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Hidden Subgroup ProblemSuppose• K is a subgroup of a group G• f : g → x function from G to discreteset X• f (g 1 )= f (g 2 ) ⇔ g 2 ∈ g 1 + KK+K+K+Kc m−1c m−2c 1x m−1x m−2x 1x0GfXProblem: Find generating set for KExample (Simon’s Problem): K = {0,β}
More Group TheoryRepresentation ρ(G): Mapping from G ⇒ group of complex matricespreserving group pro<strong>per</strong>ties (homomorphism)e.g. ρ(g i )ρ(g j )=ρ(g i + g j )Irreducible Representation ρ: ρ cannot be written as ρ 1 ⊕ ρ 2Blank Example: The irreducible representations of Z 4ρ(0) ρ(1) ρ(2) ρ(3)ρ 1 1 1 1 1 ⇐ trivial representationρ 2 1 −i −1 iρ 3 1 −1 1 −1ρ 4 1 i −1 −iCharacter χ(ρ): Mapping defined by g i → trace(ρ(g i ))For Abelian groups irr. representations are 1-D ⇒ irreducible χ(ρ)=ρ
Page 1 and 2: The Hidden Subgroup ProblemKetan Pa
Page 3 and 4: Groups• Group (G,+): closure, ass
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: Order Finding Algorithm (cont.)Step
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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