Is there any NP-hard problem which was proven to be solved in polynomial time or at least close to polynomial time?

I know this could be a strange question. But was there any algorithm ever found to compute an NP-problem, whether it be hard or complete, in polynomial time. I know this dabbles into the "does P=NP" problem, but I was wondering whether there were any particular instances when people were close to it. I've also heard about the factoring number problem being solved by a quantum computer by Shor's algorithm. Would this count as an algorithm which solved an NP-problem in polynomial time please ? Please do ask if you want me to clarify any parts in my questions. Thanks in advance.

By definition, if you were to find a polynomial time algorithm for an NP-hard (or NP-complete) problem, then $P=NP$. So, short answer is - no.

However, its possible to think instead of solving the problems fully, to approximate a solution, or to solve them randomly. There are attempts at attacking from those points of view, but they are not perfect at all. Randomization isn't known to help "as-is", and many NP-hard problems are known that are also hard to approximate (finding an approximation will yield $P=NP$)

There are also attemps taking a look at notions much stronger than the usual turing machine. For instance, the quantum computer can be considered one. Another obvious example is the non-deterministic turing machines, in which - by definition - $NP$ would be in polynomial time.

StackExchange Computer Science Q#139829, answer score: 10