> With an error rate of 1% the surface error correction code requires ~ 500 physical qubits required to encode one logical qubit.
This was the correction near the end:
> With an error rate of 0.3% the surface error correction code requires ~ 10 thousand physical qubits to encode one logical qubit to achieve 10^-10 logical qubit error rate.
The qubit count at an error rate of 1% was clearly off because the threshold of the surface code under circuit noise is a bit below 1%. Meaning at 1% it would have infinite cost; way more than 500. To get good numbers you need be well below the threshold. At a 0.1% error rate, assuming a square grid of qubits with local connections, the best physical-per-logical estimate that I'm aware of is 600, from surface codes plus a few extra parity checks layered on top [1][2]. Another code that achieves a teraquop footprint of ~600 on a planar grid is the honeycomb code [3][4] but that number requires a dissipative two qubit gate which seems to be harder to build than the usual unitary ones.
[1]: https://www.youtube.com/watch?v=Ge7fEaXjvq4
[2]: https://arxiv.org/pdf/2312.04522
I had a brief run-in with error-correcting gates. My research topic was about a simple bit-flip operation (quantum dot qubits) under random "telegraph" noise -- different laser pulse shapes result in different error rates which are actually a function of time (on the scale of the pulse itself). Point being that even given a particular physical qubit framework, individual gates are actually a family of (physical) operations with different error rates (and other features, of course) to be optimized for the same logical operation. Not too unlike the idea of NAND vs NOR flash.
Personally, I feel that's slightly "good" in the sense that there are many paths we can take to get quantum computation "good enough".
