Hardware noise and finite sampling perturb the fidelity estimates forming a quantum-kernel Gram matrix. We measured how far three hardware-reconstructed Gram matrices depart from an exact statevector reference for one frozen four-qubit ZZ feature map on N=24 indoor air-quality windows, executed on ibm_fez at 1024 shots per circuit in three single, non-interleaved jobs: baseline, dynamical decoupling alone, and gate twirling alone. All were complete, finite, and positive-semidefinite. Off-diagonal root-mean-squared error (RMSE) against the reference was 0.0878, 0.0864, and 0.0427; full-matrix centered kernel alignment (CKA) ranged 0.933-0.989 and the post hoc diagonal-excluded (U-centered) CKA 0.816-0.986. The gate-twirled job deviated least on every reported geometry axis; its baseline contrasts are deletion-stable for the Spearman, mean-absolute-error, RMSE, and full-matrix CKA diagnostics, while the Pearson and diagonal-excluded contrasts fall just below that convention. Dynamical decoupling was not separated from the baseline. The observed error exceeded both finite-shot reference scales, so, under those sampling-only models, sampling does not explain it. Centered kernel-target alignment did not track reconstruction fidelity and stayed at or below each label-permutation reference: implementation fidelity and task relevance are distinct diagnostic axes. All configuration-level statements describe three realized jobs on one backend; no mitigation-efficacy, classifier-superiority, forecasting, or quantum-advantage claim is made.
14 pages, 2 figures. v2: retitled and restructured after journal peer review as a single-backend, fixed-kernel, three-job diagnostic case study; main text shortened by about 75%; claims restricted to the three observed jobs; ancillary files contain the Supplementary Methods and Supplementary Notes S1-S3 (four PDFs). Code: https://doi.org/10.5281/zenodo.21438523
Hardware noise and finite sampling perturb the fidelity estimates forming a quantum-kernel Gram matrix. We measured how far three hardware-reconstructed Gram matrices depart from an exact statevector reference for one frozen four-qubit ZZ feature map on N=24 indoor air-quality windows, executed on ibm_fez at 1024 shots per circuit in three single, non-interleaved jobs: baseline, dynamical decoupling alone, and gate twirling alone. All were complete, finite, and positive-semidefinite. Off-diagonal root-mean-squared error (RMSE) against the reference was 0.0878, 0.0864, and 0.0427; full-matrix centered kernel alignment (CKA) ranged 0.933-0.989 and the post hoc diagonal-excluded (U-centered) CKA 0.816-0.986. The gate-twirled job deviated least on every reported geometry axis; its baseline contrasts are deletion-stable for the Spearman, mean-absolute-error, RMSE, and full-matrix CKA diagnostics, while the Pearson and diagonal-excluded contrasts fall just below that convention. Dynamical decoupling was not separated from the baseline. The observed error exceeded both finite-shot reference scales, so, under those sampling-only models, sampling does not explain it. Centered kernel-target alignment did not track reconstruction fidelity and stayed at or below each label-permutation reference: implementation fidelity and task relevance are distinct diagnostic axes. All configuration-level statements describe three realized jobs on one backend; no mitigation-efficacy, classifier-superiority, forecasting, or quantum-advantage claim is made.
H. Billah and M.M.H. Sarker contributed equally to this work. Interpreting a fixed quantum circuit's output as evidence of a hardware effect requires ruling out two things usually conflated with it: finite-sample fluctuation, and structure already present in the circuit's own noiseless output. Less commonly checked is a third: whether the reference circuit compared against is the one actually executed, once compiler-level qubit routing is accounted for. We execute a fixed 35-qubit, 243-gate diagnostic circuit on IBM Heron-family hardware (three runs, two backends) and show this third check is essential: the routed circuit submitted to hardware has different noiseless predictions than the logical circuit intended, for several qubits. Against the correct, routed-circuit reference, the aggregate deviation is significant in two of three runs, and four registers (0, 9, 14, 27) show a significant output-probability deviation in every run — though we distinguish a high-confidence subset, {0, 9, 27}, with tightly time-matched calibration, from qubit 14, included only on an exploratory basis given its calibration was retrieved roughly two weeks post-execution. Same-backend replicates correlate more strongly by the Pearson measure, though a rank-based (Spearman) test shows this gap does not survive at the level of register ordering, indicating substantial leverage from a small number of registers rather than a broadly robust pattern; these three executions also share a byte-identical payload and are not independent replications in the strictest sense. The mechanism is only partially resolved: independent single-qubit readout error and raw gate count are ruled out as complete explanations for most of these registers, but the simplest version of the leading hypothesis for the single most-affected register does not reproduce the observed cross-backend pattern even with live, backend-specific timing data. A second, structurally different circuit validated only in simulation calibrates the statistical procedure's behavior under one synthetic noise model; it does not itself establish generality to real hardware, and the specific register set found here should not be expected to generalize beyond this study. Note on scope: this is a methodology and case-study contribution centered on executed-circuit references for fixed-circuit diagnostics, not a characterization of IBM Heron hardware in general or a randomized-benchmarking experiment in the formal sense. Supporting data, the routed OpenQASM circuits, raw measurement counts, and analysis code for this study are available on request from the corresponding author. Affiliation: Sonicium Quantum Lab Ltd., Dhaka, Bangladesh.
Quantum kernel estimation (QKE) lets a quantum computer supply the similarity values a classical support vector machine needs, but the original hardware demonstration is hard to reproduce and gives little guidance on the number of shots a usable kernel requires. This study implements QKE on a two-qubit simulator and quantifies its shot and noise requirements. A second-order ZZ feature map encodes each data point, kernel entries are measured from an overlap circuit, and a classical support vector machine is trained on the resulting matrix. An artificial dataset of 40 training and 40 test points was generated with a separation gap of 0.3, the shot count was varied from 128 to 50,000, and a depolarising and readout noise model was applied; classical kernels and a variational classifier served as baselines. With an exactly computed kernel the classifier labels every training and test point correctly using 12 support vectors. Perfect test accuracy is reached at 1,024 shots per pair, far below the 50,000 used on hardware, and the deviation from the exact kernel decays with a fitted exponent of -0.52. The weight of negative eigenvalues falls from 2.019 to 0.096 as shots increase, making it a practical warning sign of too few shots. Under noise the kernel values drop by 0.014 on average, yet accuracy stays at 100% because the drop is nearly uniform. Classical kernels reach at most 70% test accuracy and the variational classifier 98.3%. The work contributes a reproducible open-source procedure and shot-budget guidance for quantum kernel experiments.
Quantum kernel methods retain the standard supervised-learning interface while defining similarities through overlaps of quantum feature states. In practice, the resulting Gram matrices can develop strong global components that dominate the spectrum and make geometric comparisons to classical kernels difficult. This motivates separating global effects from task-relevant variation and testing whether classical postprocessing can reveal and reshape the remaining structure without changing the embedding circuit. Kernel centering (double-centering) is therefore used to remove the constant component while preserving positive semidefiniteness. In parallel, a controlled set of classical extensions is examined at three levels, namely elementwise transforms on datapoints before embedding, spectrum-level eigenvalue mappings, and entrywise kernel transformations, with trace normalization used for consistent comparison. Overall, the results show that the centering reshapes kernel spectra and geometry-based comparisons, often amplifying the separation from a classical RBF reference, even though model predictions are invariant. In contrast, circuit-independent classical extensions can affect predictive performance, narrowing the classification gap for fidelity kernels and improving regression performance, in some settings matching or exceeding the classical baseline. These findings position kernel postprocessing as a practical and low-cost way to analyze quantum kernel geometry and to improve kernel-based supervised learning pipelines (e.g., SVC/SVR) that use quantum kernels.
Quantum kernel estimation on near-term hardware is shot-budgeted: every entry of the kernel Gram matrix is a Bernoulli expectation that must be sampled with a finite number of circuit executions. Recent work on quantum kernel classification has shown that allocating shots non-uniformly across kernel entries, weighted by their downstream task sensitivity, can reduce the shot budget required to reach a target accuracy. We extend this idea to Gaussian process (GP) regression, a setting whose downstream quantities (full-spectrum posterior variance, log-determinant, marginal likelihood) couple to kernel error more tightly than the sign-only outputs of classification. We derive three closed-form pair-level sensitivities predictive coupling ∣αiαj∣, leave-one-out residual, and marginal-likelihood gradient and plug them into a Neyman-style minimum-variance allocation rule. To prevent catastrophic over-concentration when the warm-up sensitivity estimate is itself noisy, we add a high uniform coverage floor justified by a Frobenius lower bound on the missing-entry perturbation. On four UCI benchmarks and two synthetic RBF + Bernoulli controlled studies, the resulting allocator delivers 10--21% test-RMSE improvement over uniform allocation across the moderate-budget regime. The gain transfers (i) to genuine ZZ and Pauli-Z quantum kernels on quantum-natural data (−13--15% at low budget, p<0.05 paired) and (ii) to four downstream tasks (Bayesian quadrature, heteroscedastic regression, hyperparameter learning, multi-output Cokriging). On UCI features embedded into a ZZ kernel the gain disappears, consistent with the exponential-concentration regime where shot allocation has nothing to exploit.
Abstract Modern Industrial Control Systems (ICS) face sophisticated cyber-physical attacks that exploit nonlinear correlations between process variables, rendering traditional linear classifiers ineffective. This paper presents a hardware-agnostic Quantum Support Vector Machine (QSVM) framework employing an 8-qubit ZZFeatureMap kernel for anomaly detection in critical water treatment and thermal power infrastructure. Performance benchmarks were established via noise-free statevector simulation to determine theoretical quantum advantage. Physical realizability was separately validated via circuit execution on IBM's 156-qubit ibm_fez processor, with completed hardware jobs enabling quantitative analysis of the simulation-to-hardware performance gap. Through rigorous cross-testbed validation on the SWaT (Secure Water Treatment) and HAI (Hardware-in-the-Loop Augmented ICS) datasets, our simulated approach achieves an AUC-ROC of 0.9912 ± 0.004 on SWaT and 0.8309 ± 0.050 on HAI, demonstrating consistent quantum advantage of +10.8% AUC over classical RBF-kernel SVMs on the more challenging HAI testbed. Statistical robustness is established through 5-seed cross-validation with stratified sampling. Hardware execution on ibm_fez confirms physical realizability with circuit depth 76 and 28 CNOT gates, while revealing an expected fidelity degradation of approximately 17–20% compared to ideal simulation due to gate errors and decoherence. The quantum kernel's Z i Z j entanglement structure geometrically captures pairwise feature correlations, providing inductive bias well-suited for coupled sensor dynamics. Unlike deep learning approaches that require massive attack datasets unavailable in critical infrastructure domains, our kernel method generalizes effectively from limited training samples. All experimental artifacts are publicly available at https://github.com/Ali-Badami/Quantum-IDS. IBM Quantum job identifiers: d5l9htjh36vs73bgsi3g (SWaT), d5l9huk8d8hc73cfb0pg (HAI).
Two recent results have reshaped quantum Gaussian processes (QGPs). On the one hand, \citet{lowe2025assessing} rule out the exponential speedups claimed by HHL-based QGP regression in the typical, well-conditioned regime; on the other, an independent line of work shows that highly expressive quantum kernels suffer posterior pathologies that break Bayesian optimization. We show that these seemingly unrelated phenomena are governed by the same quantity: the normalized spectral entropy S(K)/logn of the kernel Gram matrix. We prove a Cauchy--Schwarz tail bound on Nyström approximation error, a finite-sample variance-contraction identity in terms of Bach's degrees of freedom dσ(K), and a characterization of the \emph{target-dependent} optimal entropy via the intrinsic dimension of the target in the kernel eigenbasis. Empirically, the diagnostic is kernel-agnostic: hardware-efficient, matchgate, IQP \emph{and} RBF/Matérn/RFF/deep-kernel families all collapse onto identical S/logn curves on dequantization, ECE, and variance-contraction panels. The NLL sweet spot lives at high entropy for smooth targets and at low entropy for band-limited quantum-data targets. The diagnostic transfers from simulator to IBM Heron hardware with median absolute error 3.2% and mean 5.2% in S/logn across 24 configurations at nq=4, with matchgate and IQP within 5% mean and a single HE configuration returning a 30% outlier that drops to 0.5% on rerun (attributed to calibration drift); the same diagnostic transfers to a second Heron backend (mean error 2.7%) and to a nq=6 scale-up on the original backend (mean error 1.7%). No error mitigation is applied throughout.