How much information flows through a system? This fundamental question is at the heart of communication complexity. Techniques from this field have turned out to be immensely powerful and fairly universal tools to understand the power of different kinds of algorithms.

In this talk, I will describe new methods that I have developed to analyze communication which offer fresh insights into quantum computing and optimization. Using these techniques, I will answer a question of Aaronson and Ambainis regarding the maximal advantage that quantum algorithms can have over classical algorithms in the “blackbox” model, and another conjecture due to Rothvoss about optimal linear programs for approximately solving the matching problem.

Looking forward, I also propose new directions to explore further connections among these areas with the intention of answering key questions regarding quantum speedups and more powerful optimization approaches, such as semidefinite programming.