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Quantum and Classical Complexity Classes:Separations, Collapses, and Closure Properties



ZPP, WPP, C=P. The contributions of this paperare threefold. First, we show that relative to an oracle, ZPP is not containedin WPP. As an immediate consequence, this implies that no relativizable prooftechnique can improve the best known classical upper bound for BQP (BQP ` AWPP [16]) to BQP ` WPP and the best known classical lower bound for EQP (P ` EQP) to ZPP ` EQP. Second, we extend some known oracle con-structions involving counting and quantum complexity classes to immunity separations. Third, motivated by the fact that counting classes (like LWPP, AWPP,etc.) are the best known classical upper bounds on quantum complexity classes, we study properties of these counting classes. We prove that WPP is closedunder polynomial-time truth-table reductions, while we construct an oracle relative to which WPP is not closed under polynomial-time Turing reductions.This shows that proving the equality of the similar appearing classes LWPPand WPP would require nonrelativizable techniques. We also prove that both AWPP and APP are closed under ^UPT reductions, and use these closure prop-erties to prove strong consequences of the following hypotheses: NQP ` BQPand EQP = NQP. 1 Introduction Quantum complexity classes like EQP, BQP [4] (quantum analogs, respectively, of Pand BPP [17]), and NQP [1] (the quantum analog of NP) are defined using quantumTuring machines, the quantum analog of classical Turing machines

Year: 2009
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