Optimal algorithmic complexity of inference in quantum kernel methods

2026-04-16

AI Summary

This work addresses the query complexity of quantum kernel methods during inference, which typically scales linearly with the number of training samples $N$. By jointly optimizing the strategy for kernel value estimation—choosing between sampling and quantum amplitude estimation—and the summation approach—either estimating terms individually or encoding the entire sum into a single observable—we propose the first inference algorithm with query complexity $O(\|\alpha\|_1/\varepsilon)$, thereby eliminating any dependence on $N$. We prove this complexity is theoretically optimal by establishing a matching lower bound of $\Omega(\|\alpha\|_1/\varepsilon)$, achieving quadratic speedups in both $\|\alpha\|_1$ and $\varepsilon$. Furthermore, we uncover a fundamental trade-off between query-optimal and gate-complexity-optimal strategies, offering practical guidance tailored to different hardware constraints.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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 $\sum_{i=1}^N α_i k(x,x_i)$ of $N$ kernel values to additive precision $\varepsilon$, where $α$ is the vector of trained coefficients. The standard approach estimates each term independently via sampling, yielding a query complexity of $O(N\lVertα\rVert_2^2/\varepsilon^2)$. 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 $O(\lVertα\rVert_1/\varepsilon)$, removing the dependence on $N$ from the query count and yielding a quadratic improvement in both $\lVertα\rVert_1$ and $\varepsilon$. We prove a matching lower bound of $Ω(\lVertα\rVert_1/\varepsilon)$, 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.

Problem

• quantum kernel methods
• inference complexity
• query complexity
• amplitude estimation
• supervised learning

Innovation

• quantum kernel methods
• query complexity
• amplitude estimation
• optimal inference
• quantum advantage