Construction of Generalized Quantum Boolean Functions

The existing construction methods of Quantum Boolean functions (QBFs) are extended and simplified. All QBFs with one qubit and all local QBFs with any qubits are constructed. And we propose the concept of Generalized quantum Boolean functions (GQBFs). We find all GQBFs with one qutrit and all kinds of local GQBFs with any qutrits. The number of each of the four kinds of functions above is uncountably infinitely many. By using diagonal matrices, we obtain uncountably infinitely many non-local QBFs with any qubits and GQBFs with any qutrits. Infinitely many families of GQBFs with any qudits are obtained from the properties of projection matrices of known saturated orthogonal arrays.