So the "useful quantum computing" that is "imminent" is not the kind of quantum computing that involves the factorization of nearly prime numbers?
Either this relation is not that strong, or factoring should "imminently" become a reasonable benchmark, or useful quantum computing cannot be "imminent". So which one is it?
I think you are the author of the blogpost I linked to? Did I maybe interpret it too negatively, and was it not meant to suggest that the second option is still quite some time away?
the other problem is that factoring 21 is so easy that it actually makes it harder to prove you've factored it with a functional quantum computer. for big numbers, your program can fail 99% of the time because if you get the result once, you prove that the algorithm worked. 21 is small enough that it's hard not to factor, so demonstrating that you've factored it with a qc is fairly hard. I wouldn't be surprised as a result if the first number publicly factored by a quantum computer (using error correction) was in the thousands instead of 21. By using a number that is not absolutely tiny, it becomes a lot easier to show that the system works.
Google have been working on this for years
Don't ask me if they've the top supercomputers beat, ask Gemini :)
Like if you were building one of the first normal computers, how big numbers you can multiply would be a terrible benchmark since once you have figured out how to multiply small numbers its fairly trivial to multiply big numbers. The challenge is making the computer multiply numbers at all.
This isn't a perfect metaphor as scaling is harder in a quantum setting, but we are mostly at the stage where we are trying to get the things to work at all. Once we reach the stage where we can factor small numbers reliably, the amount of time to go from smaller numbers to bigger numbers will be probably be relatively short.
In QC systems, the engineering "difficulty" scales very badly with the number of gates or steps of the algorithm.
Its not like addition where you can repeat a process in parallel and bam-ALU. From what I understand as a layperson, the size of the inputs is absolutely part of the scaling.
So it seems like it takes an exponentially bigger device to factor 21 than 15, then 35 than 21, and so on, but if I understand right, at some point this levels out and it's only relatively speaking a little harder to factor say 10^30 than 10^29.
Why are we so confident this is true given all of the experience so far trying to scale up from factoring 15 to factoring 21?
I don't think we have any "real" experience scaling from 15 to 21. Or at least not in the way shor's algorithm would be implemented in practise on fault tolerant qubits.
We haven't even done 15 yet in a "real" way yet. I susect the amount of time to factor 15 on fault tolerant qubits will be a lot longer than the time to go from 15 to 21.
In the case of quantum algorithms in BQP, though, one of those properties is SNR of analog calculations (which is assumed to be infinite). SNR, as a general principle, is known to scale really poorly.
As far as i understand, that isn't an assumption.
The assumption is that the SNR of logical (error-corrected) qubits is near infinite, and that such logical qubits can be constructed from noisey physical qubits.
This is an argument I've heard before and I don't really understand it[1]. I get that you can make a logical qubit out of physical qubits and build in error correction so the logical qubit has perfect SNR, but surely if (say the number of physical qubits you need to get the nth logical qubit is O(n^2) for example, then the SNR (of the whole system) isn't near infinite it's really bad.
[1] Which may well be because I don't understand quantum mechanics ...
The hard problem then remains how to connect those qubits at scale. Using a coaxial cable for each qubit is impractical; some form of multiplexing is needed. This, in turn, causes qubits to decohere while waiting for their control signal.
In comparison, there is no realistic path forward for scaling quantum computers. Anyone serious that is not trying to sell you QC will tell you that quantum systems become exponentially less stable the bigger they are and the longer they live. That is a fundamental physical truth. And since they're still struggling to do anything at all with a quantum computer, don't get your hopes up too much.
If what you are saying is that error rates increase exponentially such that quantum error correction can never correct more errors than it introduces, i don't think that is a widely accepted position in the field.