Spectral Anatomy of Quantum Gaussian Process Kernels

2026-05-29

AI Summary

This work uncovers a common origin underlying the absence of exponential speedup in quantum Gaussian process regression and the failure of Bayesian optimization: the low intrinsic dimensionality of the target function in the kernel feature space leads to a diminished normalized spectral entropy of the quantum kernel matrix, $S(K)/\log n$, which in turn induces performance degradation and "dequantization" behavior. The study introduces normalized spectral entropy as a universal diagnostic metric, providing a unified explanation for diverse quantum kernel pathologies and enabling consistent performance prediction across kernel families and hardware platforms. By integrating Nyström approximation error bounds, Bach’s effective degrees of freedom analysis, and empirical validation on both simulators and IBM Heron hardware, the approach achieves average prediction errors of only 1.7%–5.2% and accurately identifies the region of optimal negative log-likelihood.

Abstract

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 $S(K)/\log n$ 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 $d_σ(K)$, 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 $S/\log n$ 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 $S/\log n$ across $24$ configurations at $n_q = 4$, 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 $n_q = 6$ scale-up on the original backend (mean error $1.7\%$). No error mitigation is applied throughout.

Problem

• quantum Gaussian processes
• spectral entropy
• posterior pathologies
• exponential speedups
• kernel Gram matrix

Innovation

• spectral entropy
• quantum Gaussian processes
• Nyström approximation
• Bayesian optimization
• kernel diagnostics