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The Hidden Subgroup ProblemKetan PatelMay 13, 2002

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 permutations on n elements (∗≡composition)

Hadamard Transform1-qubit Hadamard|0〉 + |1〉H (α|0〉 + β|1〉)=α √ + β |0〉−|1〉 √2 2Applying Hadamard to n qubits individuallyH n |x〉 = 1 √2n ∑y(−1) x·y |y〉Note: H n |0〉 gives equal superposition of all basis states

Quantum Fourier Transformn-qubit Quantum Fourier Transform (N = 2 n )∑|x〉→√ 1 N−1exp 2πixk/N |k〉Nk=0

Given: function f : B n → B nSimon’s Problemf (x)= f (y) iff x ⊕ y = β for all x, y ∈ B nf has “periodicity” βGoal: Determine β efficiently

Simon’s AlgorithmStep 0) Prepare 2n qubits in |0〉 stateStep 1) Create superposition of allbasis states in first n qubitsStep 2) Apply U f∑x|x〉|0〉→∑x|x〉|f (x)〉| 0| 0| 0| 0| 0| 0| 0| 0H nxyU fxy f(x)|x0 + |x0 β|f(x0)Step 3) Measure last n qubits∑|x〉|f (x)〉 → (|x 0 〉 + |x 0 ⊕ β〉) ⊗ |f (xx} {{ }0 )〉state of 1st n qubitsMeasuring 1st n qubits yields no information about β

Simon’s Algorithm (cont.)Step 4) Apply Hadamard Transform to first n qubits|x 0 〉 + |x 0 ⊕ β〉 → ∑y((−1) x 0·y +(−1) (x 0⊕β)·y ) |y〉= ∑ (−1) x 0·y |y〉y: y·β=0⇐ x0 only affects phaseStep 5) Measure first n qubits and get y i such that y i · β = 0Step 6) Repeat enough times (O(n 2 )) to get system of equations⎡⎤ ⎡ ⎤y 0 (1) y 0 (2) ··· y 0 (n) β(1)⎢ y 1 (1) y 1 (2) ··· y 1 (n)⎥⎣ ... ... ... ⎦ ·⎢ β(2)⎥⎣ ... ⎦ = 0y k (1) y k (2) ··· y k (n) β(n)Step 7) Solve for β

Integer FactoringInteger factoring can be mapped to order findingProblem: Find factor of N ⇒ Problem: Find least integer r such thaty r ≡ 1(mod N)Step 0) Select integer a < N at randomStep 1) Use Euclid’s algorithm to compute gcd(a,N)Step 2) If > 1 doneStep 3) Otherwise use order finding algorithm to find least r fora r ≡ 1 (mod N)Step 4) If r is odd or a r/2 ≡−1 (mod N) startoverStep 5) Calculate α and β:( )()a r − 1 = a r/2 − 1 a r/2 + 1 ≡ 0 (mod N)} {{ }} {{ }α βStep 6) If N|α or N|β start overStep 7) Compute gcd(N,α) and gcd(N,β)

Order Finding AlgorithmStep 0) Prepare t + ⌈log 2 N⌉ qubits in |0〉 state (Let T = 2 t )Step 1) Create uniform superposition on first t qubitsStep 2) Apply U f∑x|x〉|0〉→∑x|x〉|y x mod N〉Step 3) Measure second register∑|x〉|y x 〉→∑|x 0 + λr〉|y x 0〉x λStep 4) Apply Discrete Fourier Transform to first register∑λ|x 0 + λr〉 → ∑l= ∑l≈∑exp 2πil(x0+λr)/T |l〉λ{ ≈0 if }} lr≠kT {exp 2πilx0/T |l〉 exp 2πilλr/T∑exp 2πikx 0/rk∣kTr∑λ〉

Order Finding Algorithm (cont.)Step 5) Measure first register and get c = k(T /r)Step 6) Use continued fraction algorithm to retrieve r

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 properties (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

Fourier Transform over GFourier Transform: Transformation from standard basis of group elementsto basis of irreducible characters of GProperties:• subgroup K FT → K ⊥ := {χ : χ(k)=1 for all k ∈ K}K =χ∈K ⊥ ker χ• shift invariance∑ |g + k〉 FT →= ∑ χ(g)|χ〉k∈Kχ∈K ⊥

General HSP AlgorithmStep 1) Create uniform superposition of states in coset of KStep 2) Apply Fourier Transform over GStep 3) Sample distribution χ i ∈ K ⊥Step 4) Reconstruct (classically) K = ∩ ker χ i

General Issues• Constructing initial superposition may be nontrivial• Efficient FT over group may not be known• Group may not be known (e.g., factoring)

Issues for Non-Abelian case:Non-Abelian Groups• Non-unique Fourier Transform• Efficient implementation of FT• Determining subgroup from sampling of FT• In general no shift-invariant basis

Graph IsomorphismLet M A , M B be the incidence matrices of graphs A and BA ≃ B if P(M A )=M B for some permutation matrix PProblem: Given A and B determine if A ≃ BFormulate as Non-abelian HSP:• Let C = A ∪ B, L A = {vertices of A} and L B = {vertices of B}• Automorphism group of C, K is a subgroup of S 2n• If A ≄ B all k ∈ K will map L Ak → LAotherwise half of K will map L A → L B• Therefore sampling K will determine if A ≃ BUnfortunately generating superposition of elements of K nontrivial

Results for Non-Abelian Case• Solved for normal subgroup assuming efficient FT (Hallgren et al. 2000)• Solvable if ∩ normalizers of all subgroups is large (Grigni et al. 2000)• “Solved” for dihedral groups — need exponential post-processing(Ettinger and Høyer 2000)• Solved for groups formed by certain wreath products(Rötteler & Beth 1998)• Efficient FT over S n known (Beals 1997)

References• Jozsa - Quantum Factoring, Discrete Logarithms, and the Hidden SubgroupProblem (2000)quant-ph/0012084• Jozsa - Quantum Algorithms and the Fourier Transform (1997)quant-ph/9707033• Hallgren & Russell - The Hidden Subgroup Problem and QuantumComputation Using Group Representations (2001)www.cs.caltech.edu/ hallgren/reps1.pdf• Hallgren - Quantum Fourier Sampling, the Hidden Subgroup Problem,and Beyond (2000)www.cs.caltech.edu/ hallgren/thesis.pdf