# Matrix permanent, #P-hard problems and NP

First of all, I'm not a computer scientist so I apologize if this is a stupid question.

I know that the problem of computing the permanent of a matrix is #P-hard, which as I understand it this implies that if you can solve an arbitrary instance of the problem in polynomial time then you can in principle solve any problem in #P in polynomial time.

What I'm wondering is if it also implies being able to solve NP problems in polynomial time (I'm guessing not).

• Thank you. That's very interesting, since it is possible to engineer quantum mechanical systems the measurement probabilities of which are described by the permanent of the system's matrix. That's why I doubted it, it seemed like it would make it theoretically possible to efficiently solve NP-Complete problems on a quantum computer. I suppose the big challenge there is doing it in the general case and not just for once specific instance. Is your statement related to the fact that $NP \subseteq P^{\#P}$? Feb 15, 2016 at 7:42
• Shouldn't #2-sum be #P-hard if $P=NP$? Dec 9, 2019 at 22:47