Multidimensional Quantum Fourier Transformation

The Quantum Fourier Transformation (QFT) is a well-known subroutine for algorithms on qubit-based universal quantum computers. In this work, the known QFT circuit is used to derive an efficient circuit for the multidimensional QFT. The complexity of the algorithm is $\mathcal{O}( \log^2(M)/d )$ for an array with $M=(2^n)^d$ elements $(n \in \mathbb{N})$ equally separated along $d$ dimensions. Relevant properties for application are discussed. An example on current hardware is depicted by a 6 qubit 2D-QFT with an IBM quantum computer.