Quantum Error Correcting Codes Using Qudit Graph States

Understand

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 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.