WP-2026-006 · quantum · ml

Quantum Kernel Methods vs Classical SVMs on MNIST

Ibrahim Laklaa · 2026

0.990

best classical accuracy, 3 vs 5

0.823

best quantum accuracy, 3 vs 5

seeds

report pages

Abstract

Fidelity quantum kernels in Qiskit (Z, ZZ, Pauli feature maps) and a projected quantum kernel, benchmarked against RBF, linear, polynomial and random Fourier feature baselines over 10 seeds with bootstrap intervals and McNemar tests. Classical kernels win by about 17 points on MNIST 3 vs 5, and kernel concentration and barren plateaus are reproduced empirically.

Question

Do quantum kernels give an advantage on a real classification task, once the classical baselines and the statistics are done properly?

The quantum kernel

An input $x \in R^{n}$ is encoded by a parametric circuit, $∣ ϕ (x)⟩ = U_{ϕ} (x) ∣0 ⟩^{\otimes n}$ , and compared by fidelity (Havlicek et al., 2019):

K (x, y) = ⟨ ϕ (x) ∣ ϕ (y)⟩^{2} = Tr (ρ (x) ρ (y))

which is symmetric, positive semi-definite and has a unit diagonal. Three feature maps (Z, ZZ, Pauli) are built in Qiskit and fed to an SVM with precomputed Gram matrices. A projected quantum kernel (Kübler et al., 2021) trades expressivity for slower concentration:

K_{P Q K} (x, y) = exp (- γ i \sum P \in {X, Y, Z} \sum (⟨ P_{i} ⟩_{ϕ (x)} - ⟨ P_{i} ⟩_{ϕ (y)})^{2})

Protocol

MNIST 0 vs 1 and the harder 3 vs 5 (PCA keeps only 44.7 % of the variance), against RBF, linear, polynomial and random Fourier feature baselines. 10 seeds with percentile bootstrap intervals, and McNemar paired tests combined across seeds with Fisher's method:

χ^{2} = \frac{( ∣ n _{01} - n _{10} ∣ - 1 ) ^{2}}{n _{01} + n _{10}}

Findings

On 3 vs 5, the best classical kernel reaches 0.990 accuracy and the best quantum kernel 0.823. The confidence intervals do not overlap and the gap is significant at $p < 1 0^{- 3}$ .
Kernel concentration is measured, not just cited: $Var (K) \propto 2^{- 1.54 n}$ for the ZZ map, consistent with Thanasilp et al. (2024).
Barren plateaus appear as predicted by McClean et al. (2018): $Var (\partial C) \propto 2^{- 0.98 n}$ .

Takeaway

A negative result, made solid: on this task the classical kernels win clearly, and the two pathologies that explain why are reproduced empirically. 30 unit tests and an 18-page report.

Read the paper Code