32,897 research outputs found
From Skew-Cyclic Codes to Asymmetric Quantum Codes
We introduce an additive but not -linear map from
to and exhibit some of its interesting
structural properties. If is a linear -code, then is an
additive -code. If is an additive cyclic code then
is an additive quasi-cyclic code of index . Moreover, if is a module
-cyclic code, a recently introduced type of code which will be
explained below, then is equivalent to an additive cyclic code if is
odd and to an additive quasi-cyclic code of index if is even. Given any
-code , the code is self-orthogonal under the trace
Hermitian inner product. Since the mapping 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
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
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
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