Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2^n first (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of non-abelian 2-groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2^n is O(n^2) in all cases.Comment: 16 pages, LaTeX2

Computability Theory and Ordered Groups

Ordered abelian groups are studied from the viewpoint of computability theory. In particular, we examine the possible complexity of orders on a computable abelian group. The space of orders on such a group may be represented in a natural way as the set of infinite paths through a computable tree, but not all such sets can occur in this way. We describe the connection between the complexity of a basis for a group and an order for the group, and completely characterize the degree spectra of the set of bases for a group. We describe some restrictions on the possible degree spectra of the space of orders, including a connection to algorithmic randomness

Computational problems in the theory of Abelian groups

In this thesis, the worst-case time complexity bounds on the algorithms for the problems mentioned below have been improved. A. Algorithms on abelian groups represented by a set of defining relations for computing: (I) a canonical basis for finite abelian groups (II) a canonical basis for Infinite abelian group B. Algorithms for computing: (I) Hermite normal form of an Integer matrix (II) The Smith normal form of an Integer matrix (III) The set of all solutions of a system of Diophantine Equations C. Algorithms on abelian groups represented by an explicit set of generators for computing: (I) the order of an element (space complexity 1s only improved) (II) a complete basis for a finite abelian group (III) membership-Inclusion testing (IV) the union and Intersection of two finite abelian groups D. A classification of the relative complexity of computational problems on abelian groups (as above) factorization and primility testing. E. Algorithms on abelian subgroups of the symmetric group for computing: (I) the complete structure of a group (II) membership-Indus Ion testing (III) the union of two abelian groups (IV) the Intersection of two abelian groups

A monomial matrix formalism to describe quantum many-body states

We propose a framework to describe and simulate a class of many-body quantum states. We do so by considering joint eigenspaces of sets of monomial unitary matrices, called here "M-spaces"; a unitary matrix is monomial if precisely one entry per row and column is nonzero. We show that M-spaces encompass various important state families, such as all Pauli stabilizer states and codes, the AKLT model, Kitaev's (abelian and non-abelian) anyon models, group coset states, W states and the locally maximally entanglable states. We furthermore show how basic properties of M-spaces can transparently be understood by manipulating their monomial stabilizer groups. In particular we derive a unified procedure to construct an eigenbasis of any M-space, yielding an explicit formula for each of the eigenstates. We also discuss the computational complexity of M-spaces and show that basic problems, such as estimating local expectation values, are NP-hard. Finally we prove that a large subclass of M-spaces---containing in particular most of the aforementioned examples---can be simulated efficiently classically with a unified method.Comment: 11 pages + appendice

Complexity volumes of splittable groups

Using graph of groups decompositions of finitely generated groups, we define Euler characteristic type invariants which are non-zero in many interesting classes of finitely presented, hyperbolic, limit and CSA groups, including elementarily free groups and one-ended torsion-free hyperbolic groups whose JSJ decomposition contains a maximal hanging Fuchsian vertex group.Comment: Major revisions following the suggestions of the referee. To appear in Journal of Algebr