A Simple Proof of Vyalyi's Theorem and some Generalizations

摘要

In quantum computational complexity theory, the class QMA models the set of problems efficiently verifiable by a quantum computer the same way that NP models this for classical computation. Vyalyi proved that if QMA = PP then PH QMA . In this note, we give a simple, self-contained proof of the theorem, using only the closure properties of the complexity classes in the theorem statement. We then extend the theorem in two directions: (i) we strengthen the consequent, proving that if QMA = PP then QMA = PH PP , and (ii) we weaken the hypothesis, proving that if QMA = coQMA then PH QMA . Lastly, we show that all the above results hold, without loss of generality, for the class QAM instead of QMA. We also formulate a ``Quantum Toda's Conjecture''.