  We propose three core ideas: 1. the wave-particle duality of the qudit
quantum space; 2. the classification of all elementary quantum gates by ordered
pairs of qudit functionals; 3. a new type of quantum gates called the "quantum
wave gates". We first study the quantum functionals whose relation to the
quantum states is analogous to that between the momentum and position
wavefunctions in fundamental quantum physics: a Fourier transform and an
entropic uncertainty principle can be defined between the dual representations.
The quantum functionals are not just mathematical constructs but have clear
physical meanings and quantum circuit realizations. Connecting the partition
interpretation of the qudit functionals to the effects of quantum gates we
classify all elementary quantum gates by ordered pairs of qudit functionals. By
generalizing the qudit functionals to quantum functionals, the new type of
"quantum wave gates" are discovered as quantum versions of the conventional
quantum gates.
