Then the Controlled - operation acts as
Using these operators we can make the Qudit Swap Gate.
Now consider the following gates
Call this as Gate M1.
Call this as Gate M2
Now we can see that operation M1 is
And operation M2 is
Now by adding an operation at the end we can form the Qudit swap gates as shown
Since the inverse of Swap operation is also Swap operation we have the following implementations also