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Direct Product Theorems for Randomized Query Complexity

Shalev Ben-David, Eric Blais

TL;DR

This work advances the theory of direct product theorems for randomized query complexity by introducing discounted score as a unifying complexity lens. It proves two main results: an optimal direct product theorem in terms of the maximum distributional complexity $_\gamma(f)$ and a comprehensive list-decoding direct product framework that encompasses threshold and labelled-threshold variants. The approach hinges on the tensorization of discounted score and the equivalence between discounted score and score-weighted distributional complexity, enabling tight lower and upper bounds across bounded-error and small-error regimes. Together, these results unify prior theorems (Drucker, Blais–Brody) and deliver new variants with sharp, information-theoretic proofs. The framework provides robust tools for hardness amplification in query complexity and related models.

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 $γ^n$, requires $Θ(n)$ times the "maximum distributional" query complexity of $f$ with success parameter $γ$. This result holds for all success parameters $γ$, even when $γ$ is very close to $1/2$ or to $1$. As a result, it unifies and generalizes Drucker's direct product theorem (2012) for $γ$ bounded away from $\frac12$ and $1$ as well as the strong direct sum theorem of Blais and Brody (2019) for $γ\approx 1-1/n$. 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.

Direct Product Theorems for Randomized Query Complexity

TL;DR

This work advances the theory of direct product theorems for randomized query complexity by introducing discounted score as a unifying complexity lens. It proves two main results: an optimal direct product theorem in terms of the maximum distributional complexity and a comprehensive list-decoding direct product framework that encompasses threshold and labelled-threshold variants. The approach hinges on the tensorization of discounted score and the equivalence between discounted score and score-weighted distributional complexity, enabling tight lower and upper bounds across bounded-error and small-error regimes. Together, these results unify prior theorems (Drucker, Blais–Brody) and deliver new variants with sharp, information-theoretic proofs. The framework provides robust tools for hardness amplification in query complexity and related models.

Abstract

We establish two new direct product theorems for the randomized query complexity of Boolean functions. The first shows that computing copies of a function , even with a small success probability of , requires times the "maximum distributional" query complexity of with success parameter . This result holds for all success parameters , even when is very close to or to . As a result, it unifies and generalizes Drucker's direct product theorem (2012) for bounded away from and as well as the strong direct sum theorem of Blais and Brody (2019) for . The second establishes a general "list decoding" direct product theorem that captures many different variants of partial computation tasks related to the function consisting of copies of . 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.

Paper Structure

This paper contains 47 sections, 37 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. An Optimal Direct Product Theorem
  3. List-Decoding Direct Product Theorem
  4. Discounted Score
  5. Tangential remarks.
  6. Motivation.
  7. Tensorization.
  8. Success-conditioned score
  9. Overview of the Other Proofs
  10. Direct Product Theorem Upper Bound
  11. List-Decoding Direct Product Theorem
  12. Definitions and Foundations
  13. Score Measures
  14. Examples of Score Measures
  15. Success probability score.
  16. ...and 32 more sections

Key Result

Theorem 1

For every (possibly partial) function $f \colon \{0,1\}^m \to \{0,1\}$, success probability $\gamma$ in the range $\frac{1}{2} < \gamma < 1$, and parameter $n \ge 1$, This lower bound is best possible when $n$ is large enough: there is a universal constant $c$ such that for every function $f$, success probability $\gamma$ in the range $\frac{1}{2} < \gamma < 1$, and parameter $n \ge \frac{c}{(\mi

Theorems & Definitions (64)

  • Theorem 1: Optimal Direct Product Theorem
  • Theorem 2: List-Decoding Direct Product Theorem
  • Corollary 3: Threshold Direct Product Theorem
  • Corollary 4: Labelled Threshold Direct Product Theorem
  • Lemma 4: Tensorization Lemma
  • Lemma 4: Equivalence Lemma
  • Theorem 5: Direct Product Theorem for sR
  • Lemma 5
  • Lemma 5
  • Proposition 5
  • ...and 54 more