Limitations of quantum advice and one-way communication

Abstract

Although a requires exponentially many classical bits to describe, the laws of mechanics impose severe restrictions on how that can be accessed. This paper shows in three settings that messages have only limited advantages over classical ones. First, we show that $\mathsf{BQP/qpoly}\subseteq\mathsf{PP/poly}$, where $\mathsf{BQP/qpoly}$ is the class of problems solvable in polynomial time, given a polynomial-size quantum advice state that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between and classical advice. Underlying our complexity result is a general new relation between deterministic and one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which $\mathsf{NP}\not \subset \mathsf{BQP/qpoly}$. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for search. This theorem has other applications; for example, it can be used to fix a result of Klauck about time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on one-way communication complexity. Using this method, we obtain optimal lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols. A preliminary version of this paper appeared in the 2004 Conference on Computational Complexity (CCC).