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