1. Shor’s Algorithm -basic introduction –

2. The main goal for Shor’s Alg. Find the factors of a certain integer – Why?... Long story short: prime factors, RSA algorithm, digital security. – How?... Again, long story short: Set up a special periodic function, find period, get factors. – Q-CPU?... We can execute this algorithm a lot faster on Q-CPU because of the superposition principle.

3. Structure for today... Give an example of Shor’s Algorithm – See which definitions we need for this.. Back to Quantum Computing – See how QFT is defined

4. Example of Shor’s Algorithm

7. Periodicity

8. Recap Choose a number you want to factor (N) Choose a number which will help you with this process (a) Find the order r of a (mod N) – Order r odd -> Throw it away, choose different a – Order r even -> Ca-ching! Calculate x = a^(r/2), this is your NTSR Factors of N are gcd(N,x-1) and gcd(N,x+1)

9. Crux of the problem: finding the order To find the order we will use the quantum fourier transform of our modulo function. But we first need some introduction to this...

13. Now that we have this tool.. We can prove that the QFT – is unitary; – and that linear shifts of state-vectors cause relative phase shifts of their Fourier Transform.

19. We can repeat this process a couple of times, until we measure both |0> and |4>. Since we are using an N qubit system, with unkown period r, we can say that N/r = 4.. And since N = 8 we get that r = 2.