Limitations of quantum advice and one-way communication

Abstract: Although a quantum state requires exponentially many classical bits to de-scribe, 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 BQP/qpoly⊆PP/poly, where 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. Un-derlying 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 NP 6 ⊂ 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 quan-tum one-way communication complexity. Using this method, we obtain optimal quantu