Week 5, summary 
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1. Proof of Kitaev's lemma: 
   Given $U$ which can be calculated efficiently, together with its 
   powers $U^{2^i}$, we can approximate the transformation 
   |\phi>|0> --> |\phi>|\theta> where |\phi> is an eigenvector of U 
   with eigenvalue $e^{i\theta}$.

   This lemma given an alternative factorization algorithm, as we will 
   see in ex3. 

2. I will define group representations and 
   fourier transforms over any Abelian group. 
   I then show how to use Kitaev's lemma to give 
   a quantum efficient algorithm 
   for Fourier transform over any Abelean group. 
 
3. The Hidden Subgroup problem, a general framework for 
   all quantum algorithms presented so Far. 
   The Graph Automorphism problem as a hidden subgroup 
   problem.