Monte Carlo to Las Vegas for Recursively Composed Functions

TL;DR

This paper investigates the behavior of query complexity measures under recursive function composition, revealing that the composition limit of randomized and quantum complexities can be characterized and transformed, with implications for algorithm design.

Contribution

It introduces the composition limit concept for query complexity measures, proves convergence under certain conditions, and establishes a key relationship for randomized and quantum complexities on recursively composed functions.

Findings

The composition limit of randomized query complexity equals the maximum of the limits of zero-error and classical complexities.

Any bounded-error recursive 3-majority algorithm can be converted into a zero-error algorithm.

The results extend to quantum algorithms, enabling high-probability certificate finding.

Abstract

For a (possibly partial) Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ as well as a query complexity measure $M$ which maps Boolean functions to real numbers, define the composition limit of $M$ on $f$ by $M^*(f)=\lim_{k\to\infty} M(f^k)^{1/k}$. 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.