No reduction from BQP to NP

A reduction of class unstated from BQP to NP would imply a contradiction.

Statement

Migrated verbatim from bounded-error-quantum-polynomial-time § Known relationships:

• $\mathbf{BQP}$ vs $\mathbf{NP}$: the two classes are believed incomparable. Simon’s problem is in $\mathbf{BQP}$ but not in $\mathbf{NP}$ relative to a random oracle; conversely, NP-complete problems are not believed to be in $\mathbf{BQP}$.

Notes

source: folklore: the claim carried no citation on the page it was migrated from, and none was invented.

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• TYPING LOSS: as a bare {BQP} ⇒ {NP} pair this is indistinguishable from an inclusion claim, which is the opposite of what the sentence asserts.
• Uncited (Simon94 / BBBV93 would be the standard sources).
• SUSPECTED IMPRECISION (recorded, not fixed): Simon’s problem is defined relative to a random 2-to-1 function with a hidden XOR mask, not a uniformly random oracle; the random-oracle statement usually cited in this area is BBBV, which separates in the OTHER direction.
• No citation for the oracle separation (BBBV93 / Simon94 would be the standard ones).
• SUSPECTED IMPRECISION (report only): Simon’s problem is defined relative to a random 2-to-1 function with a hidden XOR mask, not a uniformly random oracle; ‘relative to a random oracle’ overstates what the Simon separation gives. The random-oracle statement usually cited in this direction is BBBV (NP not in BQP relative to a random oracle), which is the OTHER direction.
• Two belief statements plus one oracle-separation claim in one bullet; beliefs are not theorems and should not migrate as reductions.