Quantum-error-correcting codes using qudit graph states

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.

Figure 1

Action of $X$ and $X^2$ on graph state $(D=4)$.

Figure 2

Examples from different graph sequences: (a) bar (odd $n$), (b) star, (c) cycle, (d) wheel, (e) $n=16$ hypercube.