No relativizing reduction from BQP to NP

Unconditional · against relativizing reductions · RT19

Statement

No relativizing reduction from BQP to NP exists: there is an oracle, built from a variant of the Forrelation problem, relative to which $\mathbf{BQP}⊈\mathbf{PH}$, and hence $\mathbf{BQP}⊈\mathbf{NP}$ — RT19.

Notes

• Unrelativized, neither $\mathbf{BQP}\subseteq \mathbf{NP}$ nor $\mathbf{NP}\subseteq \mathbf{BQP}$ is known — standard.