A Strong Direct Sum Theorem for Distributional Query Complexity

TL;DR

This paper proves a strong direct sum theorem for distributional query complexity, showing that computing multiple copies of a function simultaneously requires proportionally more resources, and introduces novel techniques involving the Hardcore Theorem.

Contribution

It establishes the first strong direct sum theorem for distributional query complexity, demonstrating the optimality of naive parallel computation strategies.

Findings

Naive sequential computation is essentially optimal for direct product problems.

Computing a direct product requires proportional increases in query complexity and error.

Introduces the first use of the Hardcore Theorem in query complexity analysis.

Abstract

Consider the expected query complexity of computing the $k$-fold direct product $f^{\otimes k}$ of a function $f$ to error $\varepsilon$ with respect to a distribution $\mu^k$. One strategy is to sequentially compute each of the $k$ copies to error $\varepsilon/k$ with respect to $\mu$ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error. Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem had been elusive. A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new "resilience lemma" that accompanies it, showing that the hardcore of $f^{\otimes k}$ is likely to remain dense under arbitrary partitions of the input space.