Whatever you're encoding into the light that gets shined on the prism... that process is probably not gonna be O(1).

It's not, no. Whats your point? The factorization step is still O(1).

You can't factorize a number in O(1) if it takes O(n) or whatever to set up the problem.

Everything is O(1) if you can ignore steps.

There's a limited amount of information in the signal to a photonic machine of a given size though. If the inputs are bounded in some sense then a digital computer can calculate Fourier in O(1) too. That doesn't make the problem any easier asymptotically -- you still need more machine or more time to calculate arbitrary Fourier.

That said, recent work on photonic Fourier transforms does look interesting from the perspective of a co-processor. They offer meaningful gains over alternatives.

How can you calculate Fourier in O(1) if the input is bound?

If the input maxes out at 64 bits, then calculating the Fourier transform is O(64), which is O(1).

I don't follow.

O(whatever) refers to the asymptotic behavior as n gets arbitrarily large. If n is bounded, big O notation doesn’t really mean anything because everything is O(1).

I understand big O however Fourier bio O runtime is bound by the length of the signal, not by maximum value of the signal.

The photonic processor in question can only process some maximum M number of bits worth of signal in a given time T though. To have a throughput of better than M/T you need more time or more/better processors (i.e., for large enough inputs the runtime isn't actually O(1) ).

What I was pointing out is that digital implementations have this exact same property -- if the input is bounded in the sense that there are a maximum number of bits being fed into your digital computer then the maximum runtime is a constant equal to the maximum runtime for any of those finite inputs -- i.e., the runtime is O(1) for the inputs you're actually able to provide (the exact same inputs you could provide to the photonic processor to achieve a constant-time Fourier transform). To scale up beyond that threshold you still need more time or more/better processors.

Are we talking about analog computers?

Yes.

Photons are discrete.

What's your point?

You cannot do continuous computation with discrete components.