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

In classical deep learning, mode connectivity—the existence of low-loss paths between different minima—is a well-studied phenomenon. In quantum machine learning (QML), however, the loss landscape is defined over parameterized quantum circuits, where the parameters are merely coordinates for an underlying unitary transformation. This paper seeks to provide a rigorous theoretical framework for mode connectivity in QML by shifting the focus from parameter space to the geometry of the reachable unitary Lie group.

The author models the QML loss landscape using the dynamical Lie algebra (DLA) generated by the circuit's operators. By defining the reachable unitary group as the Lie group generated by this algebra, the study treats the loss function as a map from this group to real numbers. Using tools from Morse theory, the author proves that if a low-loss band is free of critical values and the region near the global minimum is connected, then the entire low-loss sublevel set on the Lie group is path-connected. The paper further argues that overparameterization acts as a mechanism to "lift" these Lie-group paths into the circuit's parameter space, making the connectivity observable in practice.

The study establishes that mode connectivity is a geometric property of the reachable Lie group. The author shows that when the DLA is sufficiently expressive, the low-loss regions are connected, and this connectivity can be realized in parameter space provided the circuit is overparameterized. Numerical experiments on 5-qubit circuits support this, showing that geodesic interpolations between trained unitaries exhibit near-zero loss barriers, consistent with the theoretical prediction that the low-loss manifold is connected.

This work provides a foundational geometric interpretation for empirical observations of mode connectivity in QML. By linking the phenomenon to the DLA and Lie-group topology, it offers a more robust way to understand the loss landscapes of quantum circuits, potentially guiding the design of more efficient training strategies and helping researchers distinguish between connectivity arising from the quantum model's inherent geometry versus artifacts of specific parameterizations.