Twirlator: A Pipeline for Analyzing Subgroup Symmetry Effects in Quantum Machine Learning Ansatzes

Abstract

Symmetry is a strong inductive bias in geometric deep learning and its quantum counterpart, and has attracted increasing attention for improving the trainability of QML models. Yet incorporating symmetries into quantum machine learning (QML) ansatzes is not free: symmetrization often adds gates and constrains the circuits. To understand these effects, we present Twirlator, which is an automated pipeline that symmetrizes parameterized QML ansatzes and quantifies the trade-offs as the amount of symmetry increases. Twirlator models partial symmetries by the size of a subgroup of the symmetric group, enabling analysis between the ``no symmetry'' and ``full symmetry'' extremes. Across 19 common ansatz patterns, Twirlator symmetrizes circuits with respect to any subgroup of $S_n$ and measures (1) generator drift, (2) circuit overhead (depth and size), and (3) expressibility and entangling capability. The experimental evaluation focuses on subgroups of $S_4$ and $S_5$. Twirlator reveals that larger subgroups typically increase circuit overhead, reduce expressibility, and often increase entangling capability. The pipeline and results provide practical guidance for selecting ansatz patterns and symmetry levels that balance hardware cost and model performance in symmetry-aware QML applications.