The General Adversary Bound: A Survey

TL;DR

This survey reviews the general adversary bound's role in characterizing quantum query complexity, covering foundational concepts, proofs, and applications to formula size lower bounds.

Contribution

It consolidates background, definitions, and examples to clarify the graph-theoretic intuition behind the general adversary bound and its applications.

Findings

The general adversary bound characterizes quantum query complexity.

Provides detailed explanations of span programs and semi-definite programs.

Applies the bound to derive lower bounds on DeMorgan formula size.

Abstract

Ben Reichardt showed in a series of results that the general adversary bound of a function characterizes its quantum query complexity. This survey seeks to aggregate the background and definitions necessary to understand the proof. Notable among these are the lower bound proof, span programs, witness size, and semi-definite programs. These definitions, in addition to examples and detailed expositions, serve to give the reader a better intuition of the graph-theoretic nature of the upper bound. We also include an applications of this result to lower bounds on DeMorgan formula size.