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arXiv:2603.18211

Finite-size resource scaling for learning quantum phase transitions with fidelity-based support vector machines

Aaqib Ali, Giovanni Scala, Cosmo Lupo, Antonio Mandarino

quant-phSubmitted Mar 18, 2026• Announced Mar 20, 2026• 0 citations• Revised Sep 7, 2026 (v2)
v1 · Mar 18, 2026v2 · Sep 7, 2026
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Abstract

Quantum kernels offer a valid procedure for learning quantum phase transitions on quantum processing devices, yet issues on the scalability of the learning strategy in connection with the symmetry of the critical model have not been clarified. We derive a link between model symmetry and fidelity-kernel resource scaling. We quantify the measurement resources required to estimate fidelity-based quantum kernels for many-body ground states while preserving the structure of the resulting Gram matrix under finite-shot sampling. Crucially, we show that increasing symmetry in the underlying spin model systematically amplifies these shot requirements. Moving from the $\mathbb{Z}_2$-symmetric Ising/XY regimes to the $U(1)$-symmetric XX (and XXZ) regimes leads to stronger kernel concentration and therefore substantially larger shot costs under the same bounds. We consider a tunable one-dimensional spin-$\tfrac{1}{2}$ Hamiltonian spanning the transverse-field Ising, XY, XX, and XXZ limits, and define the kernel as the ground-state fidelity. Kernel entries are estimated using a SWAP-test estimator with $S$ shots, and we adapt the ensemble spread and concentration-avoidance shot bounds to obtain practical shot requirements in terms of the interquartile range of kernel values and a representative kernel magnitude. For the free-fermion XY/XX family, we use the closed-form Bogoliubov-angle fidelity, while for the interacting XXZ chain we compute fidelities by exact diagonalization and benchmark shot-noise effects. Our symmetry-aware bounds provide a pragmatic procedure for physics-informed quantum machine learning.

Comments
10 figures, 1 table

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Quantum support vector machines are classification algorithms that rely on quantum-generated kernels. The fidelity quantum kernel commonly used in quantum support vector machines suffers from exponential concentration as system size increases, preventing an efficient scaling beyond fewqubit systems. We introduce the Hamming quantum kernel, a classical post-processing method that is based on the same measurement outcomes as the fidelity quantum kernel. However, it avoids the exponential concentration problem by using the full measurement statistics rather than a single fidelity value. We evaluate the approach on both classical data (MNIST) and synthetic data generated from quantum circuits, using systems ranging from 2 to 27 qubits. Throughout the simulations, the Hamming quantum kernel outperforms the fidelity quantum kernel whenever 15 or more qubits are used. Furthermore, for synthetic quantum data, our method consistently outperforms the classical Gaussian kernel. This demonstrates that the Hamming quantum kernel improves the expressivity and robustness at larger qubit scales without requiring any additional quantum ressources.
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