty-four; a blocked entangler gives $1.984$ at a fixed cone width against $3.034$ at a proportional one. Fixed device connectivity fixes the cone, so a cubic budget from a small simulation overestimates a large machine, on top of hardware multipliers $2.07\times$ ($1.41$--$3.02$), $2.38\times$ and $1.91\times$ on ibm_marrakesh, ibm_fez and ibm_kingston. Cost-optimal readout weights cut the measured shot budget by $2.67\times$ ($1.33$--$4.00$) on hardware, flat from four to twelve qubits. A discrepancy model fitted on cheap circuits transfers its mean inside the calibration grid and, at six larger sizes named before the data, does not: nominal 90% intervals cover 36%, and split conformal is the only rung that stays near nominal.

Related papers

Input-Data Information Preservation and Protocol-Specific Error Mitigation in Noisy Fixed-Parameter Quantum Front Ends: A Reproducible Simulation Study

Hongmin Jung

I examine whether a noisy fixed-parameter quantum front end preserves class information from input data before variational optimization. Only the input vector $x$ is encoded; the label is used after circuit execution to estimate training distributions and score disjoint held-out inputs. The main density-matrix grid contains 10,800 directly simulated configurations over two binary tasks, widths $n\in \{2,3,4\}$ , every integer depth $L=1,\ldots,12$ , five depolarizing strengths, and 30 circuit seeds. At $n=4$ , mean held-out mutual information has a finite-depth maximum that shifts from $L=6$ to 5 for Gaussian data and from $L=7$ to 6 for the Wisconsin Diagnostic Breast Cancer (WDBC) benchmark as $p$ increases from 0 to 0.04. In five-fold complete-seed validation, including training-distribution mutual information with declared design variables raises held-out-accuracy $R^{2}$ from 0.724 to 0.961 for Gaussian data and from 0.714 to 0.860 for WDBC. Training total variation is the stronger binary comparator in complete-seed validation; test-set total variation is reported only as a non-deployable post-outcome oracle. Paired seed-level RMSE improvements from mutual information are 0.0185 (95% bootstrap interval 0.0179–0.0190) and 0.00634 (0.00592–0.00675), and finite-shot training estimates retain most of that gain. Twenty independently seeded Gaussian realizations, twenty repeated WDBC partitions, per-input 256-shot tests, and explicit rotation-schedule ablations expose both transfer and limitations. A family-matched damped-alignment response outperforms an equal-parameter-count polynomial comparator under identical grouped folds but transfers weakly to some widths, encodings, and schedules. Explicit local-folding zero-noise extrapolation and tensor-product measurement-error mitigation improve information and held-out likelihood accuracy in selected simulated budgets; small-cluster sensitivity and failure cases are retained. The accompanying reproducibility archive contains the complete source, raw counts, parameter registries, 2,000-replicate bootstrap vectors, mitigation replicates, automated tests, and resumable one-click workflow. I make no hardware, trained-VQNN, clinical, quantum-advantage, or universal-scaling claim.

IEEE Access•Jan 1, 2026•0 citations

When cheap gradients fail: the measurement cost of attacking quantum classifiers

Bacui Li, Chandra Thapa, Tansu Alpcan +1 more

Adversarial perturbations threaten machine learning classifiers, including variational quantum classifiers. We show that finite quantum measurement statistics (shot noise) act as a built-in defense against gradient-based test-time attacks whose cost scales unfavorably for the attacker. Because every gradient component must be inferred from repeated circuit executions under any unbiased gradient-estimation rule, white-box extraction consumes a dimension-dependent measurement budget that measurement grouping cannot remove in expressive circuits. Under stated assumptions, single-step attacks need at least quadratically many shots in the input dimension $d$, growing as $d^{5/2}$ under norm-concentration scaling, with a sufficient-budget analysis for iterative attacks via stochastic gradient Langevin dynamics. Simulations up to 784 input dimensions validate the law: the realized total budget is the $d^{5/2}$ geometric floor for plateau-mitigated models and grows as $d^{3.00}$ for the tested deep circuits, whose gradient norms decay with dimension absent barren-plateau mitigation; folding the measured gradient norm back in recovers the parameter-free $d^{3/2}$ shot-noise geometry. Against a matched classical baseline whose attack overhead is dimension-independent (the cheap-gradient principle of automatic differentiation), the quantum gradient cost ratio grows empirically as $d^{3.00}$, so the attacker's relative cost diverges as the model scales. Experiments on a 156-qubit IBM processor (ibm_boston, 4-qubit circuits, $d=12$) reproduce the effect: at matched budgets the device attack tracks the ideal within a few percent, with the high-shot gradient faithful to the exact one. The defense operates precisely when the forward map is classically hard to simulate: only then is a white-box attacker denied the simulate-and-backpropagate shortcut and must pay the measurement cost we quantify.

quant-pharXiv:2607.11095•Jul 14, 2026•0 citations

Empirical Evaluation of Quantum Kernel Methods for Classification: A Comparative Study with Classical Kernels

Ronak Bhalgami

Quantum kernel methods are the most concrete near-term proposal for quantum machine learning on classical data, yet published benchmarks disagree on whether they outperform tuned classical kernels, and the disagreement cannot be resolved from the literature because the comparisons are not protocol-controlled. We argue that the obstacle is one of measurement, not theory: the usual setup fixes the quantum feature map's data scaling while grid-searching the classical kernel's hyperparameters, recomputes nothing per fold, and reports cost in simulator time. We therefore treat the comparison itself as the object of study and impose three controls. The bandwidth is tuned on the same grid budget as the classical $γ$, the kernel is recomputed inside every fold from fold-fitted preprocessing, and cost is counted in hardware circuit evaluations. On top of these we run a factorial design of 324 cells in which dimensionality, training size, class separability and data structure are explicit covariates, together with four standard datasets, a shots-versus-gate-noise ablation, and a cost sweep. The dominant effect we measure is not quantum: tuning the feature map's bandwidth is worth 0.120 accuracy on synthetic data and 0.339 on real data, exceeding every quantum-classical difference we observe. Tuned quantum kernels are then statistically indistinguishable from a tuned RBF on all four real datasets. The advantage on synthetic data is confined almost entirely to data generated by the feature map itself, and it shrinks with dimensionality (-0.0045 per qubit) while growing with sample size. Finite sampling costs an order of magnitude more accuracy than gate and readout noise, and a fidelity kernel needs $5\times10^{7}$ circuits at n = 10,000. Protocol variance is thus sufficient to explain the field's contradictory conclusions.

quant-pharXiv:2610.42029•Nov 2, 2026

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

How Much Reconstruction Does Quantum Machine Learning Need? Late Fusion of Uncoupled Quantum Subcircuits

Prabhjot Singh, Adel N. Toosi, Rajkumar Buyya

Circuit cutting lets a large quantum neural network (QNN) run as independent subcircuits on small devices, but rebuilding its outputs by quasiprobability reconstruction carries a classical sampling overhead exponential in the number of cuts. This paper asks whether machine-learning tasks need this step. We replace it with late fusion: subcircuits are trained without cross-cut coupling, and a small classical head combines their local measurements at a cost constant in the cut count. To characterize the trade-off, we introduce a reconstruction dial $Q$, a physically executable budget that retains a fraction of the quasiprobability terms, and a cut-entanglement diagnostic that indicates, within a fixed scale, how much reconstruction a task needs (Spearman $ρ=0.59$ over $104$ runs). Across synthetic and standard datasets, uncoupled late fusion is equivalent to full reconstruction within a predefined margin of $0.02$ accuracy on most benchmarks and pooled over the controlled sweep, at exponentially lower cost; deficits of $0.02$-$0.04$ remain on the most cross-cut tasks. Fusion tops the dial's accuracy/cost frontier, is more robust to shot and device noise, and loses nothing as the coupled model's entangling depth grows. We state the exact condition under which fusion must lose accuracy, a decision-relevant cross-cut correlator, and realize it with entangled-state inputs; for classical inputs the register partition discards no information. We claim no advantage over classical machine learning.

quant-pharXiv:2608.05595•Aug 7, 2026•0 citations

Budget-Aware Neural Error Mitigation forVariational Quantum Algorithms

Guilin Zhang, Kai Zhao, Xu Chu

Abstract Variational quantum algorithms (VQAs) are limited by device noise, and error mitigation is the standard remedy on near-term hardware. Choosing a mitigation method for deployment requires comparing its true cost, yet the usual comparison hides it: it fixes the number of shots per circuit execution, while the methods differ sharply in how many executions they need per estimate. Zero-noise extrapolation (ZNE) needs three to five, Clifford data regression (CDR) more than thirty per target circuit, and a trained neural model exactly one; ZNE's accuracy further rests on a noise-scaling assumption that real devices violate. Inside a variational loop that evaluates thousands of parameter settings, it is this execution multiplicity, not the accuracy at unlimited shots, that decides which method is affordable. Our premise is that the comparison that matters is at equal total shot budget per evaluation point. We present budget-conditioned neural error mitigation (NEM): one network corrects a single shot-limited measurement from circuit and device features, conditioned on the shot-noise scale so that one model serves every budget. We evaluate every method—including each ZNE fold point and each CDR training circuit—under a protocol that spends the same total shots per evaluation point. Under it, NEM is the most accurate method at every budget under coherent miscalibration (reducing error by 72–81%), and the only method that beats unmitigated execution at budgets up to 2^14 shots under stochastic device noise (11–20%); under miscalibration CDR is also positive but overtakes NEM only at 2^16 shots, at 32x the per-evaluation cost. We further characterize when extrapolation fails: under control-offset miscalibration, folding ZNE returns a worse answer than no mitigation on 38–63% of instances in simulation, and on up to 88% on a real IBM Heron processor. On that processor, fine-tuning the model on a small on-device set—120 circuits with classically computed labels—narrows the simulation-to-hardware gap, recovering a 39% error reduction on held-out device instances from the same calibration epoch. Improvements hold from 4 to 20 qubits. Budget-aware accounting, we argue, is the comparison that matters for deployment.

•Aug 14, 2026•0 citations
Reporting the Fisher geometry of a trained variational quantum model is routine; quoting the shot budget that would establish it is not. Certifying an empirical Fisher matrix to relative Frobenius error ε\varepsilonε under coordinate-wise parameter shift costs Θ(Bp2V/(ε2G))Θ(B p^{2} V/(\varepsilon^{2} G))Θ(Bp2V/(ε2G)) circuit executions, where VVV is the measured readout variance and GGG the measured squared gradient norm, with uniform allocation optimal in that class. One constant reproduces the cost of two circuit families whose exponents differ by a full power of ppp. The exponent is an identity in how nVnVnV and nGnGnG scale with the register, holding family by family to 0.0010.0010.001 across 624 matrix-product-state cells once the finite-ppp prefactor is removed. The cubic cost is therefore a finite-size window, set by whether the readout light cone grows with the register. A product family to 256 qubits gives 1.9661.9661.966 (95% CI 1.9341.9341.934--1.9971.9971.997); a brickwork entangler falls from 2.8532.8532.853 below ten qubits to 1.7151.7151.715 beyond sixty-four; a blocked entangler gives 1.9841.9841.984 at a fixed cone width against 3.0343.0343.034 at a proportional one. Fixed device connectivity fixes the cone, so a cubic budget from a small simulation overestimates a large machine, on top of hardware multipliers 2.07×2.07\times2.07× (1.411.411.41--3.023.023.02), 2.38×2.38\times2.38× and 1.91×1.91\times1.91× on ibm_marrakesh, ibm_fez and ibm_kingston. Cost-optimal readout weights cut the measured shot budget by 2.67×2.67\times2.67× (1.331.331.33--4.004.004.00) on hardware, flat from four to twelve qubits. A discrepancy model fitted on cheap circuits transfers its mean inside the calibration grid and, at six larger sizes named before the data, does not: nominal 90% intervals cover 36%, and split conformal is the only rung that stays near nominal.
I examine whether a noisy fixed-parameter quantum front end preserves class information from input data before variational optimization. Only the input vector xxx is encoded; the label is used after circuit execution to estimate training distributions and score disjoint held-out inputs. The main density-matrix grid contains 10,800 directly simulated configurations over two binary tasks, widths n∈{2,3,4}n\in \{2,3,4\}n∈{2,3,4} , every integer depth L=1,…,12L=1,\ldots,12L=1,…,12 , five depolarizing strengths, and 30 circuit seeds. At n=4n=4n=4 , mean held-out mutual information has a finite-depth maximum that shifts from L=6L=6L=6 to 5 for Gaussian data and from L=7L=7L=7 to 6 for the Wisconsin Diagnostic Breast Cancer (WDBC) benchmark as ppp increases from 0 to 0.04. In five-fold complete-seed validation, including training-distribution mutual information with declared design variables raises held-out-accuracy R2R^{2}R2 from 0.724 to 0.961 for Gaussian data and from 0.714 to 0.860 for WDBC. Training total variation is the stronger binary comparator in complete-seed validation; test-set total variation is reported only as a non-deployable post-outcome oracle. Paired seed-level RMSE improvements from mutual information are 0.0185 (95% bootstrap interval 0.0179–0.0190) and 0.00634 (0.00592–0.00675), and finite-shot training estimates retain most of that gain. Twenty independently seeded Gaussian realizations, twenty repeated WDBC partitions, per-input 256-shot tests, and explicit rotation-schedule ablations expose both transfer and limitations. A family-matched damped-alignment response outperforms an equal-parameter-count polynomial comparator under identical grouped folds but transfers weakly to some widths, encodings, and schedules. Explicit local-folding zero-noise extrapolation and tensor-product measurement-error mitigation improve information and held-out likelihood accuracy in selected simulated budgets; small-cluster sensitivity and failure cases are retained. The accompanying reproducibility archive contains the complete source, raw counts, parameter registries, 2,000-replicate bootstrap vectors, mitigation replicates, automated tests, and resumable one-click workflow. I make no hardware, trained-VQNN, clinical, quantum-advantage, or universal-scaling claim.
Adversarial perturbations threaten machine learning classifiers, including variational quantum classifiers. We show that finite quantum measurement statistics (shot noise) act as a built-in defense against gradient-based test-time attacks whose cost scales unfavorably for the attacker. Because every gradient component must be inferred from repeated circuit executions under any unbiased gradient-estimation rule, white-box extraction consumes a dimension-dependent measurement budget that measurement grouping cannot remove in expressive circuits. Under stated assumptions, single-step attacks need at least quadratically many shots in the input dimension ddd, growing as d5/2d^{5/2}d5/2 under norm-concentration scaling, with a sufficient-budget analysis for iterative attacks via stochastic gradient Langevin dynamics. Simulations up to 784 input dimensions validate the law: the realized total budget is the d5/2d^{5/2}d5/2 geometric floor for plateau-mitigated models and grows as d3.00d^{3.00}d3.00 for the tested deep circuits, whose gradient norms decay with dimension absent barren-plateau mitigation; folding the measured gradient norm back in recovers the parameter-free d3/2d^{3/2}d3/2 shot-noise geometry. Against a matched classical baseline whose attack overhead is dimension-independent (the cheap-gradient principle of automatic differentiation), the quantum gradient cost ratio grows empirically as d3.00d^{3.00}d3.00, so the attacker's relative cost diverges as the model scales. Experiments on a 156-qubit IBM processor (ibm_boston, 4-qubit circuits, d=12d=12d=12) reproduce the effect: at matched budgets the device attack tracks the ideal within a few percent, with the high-shot gradient faithful to the exact one. The defense operates precisely when the forward map is classically hard to simulate: only then is a white-box attacker denied the simulate-and-backpropagate shortcut and must pay the measurement cost we quantify.
Quantum kernel methods are the most concrete near-term proposal for quantum machine learning on classical data, yet published benchmarks disagree on whether they outperform tuned classical kernels, and the disagreement cannot be resolved from the literature because the comparisons are not protocol-controlled. We argue that the obstacle is one of measurement, not theory: the usual setup fixes the quantum feature map's data scaling while grid-searching the classical kernel's hyperparameters, recomputes nothing per fold, and reports cost in simulator time. We therefore treat the comparison itself as the object of study and impose three controls. The bandwidth is tuned on the same grid budget as the classical γγγ, the kernel is recomputed inside every fold from fold-fitted preprocessing, and cost is counted in hardware circuit evaluations. On top of these we run a factorial design of 324 cells in which dimensionality, training size, class separability and data structure are explicit covariates, together with four standard datasets, a shots-versus-gate-noise ablation, and a cost sweep. The dominant effect we measure is not quantum: tuning the feature map's bandwidth is worth 0.120 accuracy on synthetic data and 0.339 on real data, exceeding every quantum-classical difference we observe. Tuned quantum kernels are then statistically indistinguishable from a tuned RBF on all four real datasets. The advantage on synthetic data is confined almost entirely to data generated by the feature map itself, and it shrinks with dimensionality (-0.0045 per qubit) while growing with sample size. Finite sampling costs an order of magnitude more accuracy than gate and readout noise, and a fidelity kernel needs 5×1075\times10^{7}5×107 circuits at n = 10,000. Protocol variance is thus sufficient to explain the field's contradictory conclusions.
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)
Circuit cutting lets a large quantum neural network (QNN) run as independent subcircuits on small devices, but rebuilding its outputs by quasiprobability reconstruction carries a classical sampling overhead exponential in the number of cuts. This paper asks whether machine-learning tasks need this step. We replace it with late fusion: subcircuits are trained without cross-cut coupling, and a small classical head combines their local measurements at a cost constant in the cut count. To characterize the trade-off, we introduce a reconstruction dial QQQ, a physically executable budget that retains a fraction of the quasiprobability terms, and a cut-entanglement diagnostic that indicates, within a fixed scale, how much reconstruction a task needs (Spearman ρ=0.59ρ=0.59ρ=0.59 over 104104104 runs). Across synthetic and standard datasets, uncoupled late fusion is equivalent to full reconstruction within a predefined margin of 0.020.020.02 accuracy on most benchmarks and pooled over the controlled sweep, at exponentially lower cost; deficits of 0.020.020.02-0.040.040.04 remain on the most cross-cut tasks. Fusion tops the dial's accuracy/cost frontier, is more robust to shot and device noise, and loses nothing as the coupled model's entangling depth grows. We state the exact condition under which fusion must lose accuracy, a decision-relevant cross-cut correlator, and realize it with entangled-state inputs; for classical inputs the register partition discards no information. We claim no advantage over classical machine learning.
Abstract Variational quantum algorithms (VQAs) are limited by device noise, and error mitigation is the standard remedy on near-term hardware. Choosing a mitigation method for deployment requires comparing its true cost, yet the usual comparison hides it: it fixes the number of shots per circuit execution, while the methods differ sharply in how many executions they need per estimate. Zero-noise extrapolation (ZNE) needs three to five, Clifford data regression (CDR) more than thirty per target circuit, and a trained neural model exactly one; ZNE's accuracy further rests on a noise-scaling assumption that real devices violate. Inside a variational loop that evaluates thousands of parameter settings, it is this execution multiplicity, not the accuracy at unlimited shots, that decides which method is affordable. Our premise is that the comparison that matters is at equal total shot budget per evaluation point. We present budget-conditioned neural error mitigation (NEM): one network corrects a single shot-limited measurement from circuit and device features, conditioned on the shot-noise scale so that one model serves every budget. We evaluate every method—including each ZNE fold point and each CDR training circuit—under a protocol that spends the same total shots per evaluation point. Under it, NEM is the most accurate method at every budget under coherent miscalibration (reducing error by 72–81%), and the only method that beats unmitigated execution at budgets up to 2^14 shots under stochastic device noise (11–20%); under miscalibration CDR is also positive but overtakes NEM only at 2^16 shots, at 32x the per-evaluation cost. We further characterize when extrapolation fails: under control-offset miscalibration, folding ZNE returns a worse answer than no mitigation on 38–63% of instances in simulation, and on up to 88% on a real IBM Heron processor. On that processor, fine-tuning the model on a small on-device set—120 circuits with classically computed labels—narrows the simulation-to-hardware gap, recovering a 39% error reduction on held-out device instances from the same calibration epoch. Improvements hold from 4 to 20 qubits. Budget-aware accounting, we argue, is the comparison that matters for deployment.