Common-angle replacement for parallel Kac walk

Determine whether the independent random rotations used on the matched coordinate pairs in a parallel Kac walk can be replaced by a single common random rotation while retaining the relevant properties of the walk.

Background

The parallel Kac walk partitions the coordinates into a uniformly random perfect matching and applies independent random rotations to the matched pairs in each round. The cited works of Lu, Qin, Song, Yao, and Zhao posed the question of whether this independence is necessary, specifically asking whether all matched pairs could instead use one common random rotation or even a fixed angle.

The paper resolves the fixed-angle variant by analyzing parallel orthogonal repeated averaging, in which every matched pair is rotated by the fixed angle π/4 and near-optimal coherence is achieved after O(log d) rounds. It does not resolve the distinct variant involving a common random angle, which therefore remains an unresolved question explicitly identified in the paper.