Quantum Arthur-Merlin games

This paper studies quantum Arthur-Merlin games, which are a restricted form of quantum interactive proof system in which the verifier´s messages are given by unbiased coin-flips. The following results are proved. For one-message quantum Arthur-Merlin games, which correspond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin´s message. Previous constructions for reducing error required a polynomial increase in the length of Merlin´s message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in power over BQP and a simple proof that QMA ⊆ PP. In the case of three or more messages, quantum Arthur-Merlin games are equivalent in power to ordinary quantum interactive proof systems. In fact, for any language having a quantum interactive proof system there exists a three-message quantum Arthur-Merlin game in which Arthur´s only message consists of just a single coin-flip that achieves perfect completeness and soundness error exponentially close to 1/2. Any language having a two-message quantum Arthur-Merlin game is contained in BP · PP. This gives some suggestion that three messages are stronger than two in the quantum Arthur-Merlin setting.