Quantum error-correcting codes over mixed alphabets

Abstract

We study the quantum error-correcting codes over mixed alphabets to deal with a more complicated and practical situation in which the physical systems for encoding may have different numbers of energy levels. In particular we investigate their constructions and propose the theory of quantum Singleton bound. Two kinds of code constructions are presented: a projection-based construction for general case and a graphical construction based on a graph-theoretical object composite coding clique dealing with the case of reducible alphabets. We find out some optimal one-error correcting or detecting codes over two alphabets. Our method of composite coding clique also sheds light on constructing standard quantum error-correcting codes, and other families of optimal codes are found. © 2013 American Physical Society.