A Direct Reduction from the Polynomial to the Adversary Method

TL;DR

This paper establishes a straightforward reduction from the polynomial method to the adversary method, demonstrating that polynomial lower bounds can be directly translated into adversary bounds in quantum query complexity.

Contribution

It provides the first simple, direct reduction from the polynomial method to the adversary method, linking two major techniques for quantum lower bounds.

Findings

Any dual polynomial lower bound can be viewed as an adversary lower bound.

The reduction clarifies the relationship between polynomial and adversary methods.

This connection simplifies the process of deriving adversary bounds from polynomial bounds.

Abstract

The polynomial and the adversary methods are the two main tools for proving lower bounds on query complexity of quantum algorithms. Both methods have found a large number of applications, some problems more suitable for one method, some for the other. It is known though that the adversary method, in its general negative-weighted version, is tight for bounded-error quantum algorithms, whereas the polynomial method is not. By the tightness of the former, for any polynomial lower bound, there ought to exist a corresponding adversary lower bound. However, direct reduction was not known. In this paper, we give a simple and direct reduction from the polynomial method (in the form of a dual polynomial) to the adversary method. This shows that any lower bound in the form of a dual polynomial is actually an adversary lower bound of a specific form.