Quantum NP and Quantum Hierarchy

The complexity class NP is quintessential and ubiquitous in theoretical computer science. Two different approaches have been made to define “Quantum NP,” the quantum analogue of NP: NQP by Adleman, DeMarrais, and Huang, and QMA by Knill, Kitaev, and Watrous. From an operator point of view, NP can be viewed as the result of the ∃-operator applied to P. Recently, Green, Homer, Moore, and Pollett proposed its quantum version, called the N-operator, which is an abstraction of NQP. This paper introduces the ∃-operator, which is an abstraction of QMA, and its complement, the ∀-operator. These operators not only define Quantum NP but also build a quantum hierarchy, similar to the Meyer-Stockmeyer polynomial hierarchy, based on two-sided bounded-error quantum computation.