Kind of goes without saying when nobody has built a quantum computer of the type we are talking about. No general purpose error corrected quantum computer has been used to do anything because they don't exist yet.
I don't think that's common knowledge. It's commonly accepted truth in the industry, but particularly when most people think of military/spies as secretly X years ahead (pick a number) of what the public knows is possible, the tech sector in general can't be expected to know this. It's good to add this in a thread with a headline that sounds like anyone using ecc keys might have a big problem.
That particular demonstration is interesting, but it's not a general-purpose error-corrected quantum computer. It's a single-purpose quantum computer that simulates a quantum process with fewer gate operations than a classical computer needs to simulate the same process.
That's not true. There is such a thing as a completely general set of quantum gates, which combined with qubit memory, would make for a general quantum computer capable of computing any unitary transformation to a certain accuracy.
Not fully general, no. There is no such thing like a "Turning machine" for quantum computers. There are classes of algorithms like Shor's, Grover's, and quantum annealing which can be used to solve large many instances of related problems.
This is kinda like how matrix diagonalization can be used to solve any problem which is expressible as a linear system of equations, and to some degree any continuous function can be approximated by a linear system, so a BLAS + LAPACK accelerator is a "universal simulation engine."
You could probably build a generic Shor's algorithm quantum computer that is able to both factor integers and break elliptic curve keys. But the same quantum computer wouldn't be usable for Grover's algorithm to find the presage of a cryptographic hash. This is what I mean by there not being a "universal quantum computer" in the same way a Turing machine is a universal classical computer. Quantum computers by their very nature are ASIC implementations of specific algorithms, even though those algorithms might have some multi-domain applicability.
That really isn't true. If you have the CNOT gate, controlled rotation and phase shift, you can implement any operation on a set of qbits. If you then have quantum registers, you have a truly universal quantum computer, as you can then use registers to chain operation arbitrarily.
This computer would be able to do Shor, Grover, QAOA, as well as any classical algorithm of course. If you're interested, I can try to describe the proof of universality, it's just a bit of linear algebra. Otherwise, you can look up "Solovay-Kitaev theorem".