Stabilizer codes from modified symplectic forms

Abstract

In this paper, we construct explicit families of quantum error correcting codes based on cyclic codes. But for a crucial theoretical idea that we introduce here, namely working with a modified symplectic form as the basis of Weyl commutation, these codes would not have been possible — under the standard symplectic form, there are Galois theoretic no-go theorems on cyclic codes [7], [Corollary IV.5, page 651], [6], [Corollary 5.16, page 51] of certain lengths. More importantly, we circumvent the above no-go theorems without disturbing the one crucial property that cyclic codes have — efficient decoding algorithms. Recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true hence this is a crucial property to have. Cyclicity is a basis dependent property and for codes that we construct, is only evident in the modified symplectic setting. If we were to recast this code using the standard symplectic form, there would be a basis change on the underlying vector space that disturbs cyclicity. Hence, this change of perspective that we have here is crucial not only in the construction, where we had to circumvent the no-go theorems that we mentioned above, but also in the design of efficient decoding algorithm. Theoretically our result is also tight: we have a complete characterization (Theorem 6) of the codes that we construct here.