Title: A Spectral Analysis of Quantum Gaussian Process Kernels

Original: arXiv:2605.30952v2 Announce Type: replace Abstract: Two recent developments have significantly altered the landscape of quantum Gaussian processes (QGPs). First, \citet{lowe2025assessing} demonstrated that exponential speedups, previously asserted for HHL-based QGP regression, do not materialize in standard, well-conditioned scenarios. Second, separate research has identified that highly expressive quantum kernels can induce posterior pathologies, thereby undermining Bayesian optimization. This study reveals that these distinct issues are actually driven by a single underlying metric: the normalized spectral entropy, defined as $S(K)/\log n$, of the kernel Gram matrix.

We establish a Cauchy--Schwarz tail bound for Nystr\"om approximation errors, derive a finite-sample variance-contraction identity involving Bach’s degrees of freedom $d_\sigma(K)$, and characterize the target-dependent optimal entropy through the intrinsic dimension of the target within the kernel’s eigenbasis. Our empirical findings indicate that this diagnostic approach is independent of the specific kernel type: families such as hardware-efficient, matchgate, IQP, and classical models like RBF, Mat\'ern, RFF, and deep kernels all align along identical $S/\log n$ trajectories when evaluated against dequantization limits, Expected Calibration Error (ECE), and variance-contraction metrics. Specifically, the optimal negative log-likelihood (NLL) occurs at high entropy levels for smooth targets, whereas band-limited quantum-data targets exhibit optimal performance at low entropy.

The diagnostic’s efficacy extends from simulation to physical hardware. Across 24 configurations with $n_q = 4$ qubits on IBM Heron hardware, the median absolute error for $S/\log n$ was $3.2\%$, with a mean error of $5.2\%$. Matchgate and IQP kernels showed mean errors within $5\%$. One hardware-efficient (HE) configuration initially presented a $30\%$ outlier, but this discrepancy reduced to $0.5\%$ upon rerun, a deviation attributed to calibration drift. Furthermore, the diagnostic remained robust when applied to a second Heron backend (mean error $2.7\%$) and during a scale-up to $n_q = 6$ on the original backend (mean error $1.7\%$). These results were achieved without the application of any error mitigation techniques.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC