Can quantum computers be used to solve P = NP

I see maybe four (4) ways to interpret the question.

1. The first asks whether we can use a quantum computer to efficiently solve $mathsf{NP}$ problems. The class of problems efficiently solvable by a quantum computer is called $mathsf{BQP}$, and so the question would ask whether $mathsf{NP}subseteqmathsf{BQP}$. As is indicted in the comments, quantum computers are not known to efficiently solve arbitrary $mathsf{NP}$ problems, and are suspected not to be able to.

2. The second asks whether a proof that $mathsf{NP}subseteqmathsf{BQP}$ would suffice as a proof that $mathsf{P}=mathsf{NP}$. Although $mathsf{P}subseteqmathsf{BQP}$, such a containment does not necessarily follow. For example, there may be problems in $mathsf{BQP}$ that are not in $mathsf{P}$ or even in $mathsf{NP}$ - a famous candidate being related to knot theory. I don't think you'd win the Clay $1M for such a containment. (You might get other riches, but that's a different story).

3. The third asks whether a proof that $mathsf{NP}notsubseteqmathsf{BQP}$ would suffice as a proof that $mathsf{P}nemathsf{NP}$. However, even this does not follow.

4. The fourth interpretation asks whether a quantum computer could provide any leverage in answering the question of if the classical complexity class $mathsf{P}$ is equal to $mathsf{NP}$. That is, could one pose the $mathsf{P}$ vs. $mathsf{NP}$ problem as a a problem to be fed through a quantum computer? Although this may be interesting, I doubt one could envision a way to ask such a finite question.

The "standard model" that many people believe is that $mathsf{P}subsetneqmathsf{NP}$, and further both $mathsf{BQP}notsubseteqmathsf{NP}$ and $mathsf{NP}notsubseteqmathsf{BQP}$ - that is, $mathsf{NP}$ and $mathsf{BQP}$ are incomparable. However, we are likely far away from proving any.