Quantum Computing based on Tensor Products ... - Cinvestav

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<strong>Quantum</strong> Computation of the Discrete Fourier Transform Given f = � n−1 j=0 f(j)ej ∈ C n , its discrete Fourier transform is DFT(f) = ˆ ⎡ �n−1 1 �n−1 � � ⎤ 2πijk f = ⎢⎣ √ exp f(k) ⎥⎦ n n ej ∈ C n . j=0 k=0 DFT is linear transform and, w.r.t. � �the canonical �� basis, it is represented by 2πijk exp n the unitary matrix DFT = 1 √ n Morales-Luna (CINVESTAV) QC <strong>based</strong> on Tensor Products 5-th Int. WS App. Cat. Th. 24 / 38 jk
If n = 2 ν , Hν = C n , and by identifying each j ∈ [[0, 2 ν − 1]] with εj = εj,ν−1 · · · εj,1εj,0: DFT(eεj ) = �ν−1 � 1 √ e0 + exp 2 � � πij 2 k e1 k=0 = 1 � � πij √ e0+exp 2 20 � � e1 ⊗ 1 � � πij √ e0+exp 2 21 � � e1 ⊗···⊗ 1 √ 2 � � � πij e0+exp 2ν−1 � � e1 The products appearing in this tensor product suggest the operators Qk : H1 → H1 and their “controlled” versions: Qk = ⎡ ⎢⎣ 1 0 0 exp � πi 2k � ⎤ ⎥⎦ , Q c kj = ⎡ ⎢⎣ 1 0 0 exp � πi Morales-Luna (CINVESTAV) QC <strong>based</strong> on Tensor Products 5-th Int. WS App. Cat. Th. 25 / 38 j 2k � ⎤ ⎥⎦ . (2)
Page 1 and 2: Quantum Co
Page 3 and 4: Agenda 1 Abstract 2 Overview of the
Page 7 and 8: 5 Shor Algorithm 1 A Short Refreshm
Page 9 and 10: General references Quantum<
Page 11 and 12: Algorithmics and Crypto T. Hogg. So
Page 15 and 16: Geometrical properties: m-dimension
Page 17 and 18: Qubits and Quregisters Spaces and b
Page 23: Agenda 1 Abstract 2 Overview of the
Page 29 and 30: Biggest problem Calculate the order
Page 31 and 32: Quantum Cryptograp
Page 35 and 36: Identity Checking Exact and obvious
Page 37 and 38: Deutsch-Josza Relation Pseudotelepa

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