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

Abstract

We study the complexity of quantum complexity classes like EQP, BQP, NQP (quantum analogs of P, BPP, and NP, respectively) using classical complexity classes like ZPP, WPP, C=P. The contributions of this paper are threefold. First, we show that relative to an oracle, ZPP is not contained in WPP. As an immediate consequence, this implies that no relativizable proof technique 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 constructions 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 closed under 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 LWPP and WPP would require nonrelativizable techniques. We also prove that both AWPP and APP are closed under \({\leq}^{UP}_{T}\) reductions, and use these closure properties to prove strong consequences of the following hypotheses: NQP ⊆ BQP and EQP = NQP.