Simons Algorithm Arathi Ramani EECS 598 Class Presentation

Simon’s Algorithm Arathi Ramani EECS 598 Class Presentation

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

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

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

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

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

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.

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)

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

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

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

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

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)

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.

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

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

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