Error correction schemes for fully correlated quantum channels protecting both quantum and classical information

We study efficient quantum error correction schemes for the fully correlated channel on an $n$-qubit system with error operators that assume the form ${\sigma }_x^{\otimes n}$, ${\sigma }_y^{\otimes n}$, ${\sigma }_z^{\otimes n}$. Previous schemes are improved to facilitate implementation. In particular, when $n$ is odd and equals $2k+1$, we describe a quantum error correction scheme using one arbitrary qubit $\sigma$ to protect the data state $\rho$ in a $2k$-qubit system. The encoding operation $\sigma \otimes \rho \mapsto \Phi(\sigma \otimes \rho)$onlyrequires$3k$ CNOT gates (each with one control bit and one target bit). After the encoded state $\mathrm{\Phi }(\sigma \otimes \rho )$ goes through the channel, we can apply the inverse operation ${\mathrm{\Phi }}^{-1}$ to produce $\tilde{\sigma }\otimes \rho$ so that a partial trace operation can recover $\rho$. When $n$ is even and equals $2k+2$, we describe a hybrid quantum error correction scheme using any one of the two classical bits $\sigma \in {|ij\rangle \langle ij|: i, j \in {0,1}}$toprotecta$2k$-qubitstate$\rho$and2classicalbits\mathrm{.}Theencodingoperation$\sigma \otimes \rho \mapsto \Phi(\sigma \otimes \rho)$canbedoneby$3k+2$ CNOT gates and a single quibt Hadamard gate. After the encoded state $\mathrm{\Phi }(\sigma \otimes \rho )$ goes through the channel, we can apply the inverse operation ${\mathrm{\Phi }}^{-1}$ to produce $\sigma \otimes \rho$ so that a perfect protection of the two classical bits $\sigma$ and the $2k$-qubit state is achieved. If one uses an arbitrary $2$-qubit state $\sigma$, the same scheme will protect $2k$-qubit states. The scheme was implemented using Matlab, Mathematica, Python, and the IBM’s quantum computing framework qiskit.