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Outline• Preliminaries• Simon’s Problem• Integer Factoring / Order Finding• Hidden Subgroup Problem• Generalized Fourier Transform• Non-Abelian Groups .
Groups• Group (G,+): closure, associativity, identity, inverses• Abelian Group: group satisfying commutativity• Subgroup K of G: subset of G forming a group• Coset of K: translation of a subgroup KExamples of groups:(Z n ,+) additive group of integers mod n(R,∗) multiplicative group of nonzero real numbers(B n ,+) additive group of binary n-tuples (+ ≡ bitwise ⊕)(S n ,∗) group of <strong>per</strong>mutations on n elements (∗≡composition)
Page 1: The Hidden Subgroup ProblemKetan Pa
Page 5 and 6: Quantum Fourier Transformn-qubit Qu
Page 7 and 8: Simon’s AlgorithmStep 0) Prepare
Page 9 and 10: Integer FactoringInteger factoring
Page 11 and 12: Order Finding Algorithm (cont.)Step
Page 13 and 14: More Group TheoryRepresentation ρ(
Page 15 and 16: General HSP AlgorithmStep 1) Create
Page 17 and 18: Issues for Non-Abelian case:Non-Abe
Page 19 and 20: Results for Non-Abelian Case• Sol

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