On Query-to-Communication Lifting for Adversary Bounds

TL;DR

This paper explores query-to-communication lifting theorems related to quantum and classical adversary bounds, establishing new lower bounds and connections between different complexity measures, with implications for quantum communication and query complexity.

Contribution

It introduces strong lifting theorems for classical and quantum adversary bounds, connecting them to secure quantum computation and establishing quadratic relations in query complexity.

Findings

Classical adversary bound lifts to randomized communication lower bounds with a constant-sized gadget.

Quantum adversary bounds relate to secure 2-party quantum computation impossibility.

Positive-weight quantum adversary is quadratically related to classical adversary and approximate degree.

Abstract

We investigate query-to-communication lifting theorems for models related to the quantum adversary bounds. Our results are as follows: 1. We show that the classical adversary bound lifts to a lower bound on randomized communication complexity with a constant-sized gadget. We also show that the classical adversary bound is a strictly stronger lower bound technique than the previously-lifted measure known as critical block sensitivity, making our lifting theorem one of the strongest lifting theorems for randomized communication complexity using a constant-sized gadget. 2. Turning to quantum models, we show a connection between lifting theorems for quantum adversary bounds and secure 2-party quantum computation in a certain "honest-but-curious" model. Under the assumption that such secure 2-party computation is impossible, we show that a simplified version of the positive-weight adversary bound lifts to a quantum communication lower bound using a constant-sized gadget. We also give an unconditional lifting theorem which lower bounds bounded-round quantum communication protocols. 3. Finally, we give some new results in query complexity. We show that the classical adversary and the positive-weight quantum adversary are quadratically related. We also show that the positive-weight quantum adversary is never larger than the square of the approximate degree. Both relations hold even for partial functions.