Lie-Group Mode Connectivity in Quantum Machine Learning from a Dynamical Lie Algebra Perspective

Abstract:Mode connectivity has been widely studied in classical machine learning as a geometric property of low-loss regions in parameter space. In quantum machine learning (QML), however, the physically relevant object is not the parameter vector itself but the unitary transformation implemented by a parameterized quantum circuit. In this study, we formulate mode connectivity on the reachable unitary Lie group generated by the dynamical Lie algebra of the generators. We show that, under a near-minimum connectedness assumption and the absence of critical values in a low-loss band, the corresponding low-loss sublevel set on the reachable Lie group is path-connected. This provides a geometric interpretation of mode connectivity in QML that is independent of a particular parameterization. We further discuss how overparameterization can enable the lifting of Lie-group paths to parameter space, thereby making Lie-group connectivity observable in parameter-space experiments. Finally, we present toy numerical experiments in which geodesic interpolations between trained unitaries exhibit nearly zero loss barriers, consistent with the proposed interpretation.

arXiv:2607.17554 [quant-ph]

(or arXiv:2607.17554v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.17554 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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