Quantum algorithms for a set of group theoretic problems

Abstract

We study two group theoretic problems, GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, in the setting of black-box groups, where DOUBLE COSET MEMBERSHIP generalizes a set of problems, including GROUP MEMBERSHIP, GROUP FACTORIZATION, and COSET INTERSECTION. No polynomial-time classical algorithms are known for these problems. We show that for solvable groups, there exist efficient quantum algorithms for GROUP INTERSECTION if one of the underlying solvable groups has a smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one of the underlying solvable groups is smoothly solvable. We also study the decision versions of STABILIZER and ORBIT COSET, which generalizes GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, respectively. We show that they reduce to ORBIT COSET under certain conditions. Finally, we show that DOUBLE COSET MEMBERSHIP and DOUBLE COSET NONMEMBERSHIP have zero knowledge proof systems.