Observable-Guided Generator Selection for Improving Trainability in Quantum Machine Learning with a $ \mathfrak{g} $-Purity Interpretation under Restricted Settings

TL;DR

This paper introduces an observable-guided generator selection algorithm for quantum machine learning that improves trainability by optimizing generator properties related to sensitivity and interference, with theoretical and empirical support.

Contribution

It proposes a novel generator selection method based on observable criteria, linking it to algebraic properties like $\mathfrak{g}$-purity, and demonstrates its effectiveness in small-scale experiments.

Findings

Selected generators enable faster training compared to random selection.

The criteria relate to $\mathfrak{g}$-purity, providing a theoretical interpretation.

The method maintains expressibility while enhancing trainability.

Abstract

To study generator design for parameterized unitaries in quantum machine learning (QML), we propose an observable-guided generator selection algorithm for $ n $-qubit Pauli-string generator pools. The proposed method selects generators based on two criteria: maintaining large first-order sensitivity in the gradients and suppressing second-order interference in the Hessian matrix. Under a restricted setting with Pauli-string observables and candidate generators, the selection problem can be formulated as a binary optimization problem that favors mutually anti-commuting generators. Numerical experiments on a synthetic dataset with a small-scale five-qubit circuit show that the selected generators yield faster training than random generator selection in our setting, while exhibiting similar expressibility. Furthermore, under additional algebraic assumptions, the proposed criteria admit an interpretation in terms of the $ \mathfrak{g} $-purity of the observable: the first-order sensitivity is proportional to the $ \mathfrak{g} $-purity, whereas the second-order interference, namely the off-diagonal elements of the Hessian matrix, is upper-bounded by it. These results suggest that observable-guided generator selection is a promising direction for improving trainability in restricted QML settings.