Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems
Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $D=d^n$ with $d$ an odd integer, and is inspired by the Cooley-Tukey formalism. The `large Fourier transform' is expressed as a sequence of $n$ `small Fourier transforms' (together with some other transforms) in quantum systems with $d$-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the $n$ Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space $D=d_0...d_{n-1}$ with $d_0,...,d_{n-1}$ odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the `large Fourier transform' is expressed as a sequence of $n$ `small Fourier transforms' (that involve some constants related to the number theory that describes the formalism). The `small Fourier transforms' can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from ${\cal O}(D^2)$ to ${\cal O}(D\log D)$. The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.