Hidden Translation and Translating Coset in Quantum Computing

Katalin Friedl, Gábor Ivanyos, Frédéric Magniez, Miklos Santha, Pranab Sen
2014 SIAM journal on computing (Print)  
We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in _p^n, whenever p is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a
more » ... ul self-reducibility result: Translating Coset in a finite solvable group G is reducible to instances of Translating Coset in G/N and N, for appropriate normal subgroups N of G. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.
doi:10.1137/130907203 fatcat:rwapyh6j7ffddp5phted42l4re