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(This is a somewhat soft question.)

The quantum Fourier transform is formally quite similar to the fast Fourier transform, but exponentially faster.

The QFT is famously at the core of Shor's algorithm for period finding. It also comes up in a few other places, like the HHL algorithm for solving (certain very special) linear systems of equations.

The FFT, on the other hand is used in *countless* different applications throughout applied math, science, engineering, finance, and music (notably for signal processing and solving differential equations). Gilbert Strang called it "the most important numerical algorithm of our lifetime."

Given that the QFT is exponentially faster than the FFT, there seems to me to be a strange discrepancy between the literally thousands of known applications of the FFT and the relatively few applications of the QFT (even at the theoretical level, setting aside the obvious practical implementation challenges). I would have expected that, given the exponential speedup that the QFT delivers over the FFT, a quantum computer capable of implementing the QFT would instantly render many classical computing applications obsolete, and would also open up many more potential applications that are currently impractical. And yet the only application of the QFT that people seem to really discuss is using Shor's algorithm for decryption. (I don't mean "application" in the academic's sense of "something that will get me a paper published", but in the business person's sense of "something that there might be a commercial market for".) It's not even clear if the HHL algorithm would actually deliver a useful speedup in practice.

Is there some conceptual explanation for why the QFT doesn't seem to be as big of a deal in practice as one might expect? Is it just the usual I/O challenge of (a) efficiently reading a large data set in memory and (b) only being able to statistically sample the output amplitudes over many runs instead of being able to read them out all at once?

+1 and nice username! I hope you're a vim user :) – user1271772 – 2021-01-25T23:35:51.007