Optimal algorithmic complexity of inference in quantum kernel methods

• URL: http://arxiv.org/abs/2604.15214v2
• Date: Fri, 17 Apr 2026 07:29:39 GMT
• Title: Optimal algorithmic complexity of inference in quantum kernel methods
• Abstract summary: Quantum kernel methods are among the leading candidates for achieving quantum advantage in supervised learning.<n>Standard approach estimates each term independently via sampling, yielding a query complexity of $O(NlVertrVert2/varepsilon2)$.<n>We propose a query-optimal combination, encoding the full inference sum as the expectation value of a single observable.<n>Our results provide both a query-optimal algorithm and a practically optimal choice of strategy depending on hardware capabilities.
• Score: 0.815557531820863
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Learning Sparse Quantum States [0.8122270502556375]
  We give the first near optimal algorithm for learning $n$-qubit $k$-sparse pure quantum states.<n>We obtain fidelity at least $1-varepsilon$ with high probability using $tildeO(k/varepsilon)$ copies of the state and $tildeO(kn/varepsilon)$ time.<n>As an implication, we also obtain an algorithm with near optimal $tildeO(kr/varepsilon)$ sample complexity for learning $k$-sparse rank-$r
  arXiv  Detail & Related papers  (2026-09-10T21:28:21Z)
• Quantum Multi-Level Estimation of Functionals of Discrete Distributions [23.53427184324404]
  We propose a quantum multi-level estimation framework for a functional $sum_i=1n f(p_i)$ of a discrete distribution.<n>We present efficient quantum estimators for the $q$-Tsallis entropy of discrete distributions.
  arXiv  Detail & Related papers  (2026-05-05T12:25:17Z)
• Provably Adaptive Linear Approximation for the Shapley Value and Beyond [73.0940890296463]
  A fundamental and long-standing challenge is their efficient approximation.<n>We develop a linear-space algorithm that requires $P(|hatboldsymbol-boldsymbol|_2geq)leq $ for all commonly used semi-values.<n>Our algorithm allows explicit minimization of the mean square error for each specific utility function.
  arXiv  Detail & Related papers  (2026-04-09T16:38:14Z)
• Quantum Algorithms for Non-smooth Non-convex Optimization [30.576546266390714]
  This paper considers the problem for finding the $(,epsilon)$-Goldstein stationary point of Lipschitz continuous objective. We construct a zeroth-order quantum estimator for the surrogate oracle function.
  arXiv  Detail & Related papers  (2024-10-21T16:52:26Z)
• Information-Computation Tradeoffs for Learning Margin Halfspaces with Random Classification Noise [50.64137465792738]
  We study the problem of PAC $gamma$-margin halfspaces with Random Classification Noise. We establish an information-computation tradeoff suggesting an inherent gap between the sample complexity of the problem and the sample complexity of computationally efficient algorithms.
  arXiv  Detail & Related papers  (2023-06-28T16:33:39Z)
• Private estimation algorithms for stochastic block models and mixture models [63.07482515700984]
  General tools for designing efficient private estimation algorithms. First efficient $(epsilon, delta)$-differentially private algorithm for both weak recovery and exact recovery.
  arXiv  Detail & Related papers  (2023-01-11T09:12:28Z)
• Unitarity estimation for quantum channels [7.323367190336826]
  Unitarity estimation is a basic and important problem in quantum device certification and benchmarking. We provide a unified framework for unitarity estimation, which induces ancilla-efficient algorithms. We show that both the $d$-dependence and $epsilon$-dependence of our algorithms are optimal.
  arXiv  Detail & Related papers  (2022-12-19T09:36:33Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints [62.72309460291971]
  We introduce a variational quantum algorithm for Goemans-Williamson algorithm that uses only $n+1$ qubits. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems.
  arXiv  Detail & Related papers  (2022-06-30T03:15:23Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Best of Both Worlds: Practical and Theoretically Optimal Submodular Maximization in Parallel [17.462968952951883]
  Main algorithm is assembled from two components which may be of independent interest. A variant of LINEARSEQ is shown to have adaptive complexity of $O(log(n))$ smaller than that of any previous algorithm in the literature.
  arXiv  Detail & Related papers  (2021-11-15T17:10:40Z)
• Proper Learning, Helly Number, and an Optimal SVM Bound [29.35254938542589]
  We characterize classes for which the optimal sample complexity can be achieved by a proper learning algorithm. We show that the dual Helly number is bounded if and only if there is a proper joint dependence on $epsilon$ and $delta$.
  arXiv  Detail & Related papers  (2020-05-24T18:11:57Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.