A Fusion Algorithm for Solving the Hidden Shift Problem in Finite Abelian Groups

Abstract

It follows from a result by Friedl, Ivanyos, Magniez, Santha and Sen from 2014 that, for any fixed integer m > 0 (thought of as being small), there exists a quantum algorithm for solving the hidden shift problem in an arbitrary finite abelian group (G, +) with time complexity

poly(log vertical bar G vertical bar). 2(O) ((root log vertical bar mG vertical bar)).

As discussed in the current paper, this can be viewed as a modest statement of Pohlig-Hellman type for hard homogeneous spaces. Our main contribution is a somewhat simpler algorithm achieving the same run-time for m = 2(t)p, with t any non-negative integer and p any prime number, where additionally the memory requirements are mostly in terms of quantum random access classical memory; indeed, the amount of qubits that need to be stored is poly(log vertical bar G vertical bar). Our central tool is an extension of Peikert's adaptation of Kuperberg's collimation sieve to arbitrary finite abelian groups. This allows for a reduction, in said time, to the hidden shift problem in the quotient G/2(t)pG, which can then be tackled in polynomial time, by combining methods by Friedl et al. for p-torsion groups and by Bonnetain and Naya-Plasencia for 2(t) -torsion groups.