Lesser qubit computer doing the parts of Shor's against e.g., RSA-2048 sized prime

Question

After posting this question to Physics, it became pretty clear I should have posted here. So:

How might a (e.g.) 72-bit crypto-relevant quantum computer attack RSA-2048?

Bonus: how might that be characterized? (e.g., nn-qubit requires xxx passses, run time ~yyy)

Shor's algorithm appears to allow for parallel execution or iterative runs with a combination step. Assumption is that smaller-qubit QC might be able to perform those pieces.

However, it is suggested that a 4000-qubit/100m-gate quantum computer would be necessary. As the quantum piece of Shor's is a large transform, I assume that sets the constraint for qubit-size

Side note: there also appear to be possible speedups that may reduce the run time, such as qubit recycling? or the 4-8 passes vs. the 20-30 passes (by David McAnally)

Answer 1

Even with qubit recycling, 72 qubits will not be enough to do RSA-2048. Table 1 of the paper:

https://www.nature.com/articles/nature12290

Tells you that 1154 qubits are needed to do RSA-768 (which is much smaller than RSA-2048). This is without error correction.

Sure you can use your 72-qubit quantum computer to do a little sub-routine of Shor's algorithm, but this will not help if you have to do the rest of the algorithm on a classical computer. For any benefit, the quantum computer has to be doing the "rate-limiting step"