Quantum error correction with the semion code

Abstract

We present a full quantum error correcting procedure with the semion code: an off-shell extension of the double-semion model. We construct open-string operators that recover the quantum memory from arbitrary errors and closed-string operators that implement the basic logical operations for information processing. Physically, the new open-string operators provide a detailed microscopic description of the creation of semions at their end-points. Remarkably, topological properties of the string operators are determined using fundamental properties of the Hamiltonian, namely, the fact that it is composed of commuting local terms squaring to the identity. In all, the semion code is a topological code that, unlike previously studied topological codes, it is of non-CSS type and fits into the stabilizer formalism. This is in sharp contrast with previous attempts yielding non-commutative codes.