Efficient Implementation of Controlled Operations for Multivalued Quantum Logic

This paper presents a new quantum array that can be used to control a single-qudit hermitian operator for an odd radix r ≫ 2 by n controls using Theta(n^log_2 r + 2) single-qudit controlled gates with one control and no ancilla qudits. This quantum array is more practical than existing quantum arrays of the same complexity because it does not require the use of small roots of the operation that is being implemented. Another quantum array is also presented that implements a single-qudit operator with n controls for any radix r ≫ 2 using ceiling(log_(r - 1) n) ancilla qudits and Theta(n^(log_(r - 1) 2 + 1)) single-qudit gates with one control.