# Spectral Anatomy of Quantum Gaussian Process Kernels

arXiv: https://arxiv.org/abs/2605.30952 · HTML: https://mlsanity.com/paper/2605.30952/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 28 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 19 of 19)

- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [A quantum-inspired classical algorithm for recommendation systems](https://mlsanity.com/paper/1807.04271/) (2018)
- [Quantum assisted Gaussian process regression](https://mlsanity.com/paper/1512.03929/) (2015)
- [Bayesian Deep Learning on a Quantum Computer](https://mlsanity.com/paper/1806.11463/) (2018)
- [Quantum Bayesian Optimization](https://mlsanity.com/paper/2310.05373/) (2023)
- [Quantum Gaussian Process Regression for Bayesian Optimization](https://mlsanity.com/paper/2304.12923/) (2023)
- [Trained quantum neural networks are Gaussian processes](https://mlsanity.com/paper/2402.08726/) (2024)
- [Quantum-Assisted Hilbert-Space Gaussian Process Regression](https://mlsanity.com/paper/2402.00544/) (2024)
- [Assessing Quantum Advantage for Gaussian Process Regression](https://mlsanity.com/paper/2505.22502/) (2025)
- [Provable and scalable quantum Gaussian processes for quantum learning](https://mlsanity.com/paper/2605.00099/) (2026)
- [Quantum algorithms for training Gaussian Processes](https://mlsanity.com/paper/1803.10520/) (2018)
- [Quantum-assisted Gaussian process regression using random Fourier features](https://mlsanity.com/paper/2507.22629/) (2025)

## Abstract (arXiv)

Two recent results have reshaped quantum Gaussian processes (QGPs). On the one hand, citetlowe2025assessing 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 Nystr"om approximation error, a finite-sample variance-contraction identity in terms of Bach's degrees of freedom dσ(K), and a characterization of the 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 and RBF/Mat'ern/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 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.