32,897 research outputs found

    From Skew-Cyclic Codes to Asymmetric Quantum Codes

    Full text link
    We introduce an additive but not F4\mathbb{F}_4-linear map SS from F4n\mathbb{F}_4^{n} to F42n\mathbb{F}_4^{2n} and exhibit some of its interesting structural properties. If CC is a linear [n,k,d]4[n,k,d]_4-code, then S(C)S(C) is an additive (2n,22k,2d)4(2n,2^{2k},2d)_4-code. If CC is an additive cyclic code then S(C)S(C) is an additive quasi-cyclic code of index 22. Moreover, if CC is a module θ\theta-cyclic code, a recently introduced type of code which will be explained below, then S(C)S(C) is equivalent to an additive cyclic code if nn is odd and to an additive quasi-cyclic code of index 22 if nn is even. Given any (n,M,d)4(n,M,d)_4-code CC, the code S(C)S(C) is self-orthogonal under the trace Hermitian inner product. Since the mapping SS preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.Comment: 16 pages, 3 tables, submitted to Advances in Mathematics of Communication

    Low-complexity quantum codes designed via codeword-stabilized framework

    Full text link
    We consider design of the quantum stabilizer codes via a two-step, low-complexity approach based on the framework of codeword-stabilized (CWS) codes. In this framework, each quantum CWS code can be specified by a graph and a binary code. For codes that can be obtained from a given graph, we give several upper bounds on the distance of a generic (additive or non-additive) CWS code, and the lower Gilbert-Varshamov bound for the existence of additive CWS codes. We also consider additive cyclic CWS codes and show that these codes correspond to a previously unexplored class of single-generator cyclic stabilizer codes. We present several families of simple stabilizer codes with relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl

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

    Get PDF
    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
    • …
    corecore