Explicit relation between all lower bound techniques for quantum query complexity

TL;DR

This paper establishes a clear, explicit connection between the polynomial and adversary methods for quantum query complexity, showing they are fundamentally related and that all known techniques reduce to the multiplicative adversary method.

Contribution

It provides the first explicit reduction from the polynomial method to the multiplicative adversary method, extending the polynomial method to quantum state generation and unifying lower bound techniques.

Findings

Polynomial method extended to quantum state generation.

All known lower bound techniques reduce to the multiplicative adversary method.

The adversary method can be stronger than the polynomial method in some cases.

Abstract

The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.