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arXiv:0712.1979 · Physical Review A

Quantum-error-correcting codes using qudit graph states

Shiang Yong Looi, Li Zhi Yu, Vlad Gheorghiu, Robert B. Griffiths

Published Oct 7, 2008• 111 citations
arXivHTMLPDF

Abstract

Graph states are generalized from qubits to collections of $n$ qudits of arbitrary dimension $D$, and simple graphical methods are used to construct both additive and nonadditive, as well as degenerate and nondegenerate, quantum-error-correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large $n$ and $D$ are constructed using simple graphs, except when $n$ is odd and $D$ is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general $D$, and shown to provide a dual representation of an additive graph code.

DOI
10.1103/physreva.78.042303

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The paper investigates quantum stabilizer codes and our innovative construction methodology. At first, we began with representing fundamental concepts and theories of quantum computing. In order to identify quantum error correcting codes, we extend the method of using polynomials to represent qudits on square lattices to accommodate more complicated situations in general Cayley graphs. The paper briefly reviews essential definitions and examples related to graphs, groups, and stabilizer codes, and later we propose the novel method and demonstrate its application by using the dihedral groupD4.