Quantum Computing based on Tensor Products ... - Cinvestav

Algorithm for the Fourier transform
Input. n = 2 ν , f ∈ C n = Hν.
Output. ˆ f = DFT(f) ∈ Hν.
Procedure DFT(n, f)
1 Let x0 := H(e0).
2 For each j ∈ [[0, 2 ν − 1]], or equivalently, for each
(εj,ν−1 · · · εj,1εj,0) ∈ {0, 1} ν , do (in parallel):
1 For each k ∈ [[0, ν − 1]] do (in parallel):
� �
1 Let δ :=
�
�
Rk εj be the reverse of the chain consisting of
�
k
the (k + 1) less significant bits.
2 Let yjk := x0.
3 For ℓ = 0 to k do { yjk := Q c2 (yjk , eδ ) } j,ℓ
2 Let y j := y j0 ⊗ · · · ⊗ y j,ν−1 .
3 Output as result ˆ f = � 2 ν −1
j=0 fjy j .
Morales-Luna (CINVESTAV) QC <strong>based</strong> on Tensor Products 5-th Int. WS App. Cat. Th. 26 / 38