No. As far as we know, no realization of a quantum algorithm can solve NP-complete problems in polynomial many steps.
Some people that worked on this topic told me that there seems to be some improvements on the quasi-optimal solution found, but that due to the scale of current quantum computers, it just have been tried out on small-sized problems.
Theoretically, there are some papers suggesting that there are problems in BQP (the computational model of quantum computers) outside of NP [1] and even, the PH [2] (Polynomial Hierarchy, the infinite hierarchy of composition of NP and co-NP problems), which is why we cannot still satisfactorily say whether quantum computers can or cannot solve NP-complete problems.
The Wikipedia page for BQP [3] does a good job showing what is currently known.
[1] https://arxiv.org/abs/2209.10398 [2] https://eccc.weizmann.ac.il/report/2018/107/ [3] https://en.wikipedia.org/wiki/BQP#Relationship_to_other_comp...
I would also observe that it is a completely reasonable hypothesis that even if you can construct a problem in some higher complexity class that a quantum computer "should" be able to solve, it is possible that it would turn out that it couldn't because the complexity barrier turns out to be even more fundamental than quantum mechanics. If I understand it correctly, Scott Aaronson at least takes this possibility seriously, and points out that it is at least a possibility that this could end up being the legacy of quantum computing, along with the possibility that it could demonstrate some sort of currently-unknown fundamental limit on entanglement. And I'd add my own commentary that the experiment that shows QM breaks down and refuses to solve some problem that it "should" be able to solve would be on the instant shortlist for Nobel prizes for the experimenters, pending only extended confirmation, because it would be a big, big deal.
As to the PH result, arguments on relativized classes can be pretty inconclusive. There's both oracles for P^A = NP^A and P^B != NP^B.