> I thought the error rate has stayed pretty exponential in terms of the number of physical qubits needed to express a logical qubit
There is finite a threshold error rate (roughly 0.1-1%) at which you can produce a single logical qubit with an unbounded number of physical qubits (infinite overhead). For error rates below the threshold, the overhead becomes much less. People expect overheads in the thousands. See Fig. 1 in our paper. https://arxiv.org/pdf/2009.05045
> and it’s not actually known if we’re any closer on that metric vs other more easily achieved metrics.
We are getting lower error rates. But until we cross the error threshold, the overhead for a logical qubit is infinity.
> I was under the impression that not all QCs being built are able of executing Shor’s algorithm
Correct.
> which added additional challenges that aren’t solved.
Logical qubits enable general purpose quantum computing, which includes Shor's algorithm. As mentioned, we don't have logical qubits yet, and some people are trying to build less general devices to solve certain math problems in the meantime. But the overall goal for the field is still logical qubits, and there's steady progress on that.
> My final impression is that having the QC algorithm run faster than a classical computer doing the same operation has also not necessarily gotten better
I can't really parse the claim, but I think your impression is wrong. Supremacy has always been a fuzzy bound, since it's defined in terms of the best known classical algorithms. But the supremacy results have gotten more unambiguous over time.