> Quantum supremacy is debated, and so not something I can reasonably comment on. From the explanation above however, you can see that supremacy is achieved whenever you do something with a QC that cannot be done with a classical computer. The problem is proving that something is 100% provably beyond the scope of a classical computer (people are clever, and so it's often not).

My understanding was that quantum supremacy occurs when a quantum computer performs a computation that cannot be performed with comparable speed on a classical computer given the best available classical hardware and the best available classical algorithms of the time. In particular, it does not require a proof that no efficient classical algorithm exists. Instead, it is sufficient that nobody knows such a classical algorithm at the time of demonstration.

Otherwise even factoring a large number by running Shor's algorithm on an error corrected quantum computer would still not constitute quantum supremacy demonstration because there is no proof that factoring cannot be done efficiently on a classical computer. After all, it is not inconceivable that people just don't know how to factor integers fast on a classical computer.