Shor Algorithm Anuj Dawar

Finding the eigenvalue is the same as finding its phase 

Target register Index register to store the eigenvalue Unitary operator for which we calculate phase of eigenvalue using phase kickback we measure the phase We initialize to all states

Formulas for the phase estimation algorithm This is the input to QFT ()

Because e j  is an eigenvallue of U

We found phase Assume k an integer Number of bits

Concluding, we can calculate phase

Easy to multiply but difficult to factor big integers.

Assume: We want to find the smallest r such that the above is true

Greatest common denominator

So now what remains is to be able to find period, but this is something well done with spectral transforms.

Phase is 1/r

Reductions Solve RSA Factor big integers Find period Estimate Phase Fourier Transform