Abstract:Quantum kernel methods are among the leading candidates for achieving quantum advantage in supervised learning. A key bottleneck is the cost of inference: evaluating a trained model on new data requires estimating a weighted sum $\sum_{i=1}^N \alpha_i k(x,x_i)$ of $N$ kernel values to additive precision $\varepsilon$, where $\alpha$ is the vector of trained coefficients. The standard approach estimates each term independently via sampling, yielding a query complexity of $O(N\lVert\alpha\rVert_2^2/\varepsilon^2)$. In this work, we identify two independent axes for improvement: (1) How individual kernel values are estimated (sampling versus quantum amplitude estimation), and (2) how the sum is approximated (term-by-term versus via a single observable), and systematically analyze all combinations thereof. The query-optimal combination, encoding the full inference sum as the expectation value of a single observable and applying quantum amplitude estimation, achieves a query complexity of $O(\lVert\alpha\rVert_1/\varepsilon)$, removing the dependence on $N$ from the query count and yielding a quadratic improvement in both $\lVert\alpha\rVert_1$ and $\varepsilon$. We prove a matching lower bound of $\Omega(\lVert\alpha\rVert_1/\varepsilon)$, establishing query-optimality of our approach up to logarithmic factors. Beyond query complexity, we also analyze how these improvements translate into gate costs and show that the query-optimal strategy is not always optimal in practice from the perspective of gate complexity. Our results provide both a query-optimal algorithm and a practically optimal choice of strategy depending on hardware capabilities, along with a complete landscape of intermediate methods to guide practitioners. All algorithms require only amplitude estimation as a subroutine and are thus natural candidates for early-fault-tolerant implementations.
| Comments: | 26 pages (13+13), 4 figures, comments welcome |
| Subjects: | Quantum Physics (quant-ph); Machine Learning (cs.LG) |
| Cite as: | arXiv:2604.15214 [quant-ph] |
| (or arXiv:2604.15214v2 [quant-ph] for this version) | |
| https://doi.org/10.48550/arXiv.2604.15214 arXiv-issued DOI via DataCite |
From: Elies Gil-Fuster [view email]
[v1]
Thu, 16 Apr 2026 16:45:02 UTC (159 KB)
[v2]
Fri, 17 Apr 2026 07:29:39 UTC (159 KB)
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