The 7 faces of quantum NP

TL;DR

This paper surveys the diverse definitions of quantum NP, highlighting their differences, strengths, weaknesses, and how they fit into the broader quantum complexity landscape.

Contribution

It provides a comprehensive overview of the various quantum NP classes and analyzes their relationships and significance within quantum complexity theory.

Findings

Quantum NP has multiple definitions with distinct properties.

Most quantum NP classes fit into the existing complexity zoo.

The survey clarifies the landscape of quantum NP for theoretical computer scientists.

Abstract

When it comes to NP, its natural definition, its wide applicability across scientific disciplines, and its timeless relevance, the writing is on the wall: There can be only one. Quantum NP, on the other hand, is clearly the apple that fell far from the tree of NP. Two decades since the first definitions of quantum NP started rolling in, quantum complexity theorists face a stark reality: There's QMA, QCMA, QMA1, QMA(2), StoqMA, and NQP. In this article aimed at a general theoretical computer science audience, I survey these various definitions of quantum NP, their strengths and weaknesses, and why most of them, for better or worse, actually appear to fit naturally into the complexity zoo.