In this chapter, we learned about the Fourier transform, an algorithm that allows us to mathematically switch the representation of a periodic bounded function to represent it as a sum of sine waves of various frequencies, amplitudes, and phases. The inverse Fourier transform goes from the representation in terms of a sum of sine waves, back to the original function.
The Fourier transform has a quantum analogue, called the Quantum Fourier Transform, which is the Fourier transform of the amplitude of the quantum state. Since we can't measure the amplitude of a quantum state directly, the QFT is not typically used on its own, but rather as a useful subcomponent of many key quantum algorithms. In the next chapter, we will see the QFT in action when it is used as a component in Shor's algorithm.