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

TL;DR

This paper investigates how the symmetry of quantum models affects the measurement resources needed for fidelity-based quantum kernel learning of phase transitions, revealing that higher symmetry increases shot costs due to kernel concentration.

Contribution

It establishes a quantitative link between model symmetry and resource scaling in fidelity quantum kernels, providing practical bounds for resource estimation in quantum machine learning.

Findings

Higher symmetry amplifies shot requirements for fidelity estimation.

Kernel concentration increases with model symmetry, impacting measurement costs.

Provides bounds and methods for resource estimation in symmetry-aware quantum kernel learning.

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.