1. Simon’s Algorithm Arathi Ramani EECS 598 Class Presentation

2. Statement Given: Integer n >= 1 Function : Promise: There exists a nonzero element such that for all iff or Problem: Find s The function fulfils Simon’s promise w.r.t s

3. Group Theory Terminology : additive group of 2 elements with addition For, |X|: cardinality of X, : subgroup generated by X A subset X of G is linearly independent if no proper subset of X generates

4. More group theory terms.. Bilinear map: for g = (g 1 …. g n ), h = (h 1 …. h n ) Orthogonal subgroup:

5. Restatement Given: Integer n >= 1 Function : Promise: There exists a subgroup H 0 <= G such that is constant and distinct on each coset of H 0 Problem: Find a generating set for H 0

6. Propositions Proposition 1: There exists a classical deterministic algorithm that, given a subset, returns a linearly independent subset of G that generates the subgroup. The algorithm runs in time polynomial in n and linear in the cardinality of X

7. Proposition 2: There exists a classical deterministic algorithm that, given a linearly independent subset returns a linearly independent subset that generates the orthogonal subgroup of. The algorithm runs in time polynomial in n.

8. Theorem Let n >= 1 be an integer and : be a function that fulfils Simon’s promise for some subgroup H 0 <= G. Assume that a quantum algorithm to compute is given, together with the value of n (continued)

9. Theorem (contd) Then there exists a quantum algorithm capable of finding a random element of the orthogonal subgroup. Moreover, the algorithm runs in time linear in n and in the time required to compute

10. Simon’s Subroutine Apply a Hadamard transform to, producing the equally weighted superposition: Apply, producing a superposition of all cosets of H 0 (continued)

11. Simon’s Subroutine Apply Hadamard transform to the first register, producing a superposition over the orthogonal subgroup

12. Issues How many times do we need to run Simon’s subroutine? Will this ensure success?

13. Exact Quantum Algorithm Theorem: Given n >= 1, : being a function that fulfils Simon’s promise for some subgroup H 0 <= G; A quantum algorithm that computes without making any measurements; (continued)

14. The value of n; A linearly independent subset Y of the orthogonal subgroup ; Then there exists a quantum algorithm that returns an element of, provided Y does not generate, otherwise it returns the zero element. The algorithm runs in polynomial time.

15. Steps of the Exact Algorithm 1. Initialize generating set and counter to 0 2. Apply the theorem to get an element not in, update Y and the counter 3. Stop if the zero element is returned

16. Features of Exact Algorithm Shrinking a group Removing 0 from a subgroup

17. Conclusions The algorithm needs O(n 2 ) evaluations of The algorithm is exact, with a 100% probability of success Applications of Simon’s problem?