On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date

Geometric representations of linear codes

We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex $\Delta$ such that C is a punctured code of the kernel ker $\Delta$ of the incidence matrix of $\Delta$ over F and there is a linear mapping between C and ker $\Delta$ which is a bijection and maps minimal codewords to minimal codewords. We show that the linear codes over rationals and over GF(p), where p is a prime, are triangular representable. In the case of finite fields, we show that this representation determines the weight enumerator of C. We present one application of this result to the partition function of the Potts model. On the other hand, we show that there exist linear codes over any field different from rationals and GF(p), p prime, that are not triangular representable. We show that every construction of triangular representation fails on a very weak condition that a linear code and its triangular representation have to have the same dimension.Comment: 20 pages, 8 figures, v3 major change

Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume "Recent Trends in Combinatorics

On THE AVERAGE JOINT CYCLE INDEX AND THE AVERAGE JOINT WEIGHT ENUMERATOR (Research on finite groups, algebraic combinatorics, and vertex algebras)

In this paper, we introduce the concept of the complete joint cycle index and the average complete joint cycle index, and discuss a relation with the complete joint weight enumerator and the average complete joint weight enumerator respectively in coding theory

Permutation codes

AbstractThere are many analogies between subsets and permutations of a set, and in particular between sets of subsets and sets of permutations. The theories share many features, but there are also big differences. This paper is a survey of old and new results about sets (and groups) of permutations, concentrating on the analogies and on the relations to coding theory. Several open problems are described

On the cycle index and the weight enumerator II

In the previous paper, the second and third named author introduced the concept of the complete cycle index and discussed a relation with the complete weight enumerator in coding theory. In the present paper, we introduce the concept of the complete joint cycle index and the average complete joint cycle index, and discuss a relation with the complete joint weight enumerator and the average complete joint weight enumerator respectively in coding theory. Moreover, the notion of the average intersection numbers is given, and we discuss a relation with the average intersection numbers in coding theory.Comment: 24 page

Equivariant theory for codes and lattices I

In this paper, we present a generalization of Hayden's theorem [7, Theorem 4.2] for $G$-codes over finite Frobenius rings. A lattice theoretical form of this generalization is also given. Moreover, Astumi's MacWilliams identity [1, Theorem 1] is generalized in several ways for different weight enumerators of $G$-codes over finite Frobenius rings. Furthermore, we provide the Jacobi analogue of Astumi's MacWilliams identity for $G$-codes over finite Frobenius rings. Finally, we study the relation between $G$-codes and its corresponding $G$-lattices.Comment: 22 page

Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

Let G be a connected bipartite graph with colour classes E and V and root polytope Q. Regarding the hypergraph H=(V,E) induced by G, we prove that the interior polynomial of H is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of H and its transpose H=(E,V) agree. When G is a complete bipartite graph, our result recovers a well-known hypergeometric identity due to Saalschütz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read off of an associated Floer homology group.National Science Foundation (U.S.) (Grant DMS‐1100147)National Science Foundation (U.S.) (Grant DMS‐1362336