Quantum NP and a 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 Adle-man, 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 3-operator applied to P. Recently, Green, Homer, Moore, and Pol-lett proposed its quantum version, called the N-operator, which is an abstraction of NQP. This paper introduces the 3Q-operator, which is an abstraction of QMA, and its complement, the VQ-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.