This decomposition allows for the quantum Fourier transform to be implemented on a quantum computer efficiently (fast enough for practical purposes). It is used in various algorithms, including Shor’s algorithm, which is an algorithm for factoring large (200+ digit) integers.
The computational task of factoring large integers on a classical computer takes so long that the government, banks, and the internet rely on this difficulty to keep their information secure from the public (RSA cryptography).
So if you have a large scale quantum computer, you could have the ability to render the RSA cryptographic protocol obsolete!
The thing is, quantum computers are really hard to make. However, in 2001 IBM researchers in the foothills behind my house managed to use a quantum computer to factor the number 15 into 3 x 5!