BQP is not in NP

TL;DR

This paper proves that the class BQP, representing problems solvable by quantum computers in polynomial time, contains problems outside the classical complexity class NP, demonstrating quantum advantage over classical computing.

Contribution

It provides a rigorous proof that BQP includes problems not in NP, establishing a fundamental separation between quantum and classical computational complexity classes.

Findings

BQP contains problems outside NP.

Quantum computers can efficiently solve problems beyond classical capabilities.

Provides a formal proof of quantum advantage over classical computation.

Abstract

Quantum computers are widely believed have an advantage over classical computers, and some have even published some empirical evidence that this is the case. However, these publications do not include a rigorous proof of this advantage, which would have to minimally state that the class of problems decidable by a quantum computer in polynomial time, BQP, contains problems that are not in the class of problems decidable by a classical computer with similar time bounds, P. Here, I provide the proof of a stronger result that implies this result: BQP contains problems that lie beyond the much larger classical computing class NP. This proves that quantum computation is able to efficiently solve problems which are far beyond the capabilities of classical computers.