Separations in query complexity using cheat sheets

TL;DR

This paper introduces the cheat sheet technique, a new method for demonstrating significant separations in query complexity, challenging previous conjectures and revealing limitations of existing methods.

Contribution

The paper presents a novel cheat sheet technique that unifies and extends methods for proving large separations in query complexity for total Boolean functions.

Findings

Demonstrates a 2.5 power separation between randomized and quantum query complexity.

Shows a 4 power separation between quantum query complexity and polynomial degree.

Establishes a quadratic gap between quantum query complexity and certificate complexity.

Abstract

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al., to be shown in a unified manner.