# Are there any common applications where one can replace FFT with Quantum Fourier Transform?

I want to apply QFT to some common applications like on wave equations. I haven't found any applications of QFT except Shor's algorithm and I am yet to build an intuition for its use cases. I am a beginner in QC. It would be great if someone can help!

## 1 Answer

One example where the QFT comes up is in simulating quantum particles' position and momenta. A typical use of the discrete Fourier Transform is classical applications is to convert time-domain signals into frequency-domain spectra, and an analogous relationship between the position and momentum domains allows you to convert a wavefunction that represents the (probability) distribution of a particle in space into a wavefunction that represents that same particle's momentum spectrum.

To study the dynamics of particles using a scheme like this, you have to specify a way to discretize position space so that you'll be able to represent a continuum of position/momentum values on a system with finite-dimensional wavefunctions (this involves defining the minimum/maximum position you will represent, the grid spacing in position/momentum space, and so on). An example of such a discretization or "digitization" formalism is given in , or for a more thorough review see .

 Ronaldo D Somma. "Quantum simulations of one dimensional quantum systems". arXiv preprint: https://arxiv.org/abs/1503.06319

 Natalie Klco and Martin J. Savage. “Digitization of Scalar Fields for Quantum Computing.” Physical Review A 99.5 (2019): https://arxiv.org/abs/1808.10378