A Generalized Adversary Method for Quantum Query Complexity

Abstract

Quantum query complexity measures the minimum number of queries a quantum algorithm needs to make to some input string to compute a function of that input. Query complexity models are widely used throughout quantum computing, from setting limits on quantum algorithms to analyzing post-quantum cryptography.

This thesis studies quantum adversary methods, a group of mathematical tools that prove lower bounds on quantum query complexity. I introduce a new general-purpose framework for adversary methods that generalizes over both the negative weight and multiplicative adversary methods. This framework unifies the lower bound proofs of both methods, even in the general case of quantum state conversion.

This generalized method also gives a new formula for the multiplicative adversary method based on max-relative entropy. This new definition is more concise and easier to reason about than existing definitions in the literature. I verify this by reproving several known results about the multiplicative adversary method. I also use this to reprove the strong direct product theorem for quantum query complexity.