All Quantum Adversary Methods are Equivalent

TL;DR

This paper proves that all known variants of the quantum adversary method are fundamentally equivalent, unifying the approach and clarifying its limitations in quantum lower bound proofs.

Contribution

It demonstrates the equivalence of multiple quantum adversary methods and introduces new formulations, showing they are essentially the same underlying technique.

Findings

All variants of the quantum adversary method are equivalent.

The paper provides new formulations of the adversary method.

Limitations of these methods are derived from their equivalence.

Abstract

The quantum adversary method is one of the most versatile lower-bound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), and the Kolmogorov complexity adversary (Laplante and Magniez, 2004). We also pa few new equivalent formulations of the method. This shows that there is essentially _one_ quantum adversary method. From our approach, all known limitations of these versions of the quantum adversary method easily follow.