Controlled-U Qudit gate

Consider a Qudit Controlled-U Gate




Let the Qudit system under consideration have as the d-canonical states and let us consider the ring be the ring of integers from zero to d-1.

Now the controlled-U operation is defined for a qudit sytem as follows

where





and





Using this we see that the qudit version of Controlled NOT gate becomes





So this is in accordance with the d=2 where where had the CNOT operation doing





So the generalized CNOT operation is