A Note on Clifford Stabilizer Codes for Ising Anyons
Sanchayan Dutta
TL;DR
The paper tackles the problem of designing robust quantum codes for Ising anyons by bridging Majorana stabilizer constructions with classical binary codes. It introduces q-isotropic subspaces of \\mathbb{F}_2^{2n} as the building blocks for Clifford stabilizer codes and proves a one-to-one correspondence with punctured self-orthogonal codes in \\mathbb{F}_2^{2n+1}, enabling a purely classical viewpoint on quantum distance through dual distances. The authors leverage the Kuperberg-Weaver quantum-metric framework to classify error operators and show how Clifford Hamming codes arise from duals of classical Hamming codes, achieving distance-3 protection for even Clifford errors. By tying parity-preserving and parity-flipping errors to commuting even operators and odd Majorana products, the work unifies prior results and provides a practical route to construct families of codes with growing length, good distance, and nonzero rate via standard coding-theory bounds. The findings strengthen the toolbox for Majorana-based quantum architectures, offering a pathway to parity-aware, topological error correction grounded in well-understood classical codes.
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 $\mathbb{F}_2^{2n+1}$.