Shor Algorithm - Quantum Computing - Lecture Slides, Slides of Computer Science

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Shor Algorithm (continued)

Use of number theory and reductions

Reductions (^) Solve RSA

Factor big integers

Find period

Estimate Phase

Fourier Transform

Easy to multiply but difficult to factor big integers.

Shor knows number theory and uses it!!!

1. In many cases, we can use the knowledge from other

areas of research in a new and creative way.

2. You do not have to invent everything from scratch. You

just reuse something that was invented by other people.

3. If the two areas are not obviously linked, your invention

can be very important.

4. This is exactly what was done by Shor.
   1. We introduced modular arithmetic in last lecture as a general tool for algorithms and hardware
   2. Now we will show how creatively Shor used it in his algorithm.

Assume:

We want to find the

smallest r such that the

above is true

Greatest common denominator

More interesting case

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

Finding the smallest period r

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

So now we are quite optimistic!

Reductions (^) Solve RSA

Factor big integers

Find period

Estimate Phase

Fourier Transform

We are here

This was done earlier

Choosing the

operator U

1. It requires modulo multiplication in modular arithmetic
2. Not trivial
3. Potential research how to do this efficiently

Phase is 1/r

Now the problem is reduced to creation of certain quantum state. We published papers – see David Rosenbaum Docsity.com