A relation between quantum operations and the quantum Fourier transform
Runyao Duan, Zhengfeng Ji, Yuan Feng, Mingsheng Ying · arXiv (Cornell University) · 2003
We propose a general geometry representation of $d$-level quantum system (or qudit for short). Then we define a general class of operations for qudits. A sufficient and necessary condition of when an operation in the class would be a legal quantum operation is found. This condition is expressed in terms of the quantum Fourier transform. As an application of this condition, we recover the optimal $d$-level universal $NOT$ gate and show the optimality of qubit case.