Abstract
Quantum Fourier transform (QFT) plays a key role in many quantum algorithms, but the existing circuits of QFT are incomplete and lacking the proof of correctness. Furthermore, it is difficult to apply QFT to the concrete field of information processing. Thus, this chapter firstly investigates quantum vision representation (QVR) and develops a model of QVR (MQVR). Then, four complete circuits of QFT and inverse QFT (IQFT) are designed. Meanwhile, this chapter proves the correctness of the four complete circuits using formula derivation. Next, 2D QFT and 3D QFT based on QVR are proposed. Experimental results with simulation show the proposed QFTs are valid and useful in processing quantum images and videos. In conclusion, this chapter develops a complete framework of QFT based on QVR and provides a feasible scheme for QFT to be applied in quantum vision information processing.
TopPerfect Shuffle Permutations
The perfect shuffle permutation Pn,m is an mn×mn matrix, which shuffles n packs of m cards into m packs of n cards, and satisfies
,
(5.1) where
,
.
δx,y is the Kronecker delta function, i.e., δx,y=0 if x≠y, otherwise δx,y=1 (Fino & Algazi, 1977).
Two perfect shuffle permutations 
and 
are defined as (Hoyer, 1997),
(5.2) where
P2,2 is a Swap gate, i.e.,
.
(5.3)Since
,