Monte Carlo to Las Vegas for Recursively Composed Functions
Bandar Al-Dhalaan, Shalev Ben-David
TL;DR
The paper introduces and analyzes composition limits $M^*(f)=\lim_{k\to\infty} M(f^k)^{1/k}$ for Boolean-branch measures under recursive function composition, proving convergence under mild niceties. It shows a sharp equivalence for randomized and quantum Las Vegas certificates: $R_0^*(f)=\max\{R^*(f),C^*(f)\}$ and $Q_C^*(f)=\max\{Q^*(f),C^*(f)\}$, meaning bounded-error algorithms can be converted to zero-error certificates with only the certificate complexity dictating the cost. The work develops a Monte Carlo to Las Vegas technique to construct certificates from bounded-error evaluations and proves a general convergence theorem for composition limits via a subsequence interleaving argument. It also supplies a robust toolkit of combinatorial reductions, switchable-function analyses, and invariance properties that ensure composition bounds transfer across a broad class of measures, with implications for both classical and quantum query complexity and certificate-finding paradigms.
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.