Quantum NP - A Survey

TL;DR

This survey reviews Kitaev's 1999 foundational work on the quantum complexity class QMA, its relation to classical NP, and discusses open problems in quantum complexity, algorithms, and entanglement.

Contribution

It provides a comprehensive overview of QMA, its relation to NP, and highlights fundamental differences and open problems in quantum complexity theory.

Findings

QMA is the quantum analog of NP.

Local Hamiltonians are QMA-complete.

The survey discusses open problems in quantum complexity.

Abstract

We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3-SAT, namely local Hamiltonians, is QMA complete. The result builds upon the classical Cook-Levin proof of the NP completeness of SAT, but differs from it in several fundamental ways, which we highlight. This result raises a rich array of open problems related to quantum complexity, algorithms and entanglement, which we state at the end of this survey. This survey is the extension of lecture notes taken by Naveh for Aharonov's quantum computation course, held in Tel Aviv University, 2001.