Quantum ECE227/PHY272 Information Science
Homework #3 Due in Class 2/25/2009 1. Unitarity of Quantum Fourier Transform Give a direct proof that the linear transformation that defines the quantum Fourier transform
j
1 N
e
k =0
N 1
2 ijk N
k
is unitary. (Textbook Exercise 5.1)
2. Inverse Quantum Fourier Transform Give a quantum circuit to perform the inverse quantum Fourier transform. (Textbook Exercise 5.5) 3. Alternative Unitary Transformation for Quantum Order Finding The quantum state produced in the order-finding algorithm, before the inverse Fourier is
transform, = j U 1 = j x j mod N ,
j j =0 j =0
2t 1
2t 1
if we initialize the second register as 1 . (a) Show that the same state is obtained if we replace Uj with a different unitary transform V, which computes V j k = j k + x j mod N ,
and start the second register in the state 0 . (b) Show how to construct V using O(L3) gate...