[amluto]

I don’t really expect fancier codes to cause a huge jump in the number of logical qubits. At the end of the day, there’s some code rate (logical bits / physical bits) that makes a quantum computer work. The “FOOM” is the transition from that code rate changing from zero (lifetime of a logical bit is short) to something that is distinctly different from zero (the state lasts long enough to be useful when some credible code). Say the code rate is 0.001 when this happens. (I haven’t been in the field for a little while, but I’d expect higher because those huuuuge codes have huuuuge syndromes, which isn’t so fun. But if true topological QC ever works, it will be a different story.) The code rate is unlikely to ever be higher than 1/7 or so, and it will definitely not exceed 1. So there’s at most a factor of 1000, and probably less, to be gained by improving the code rate. This isn’t an exponential or super-exponential FOOM.

A factor of 1000 may well be the difference between destroying Shor’s-algorithm-prone cryptography and destroying it later, though.

[amluto]

I'll add some nuance here. In a classical computer, computing with coded bits is not particularly difficult. We've known how to do it with some degree of mathematical rigor for decades -- IIRC John von Neumann was interested in this topic. And we barely even need to do it explicitly: computers have accurate enough gates without explicit coding that merely error-correcting RAM (ECC-style) and data links (Ethernet, PCIe, etc) is good enough for almost all applications. Even in aerospace, usually we just have one extra majority vote over a few different computers.

Quantum computers are different. It seems quite unlikely that anyone build the equivalent of, say, an XOR gate that takes two single physical qubits in and spits out two physical qubits (this is quantum land -- the standard gates neither create nor destroy qubits, so the number of inputs and outputs is the same) that works well enough to actually represent that particular operation in whatever software is being run. Instead each logical operation will turn into multiple physical operations that work like an error correcting code. The easy classical trick where your transistor is janky at 0.9V so you run it at 1.0V amounts to moving more electrons around per operation, and this approach is analogous to correcting bit flips but not phase errors, and it makes your quantum computer stop being quantum.

And here's where it gets messy. The physical qubit technologies that are best for longish-term data storage may not be the same as the technologies that are good for computation, and those may not be the same technologies that are good for communication at a distance. (For example, photons are pretty good for transmitting quantum states, but transferring a qubit from a different technology to a photon state and back is not so easy, and demonstration of computation with photons have been pretty limited.) As an extreme example, one can, in principle, store quantum states quite robustly and even compute with them if one can find the correct kind of unobtanium (materials with the appropriate type of non-Abelian anyon surface states), but, last I heard, no one had much of an idea how to get the qubit states off the chip even if such a chip existed.

So it is possible that we'll end up with a quantum computer that doesn't scale, at least for a while. There might be 20k physical qubits, and some code rate, and some number of logical quantum operations you can do on the logical qubits before they decay or you get bored, and very little ability to scale to more than one computer that can split up a computation between them. In that case, the code rate is a big deal.