Quantum Error-Correcting Codes over Mixed Alphabets

Errors are inevitable during all kinds quantum informational tasks and quantum error-correcting codes (QECCs) are powerful tools to fight various quantum noises. For standard QECCs physical systems have the same number of energy levels. Here we shall propose QECCs over mixed alphabets, i.e., physical systems of different dimensions, and investigate their constructions as well as their quantum Singleton bound. We propose two kinds of constructions: a graphical construction based a graph-theoretical object composite coding clique and a projection-based construction. We illustrate our ideas using two alphabets by finding out some 1-error correcting or detecting codes over mixed alphabets, e.g., optimal $((6,8,3))_{4^52^1}$, $((6,4,3))_{4^42^2}$ and $((5,16,2))_{4^32^2}$ code and suboptimal $((5,9,2))_{3^42^1}$ code. Our methods also shed light to the constructions of standard QECCs, e.g., the construction of the optimal $((6,16,3))_4$ code as well as the optimal $((2n+3,p^{2n+1},2))_{p}$ codes with $p=4k$.