Monte Carlo to Las Vegas for Recursively Composed Functions

For a (possibly partial) Boolean function f : { 0, 1} n { 0, 1} as well as a query complexity measure which maps Boolean functions to real numbers, define the composition limit of on f by (f ) = k(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 0 (f ) = { (f ), (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.