Simple scheme for encoding and decoding a qubit in unknown state for various topological codes

Justyna Łodyga, Paweł Mazurek, Andrzej Grudka, Michał Horodecki

Published 2014-04-09, updated 2014-11-29Version 2

We present a scheme for encoding and decoding an unknown state for CSS codes, based on syndrome measurements. We illustrate our method by means of Kitaev toric code, defected-lattice code, topological subsystem code and Haah 3D code. The protocol is local whenever in a given code the crossings between the logical operators consist of next neighbour pairs, which holds for the above codes. For subsystem code we also present scheme in a noisy case, where we allow for bit and phase-flip errors on qubits as well as state preparation and syndrome measurement errors. Similar scheme can be built for two other codes. We show that the fidelity of the protected qubit in the noisy scenario in a large code size limit is of $1-\mathcal{O}(p)$, where $p$ is a probability of error on a single qubit. Regarding Haah code we provide noiseless scheme, leaving the noisy case as an open problem.