Gábor Ivanyos

Hidden translation and orbit coset in quantum computing

We give efficient quantum algorithms for the problems of <sc>Hidden Translation</sc> and <sc>Hidden Subgroup</sc> in a large class of non-abelian groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently <sc>Hidden Translation</sc> in Z <inf>p</inf>ⁿ, whenever <i>p</i> is a fixed prime. For the induction step, we introduce the problem <sc>Orbit Coset</sc> generalizing both <sc>Hidden Translation</sc> and <sc>Hidden Subgroup</sc>, and prove a powerful self-reducibility result: <sc>Orbit Coset</sc> in a finite group <i>G</i> is reducible to <sc>Orbit Coset</sc> in <i>G/N</i> and subgroups of <i>N</i>, for any solvable normal subgroup <i>N</i> of <i>G</i>.

EFFICIENT QUANTUM ALGORITHMS FOR SOME INSTANCES OF THE NON-ABELIAN HIDDEN SUBGROUP PROBLEM

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

Polynomial time algorithms for modules over finite dimensional algebras

We present polynomial time algorithms for some fundamental tasks from representation theory of finite dimensional algebras. These involve testing (and constructing) isomorphisms of modules as well as expressing of modules as direct sums of indecomposable modules. Over number fields the latter task seems to be difficult, therefore we restrict our attention to decomposition over finite fields and over the algebraic or real closure of number fields.

Treating the exceptional cases of the MeatAxe

We show that the Holt–Rees extension of the standard MeatAxe procedure finds submodules of modules over finite algebras with positive probability in more cases than originally claimed. For the case when the Holt–Rees method fails we propose a further, but still simple and efficient extension.

Finding the radical of an algebra of linear transformations

We present a method that reduces the problem of computing the radical of a matrix algebra over an arbitrary field to solving systems of semilinear equations. The complexity of the algorithm, measured in the number of arithmetic operations and the total number of the coefficients passed to an oracle for solving semilinear equations, is polynomial. As an application of the technique we present a simple test for isomorphism of semisimple modules.

Deterministic Polynomial Time Algorithms for Matrix Completion Problems

We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e., the problem of assigning values to the variables in a given symbolic matrix to maximize the resulting matrix rank. Matrix completion is one of the fundamental problems in computational complexity. It has numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings [N. J. A. Harvey, D. R. Karger, and K. Murota, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2005, pp. 489-498; N. J. A. Harvey, D. R. Karger, and S. Yekhanin, Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2006, pp. 1103-1111]. We design efficient deterministic algorithms for common generalizations of the results of Lovasz and Geelen on this problem by allowing linear polynomials in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. Our methods are algebraic and quite different from those of Lovasz and Geelen. We look at the problem of matrix completion in the more general setting of linear spaces of linear transformations and find a maximum rank element there using a greedy method. Matrix algebras and modules play a crucial role in the algorithm. We show (hardness) results for special instances of matrix completion naturally related to matrix algebras; i.e., in contrast to computing isomorphisms of modules (for which there is a known deterministic polynomial time algorithm), finding a surjective or an injective homomorphism between two given modules is as hard as the general matrix completion problem. The same hardness holds for finding a maximum dimension cyclic submodule (i.e., generated by a single element). For the “dual” task, i.e., finding the minimal number of generators of a given module, we present a deterministic polynomial time algorithm. The proof methods developed in this paper apply to fairly general modules and could also be of independent interest.

Finding maximal orders in semisimple algebras over Q

We consider the algorithmic problem of constructing a maximal order in a semisimple algebra over an algebraic number field. A polynomial time ff-algorithm is presented to solve the problem. (An ffalgorithm is a deterministic method which is allowed to call oracles for factoring integers and for factoring polynomials over finite fields. The cost of a call is the size of the input given to the oracle.) As an application, we give a method to compute the degrees of the irreducible representations over an algebraic number fieldK of a finite groupG, in time polynomial in the discriminant of the defining polynomial ofK and the size of a multiplication table ofG.

On the black-box complexity of sperner's lemma

We present several results on the complexity of various forms of Sperners Lemma in the black-box model of computing. We give a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an

Lattice basis reduction for indefinite forms and an application

O(\sqrt{n})