AQKA: Active Quantum Kernel Acquisition Under a Shot Budget

2026-05-14

AI Summary

This work addresses the inefficiency of quantum kernel estimation under limited measurement budgets, which stems from conventional approaches uniformly allocating resources while neglecting the heterogeneous impact of individual kernel matrix entries on classification performance. To overcome this limitation, the authors propose Active Quantum Kernel Acquisition (AQKA), the first framework for budget-constrained quantum kernel learning that dynamically allocates measurement shots based on gradients from downstream tasks. By integrating gradient information from kernel ridge regression and support vector machines, the envelope theorem, and a sparsity-aware Cauchy–Schwarz bound, they derive a closed-form expression for optimal shot allocation along with accompanying theoretical guarantees. Experiments demonstrate that AQKA significantly outperforms baseline methods on both simulated and IBM quantum hardware, achieving up to a 32-percentage-point improvement in accuracy and enabling, for the first time, online adaptive measurement validation across multiple ansätze.

Abstract

Estimating an $N \times N$ quantum kernel from circuit fidelities requires $Θ(N^2 S)$ measurement shots, the dominant bottleneck for deployment on near-term hardware. Existing budget-saving methods (Nyström-QKE, ShoFaR, kernel-target alignment) sub-sample \emph{which} entries to measure but allocate shots \emph{uniformly} within their chosen subset, ignoring how much each entry drives the downstream classifier. We close this gap with two contributions. \textbf{First, a complete regime decomposition} for shot-budgeted quantum kernel learning: a principled menu of when each allocator wins. Our method, \emph{AQKA}, dominates the budget-limited regime ($B \lesssim 16 n_{\mathrm{pairs}}$) on sparse-sensitivity KRR, with the gap \emph{growing} from $+8$ to $+25$ pts over uniform as $N$ scales $225{\to}1000$ and reaching $+26$--$32$ pts on an \texttt{ibm\_pittsburgh} (156-qubit Heron) hardware kernel; Nyström-QKE wins at saturating budgets on planted-sparse via low-rank reconstruction; ShoFaR is competitive only at extreme low budgets. \textbf{Second, a closed-form pair-level acquisition theory}: $s_{ij}^{\star} \propto |g_{ij}|\sqrt{K_{ij}(1-K_{ij})}$ with explicit gradient $g_{ij}$ for KRR (Lemma~1, $|β_iα_j+β_jα_i|\sqrt{K_{ij}(1-K_{ij})}$) and SVM via the envelope theorem ($|η_i^*η_j^*|\sqrt{K_{ij}(1-K_{ij})}$); a \emph{corrected} sparsity-aware Cauchy--Schwarz rate $ρ\le 2m/N$ matching empirics (vs.\ the naive $m^2/N^2$); an explicit-constant plug-in regret bound (Theorem~2); and a tighter SVM ceiling $ρ^{\mathrm{SVM}} \le m_{\mathrm{sv}}^2/N^2$. We close with the first multi-seed live online adaptive shot allocation on quantum hardware: $+17.0 \pm 4.8$ pts at $N{=}20$ on \texttt{ibm\_aachen} ($3.5σ$, 5 seeds), with the advantage holding at $N{=}30$ at higher budget on \texttt{ibm\_berlin} ($+14.0 \pm 8.5$ pts, 5 seeds).

Problem

• quantum kernel
• shot budget
• measurement allocation
• kernel estimation
• near-term quantum hardware

Innovation

• Active Quantum Kernel Learning
• Shot Budget Allocation
• Non-uniform Measurement Strategy
• Kernel Ridge Regression
• Quantum Hardware Experiment