Published: February 9, 2005

**Keywords:**Quantum Computation, advice, relativized complexity, direct product theorem, nonuniform, BQP, PP, communication, polynomial method, oracle

**ACM Classification:**F.1.2,F.1.3

**AMS Classification:**81P68, 81P05, 68Q10, 68Q15, 42A05

**Abstract:**
[Plain Text Version]

Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum 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 quantum 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 quantum and classical advice. Underlying our complexity result is a general new relation between deterministic and quantum 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 quantum search. This theorem has other applications;
for example, it can be used to fix a result of Klauck about quantum
time-space tradeoffs for sorting.

Third, we introduce a new *trace distance method* for proving lower
bounds on quantum one-way communication complexity. Using this method, we
obtain optimal quantum 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).