Direct Product Theorems for Randomized Query Complexity

TL;DR

This paper proves two new direct product theorems for randomized query complexity, showing how the complexity scales with multiple function copies and introducing a new 'discounted score' measure to analyze these problems.

Contribution

The paper introduces a novel 'discounted score' measure and applies it to establish general direct product theorems for randomized query complexity, unifying and extending prior results.

Findings

The first theorem relates the complexity of computing multiple copies with success probability $oldsymbol{ heta^n}$.

The second theorem provides a list decoding direct product result and new variants like the labelled-threshold theorem.

Both theorems are proved using the new 'discounted score' complexity measure.

Abstract

We establish two new direct product theorems for the randomized query complexity of Boolean functions. The first shows that computing $n$ copies of a function $f$, even with a small success probability of $\gamma^n$, requires $\Theta(n)$ times the "maximum distributional" query complexity of $f$ with success parameter $\gamma$. This result holds for all success parameters $\gamma$, even when $\gamma$ is very close to $1/2$ or to $1$. As a result, it unifies and generalizes Drucker's direct product theorem (2012) for $\gamma$ bounded away from $\frac12$ and $1$ as well as the strong direct sum theorem of Blais and Brody (2019) for $\gamma\approx 1-1/n$. The second establishes a general "list decoding" direct product theorem that captures many different variants of partial computation tasks related to the function $f^n$ consisting of $n$ copies of $f$. Notably, our list decoding direct product theorem yields a new threshold direct product theorem and other new variants such as the labelled-threshold direct product theorem. Both of these direct product theorems are obtained by taking a new approach. Instead of directly analyzing the query complexity of algorithms, we introduce a new measure of complexity of functions that we call "discounted score". We show that this measure satisfies a number of useful structural properties, including tensorization, that make it particularly suitable for the study of direct product questions.