Walks in Rotation Spaces Return Home when Doubled and Scaled

The dynamics of numerous physical systems, such as spins and qubits, can be described as a series of rotation operations, i.e., walks in the manifold of the rotation group. A basic question with practical applications is how likely and under what conditions such walks return to the origin (the identity rotation), which means that the physical system returns to its initial state. In three dimensions, we show that almost every walk in SO(3) or SU(2), even a very complicated one, will preferentially return to the origin simply by traversing the walk twice in a row and uniformly scaling all rotation angles. We explain why traversing the walk only once almost never suffices to return, and comment on the problem in higher dimensions.