Statevector-to-Hardware Reconstruction of a Four-Qubit ZZ Quantum Kernel: A Single-Backend Case Study of Three Execution Jobs
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 observation windows from an indoor air-quality time series. The corresponding circuits were executed on ibm_fez at 1024 shots per circuit in three separate, non-interleaved jobs—one per configuration: 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 from 0.933 to 0.989, and the post hoc diagonal-excluded (U-centered) CKA ranged from 0.816 to 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.