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> Its suspected P lies entirely within BQP.

It's actually known that BQP contains P[1]. It also contains BPP. What's not known is the relationship between BQP and NP (most experts suspect there's no containment in either direction).

[1] See this for an easy proof: https://people.eecs.berkeley.edu/~vazirani/f04quantum/notes/...

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Yes sorry you are right. I found this nice example of Grover's algorithm [0].

[0] http://davidbkemp.github.io/animated-qubits/grover.html

https://www.quantamagazine.org/finally-a-problem-that-only-q...

That's still only relative to an oracle, though (i.e., BQP^O vs NP^O). We also have oracle separations of P and NP, and that proves nothing about P vs NP without an oracle.

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