The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy

We consider the cost of implementing the quadratic collective phase $e^{-iγK^2}$, with $K$ a collective observable of evenly spaced spectrum, such as the permutation-symmetric Hamming weight. This gate is the cardinality-penalty layer of constrained quantum optimization, the one-axis-twisting gate o