Quantum Codes and Symplectic Matroids

TL;DR

This paper establishes a novel correspondence between symplectic matroids and quantum codes, linking matroid theory with quantum error correction and quantum secret sharing.

Contribution

It introduces the first known connection between representable symplectic matroids and quantum codes, including CSS and graph states, through isotropic subspaces.

Findings

Symplectic matroids correspond to $ extbf{F}_q$-linear quantum codes.

CSS codes are equivalent to homogeneous symplectic matroids.

Graph states relate to Lagrangian symplectic matroids.

Abstract

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 $\mathbb{F}_q$ correspond to $\mathbb{F}_q$-linear quantum codes. Although this connection is straightforward, it does not appear to have been made earlier in literature. The correspondence is made through isotropic subspaces. We also show that the popular Calderbank-Shor-Steane (CSS) codes are essentially the homogenous symplectic matroids while the 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.