A Note on Clifford Stabilizer Codes for Ising Anyons

Abstract

We provide a streamlined elaboration on existing ideas that link Ising anyon (or equivalently, Majorana) stabilizer codes to certain classes of binary classical codes. The groundwork for such Majorana-based quantum codes can be found in earlier works (including, for example, Bravyi (arXiv:1004.3791) and Vijay et al. (arXiv:1703.00459)), where it was observed that commuting families of fermionic (Clifford) operators can often be systematically lifted from weakly self-dual or self-orthogonal binary codes. Here, we recast and unify these ideas into a classification theorem that explicitly shows how q-isotropic subspaces in $\mathbb{F}_2^{2n}$ yield commuting Clifford operators relevant to Ising anyons, and how these subspaces naturally correspond to punctured self-orthogonal codes in $\mathbb{F}_2^{2n+1}$.