Concentration Bounds from Parallel Repetition Theorems

Abstract

This thesis contributes to the study of parallel repetition theorems and concentration bounds for nonlocal games and quantum interactive proofs. We make the following contributions:

- A lemma that is useful for converting parallel repetition theorems (bounds on the probability of winning all instances of a nonlocal game which is being repeated in parallel) into concentration bounds (bounds on winning a certain fraction of the instances).

- Exponentially-decaying concentration bounds for two-player games on the uniform distribution and k-player free games, against quantum strategies.

- A proof that given a quantum interactive proof system with parameters α (the probability with which the verifier can be convinced to accept when they should accept) and β (the soundness error), as long as α > β, both the soundness error and completeness error can be reduced exponentially by repeating the protocol in parallel and requiring an (α + β)/2 fraction of the repetitions to be won. Our result requires a log-factor more repetitions than are necessary in the classical case.