Twirlator: A Pipeline for Analyzing Subgroup Symmetry Effects in Quantum Machine Learning Ansatzes
Valter Uotila, Väinö Mehtola, Ilmo Salmenperä, Bo Zhao
TL;DR
This work addresses the lack of automated tools to study subgroup-based symmetries in quantum machine learning ansatzes. It introduces an automated pipeline that symmetrizes 19 common quantum circuit patterns with respect to subgroups of the symmetry group, then evaluates the impact using three metric classes: operator-norm differences, circuit-related overhead, and expressibility and entangling capability. The key findings are that increasing symmetry generally reduces expressibility but enhances entangling capability, with substantial gate overhead that varies with subgroup size; these trends inform the selection of practical, expressive, and efficient ansatzes for geometry-aware quantum learning. The approach, grounded in angle encoding and induced unitary representations, provides a principled framework to connect symmetry theory with practical quantum circuit design and is released as open-source to aid researchers and developers.
Abstract
Leveraging data symmetries has been a key driver of performance gains in geometric deep learning and geometric and equivariant quantum machine learning. While symmetrization appears to be a promising method, its practical overhead, such as additional gates, reduced expressibility, and other factors, is not well understood in quantum machine learning. In this work, we develop an automated pipeline to measure various characteristics of quantum machine learning ansatzes with respect to symmetries that can appear in the learning task. We define the degree of symmetry in the learning problem as the size of the subgroup it admits. Subgroups define partial symmetries, which have not been extensively studied in previous research, which has focused on symmetries defined by whole groups. Symmetrizing the 19 common ansatzes with respect to these varying-sized subgroup representations, we compute three classes of metrics that describe how the common ansatz structures behave under varying amounts of symmetries. The first metric is based on the norm of the difference between the original and symmetrized generators, while the second metric counts depth, size, and other characteristics from the symmetrized circuits. The third class of metrics includes expressibility and entangling capability. The results demonstrate varying gate overhead across the studied ansatzes and confirm that increased symmetry reduces expressibility of the circuits. In most cases, increased symmetry increases entanglement capability. These results help select sufficiently expressible and computationally efficient ansatze patterns for geometric quantum machine learning applications.