Monte Carlo to Las Vegas for Recursively Composed Functions

arXiv:2601.08073v1 Announce Type: new
Abstract: For a (possibly partial) Boolean function $fcolon{0,1}^nto{0,1}$ as well as a query complexity measure $M$ which maps Boolean functions to real numbers, define the composition limit of $M$ on $f$ by $M^*(f)=lim_{ktoinfty} M(f^k)^{1/k}$.
We study the composition limits of general measures in query complexity. We show this limit converges under reasonable assumptions about the measure. We then give a surprising result regarding the composition limit of randomized query complexity: we show $R_0^*(f)=max{R^*(f),C^*(f)}$. Among other things, this implies that any bounded-error randomized algorithm for recursive 3-majority can be turned into a zero-error randomized algorithm for the same task. Our result extends also to quantum algorithms: on recursively composed functions, a bounded-error quantum algorithm can be converted into a quantum algorithm that finds a certificate with high probability.
Along the way, we prove various combinatorial properties of measures and composition limits.