Quantum codes and symplectic matroids

The correspondence between linear codes and representable matroids is well known. But a similar correspondence between quantum codes and matroids is not known. We show that representable symplectic matroids over a finite field F q correspond to Fq-linear quantum codes. This connection is straightforward but it does not appear to have been made earlier in literature. This correspondence is made through isotropic subspaces. We show that Calderbank-Shor-Steane (CSS) codes correspond to homogenous symplectic matroids while graph states, which figure so prominently in measurement based quantum computation, correspond to a special class of symplectic matroids, namely Lagrangian matroids. This association is useful in that it enables the study of symplectic matroids in terms of quantum codes and vice versa. Furthermore, it has application in the study of quantum secret sharing schemes. © 2014 IEEE.