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Spectral Anatomy of Quantum Gaussian Process Kernels

arXiv 2605.30952 ↗First posted 29 May 2026cs.LGcited by 0

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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.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 19 of its 28 references that are in the map. First posted on arXiv 29 May 2026.

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References · 19

  1. 2019Similarity of Neural Network Representations Revisitedcited by 1,695
  2. 2018Supervised learning with quantum enhanced feature spacescited by 654
  3. 2018Barren plateaus in quantum neural network training landscapescited by 624
  4. 2017Deep Neural Networks as Gaussian Processescited by 617
  5. 2018Gaussian Process Behaviour in Wide Deep Neural Networkscited by 314
  6. 2015On the Equivalence between Kernel Quadrature Rules and Random Feature Expansionscited by 136
  7. 2021The Inductive Bias of Quantum Kernelscited by 90
  8. 2022Exponential concentration in quantum kernel methodscited by 89
  9. 2018A quantum-inspired classical algorithm for recommendation systemscited by 70
  10. 2015Quantum assisted Gaussian process regressioncited by 17
  11. 2018Bayesian Deep Learning on a Quantum Computercited by 12
  12. 2023Quantum Bayesian Optimizationcited by 12
7 more
  1. 2023Quantum Gaussian Process Regression for Bayesian Optimizationcited by 9
  2. 2024Trained quantum neural networks are Gaussian processescited by 5
  3. 2024Quantum-Assisted Hilbert-Space Gaussian Process Regressioncited by 4
  4. 2025Assessing Quantum Advantage for Gaussian Process Regressioncited by 2
  5. 2026Provable and scalable quantum Gaussian processes for quantum learningcited by 2
  6. 2018Quantum algorithms for training Gaussian Processescited by 1
  7. 2025Quantum-assisted Gaussian process regression using random Fourier featurescited by 1

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