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We study efficient quantum error correction schemes for the fully correlatedchannel 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 schemeusing one arbitrary qubit $\sigma$ to protect the data state $\rho$ in a$2k$-qubit system. The encoding operation $\sigma \otimes \rho \mapsto\Phi(\sigma \otimes \rho)$ only requires $3k$ CNOT gates (each with one controlbit and one target bit). After the encoded state $\Phi(\sigma\otimes \rho)$goes through the channel, we can apply the inverse operation $\Phi^{-1}$ toproduce $\tilde \sigma \otimes \rho$ so that a partial trace operation canrecover $\rho$. When $n$ is even and equals $2k+2$, we describe a hybridquantum error correction scheme using any one of the two classical bits $\sigma\in \{|ij\rangle \langle ij|: i, j \in \{0,1\}\}$ to protect a $2k$-qubit state$\rho$ and 2 classical bits. The encoding operation $\sigma \otimes \rho\mapsto \Phi(\sigma \otimes \rho)$ can be done by $3k+2$ CNOT gates and asingle quibt Hadamard gate. After the encoded state $\Phi(\sigma\otimes \rho)$goes through the channel, we can apply the inverse operation $\Phi^{-1}$ toproduce $\sigma \otimes \rho$ so that a perfect protection of the two classicalbits $\sigma$ and the $2k$-qubit state is achieved. If one uses an arbitrary$2$-qubit state $\sigma$, the same scheme will protect $2k$-qubit states. Thescheme was implemented using Matlab, Mathematica, Python, and the IBM's quantumcomputing framework qiskit.

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