On the computational complexity of the abelian permutation group structure, membership and intersection problems

AbstractAlgorithms on computations on abelian permutation groups are presented here. An algorithm for computing the complete structure, algorithms for membership-inclusion testing and an algorithm for computing the intersection of abelian permutation groups are given. Their worst-case time complexity is a polynomial of degree 4 in terms of n, the number of points moved by the group. The upper bounds on the running time of the algorithms shown here improve the bounds on the above problems cited in the literature

Parallel algorithms for solvable permutation groups

AbstractA number of basic problems involving solvable and nilpotent permutation groups are shown to have fast parallel solutions. Testing solvability is in NC as well as, for solvable groups, finding order, testing membership, finding centralizers, finding centers, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in NC, find a central composition series, and find pointwise stabilizers of sets. The latter is applied to an instance of graph isomorphism. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to NC

Non-Abelian Quantum Codes

Like their classical counterparts, quantum codes are designed to pro- tect quantum in- formation from noise. From the perspective of informa- tion theory one considers the op- erations required to restore the encoded information given a syndrome which diagnoses the noise. From a more physics perspective, one considers systems whose energetically protected groundspace encodes the information. In this work we show that standard error correction procedures can be applied to systems where the noise ap- pears as non- abelian Fibonacci anyons. In the case of a Hamiltonian with non-commuting terms, we build a theory describing the spectrum of these models, with particular focus on the 3D gauge color code model. Numerics support the conjecture that this model is gapped, which one would expect for a self-correcting quantum memory

An Exact Quantum Hidden Subgroup Algorithm and Applications to Solvable Groups

We present a polynomial time exact quantum algorithm for the hidden subgroup problem in Z(mk)(n). The algorithm uses the quantum Fourier transform modulo m and does not require factorization of m. For smooth m, i.e., when the prime factors of m are of size (log m)(O(1)), the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general m, the algorithm of Mosca and Zalka is available. Even for m = 3 and k = 1 our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as m. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups