Direct sum theorems beyond query complexity

Abstract

A fundamental question in computer science is: Is it harder to solve $n$ instances independently than to solve them simultaneously? This question, known as the direct sum question or direct sum theorem, has been paid much attention in several research fields. Despite its importance, however, little has been discovered in many other research fields. In this paper, we introduce a novel framework that extends to classical/quantum query complexity, PAC-learning for machine learning, statistical estimation theory, and more. Within this framework, we establish several fundamental direct sum theorems. The main contributions of this paper include: (i) establishing a complete characterization of the amortized query/oracle complexities, and (ii) proving tight direct sum theorems when the error is small. Note that in our framework, every oracle access needs to be performed \emph{classically} even in the quantum setting. This can be thought of one limitation of this work. As a direct consequence of our results, we obtain the following: (A) The first known asymptotic separation of the randomized query complexity. Specifically, we show that there is a function $f: \{0, 1\}^k \to \{0, 1\}$ and small error $\varepsilon > 0$ such that solving $n$ instances simultaneously requires the query complexity $\tilde{O}(n\sqrt{k})$ but solving one instance with the same error has the complexity $\tilde{\Omega}(k)$. In communication complexity this type of separation was previously given in~Feder, Kushilevitz, Naor and Nisan (1995). (B) The query complexity counterpart of the ``information = amortized communication" relation, one of the most influential results in communication complexity shown by Braverman and Rao (2011) and further investigated by Braverman (2015). We hope that our results will provide further interesting applications in the future.