Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma

Abstract

The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli \cite{AKS06}, we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko \cite{KP13}, we show that any subsystem stabilizer code can be "doubled" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis \cite{BTL10}. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.