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 $σ_x^{\otimes n}$, $σ_y^{\otimes n}$, $σ_z^{\otimes n}$. 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 $σ$ to protect the data state $ρ$ in a $2k$-qubit system. The encoding operation $σ\otimes ρ\mapsto Φ(σ\otimes ρ)$ only requires $3k$ CNOT gates (each with one control bit and one target bit). After the encoded state $Φ(σ\otimes ρ)$ goes through the channel, we can apply the inverse operation $Φ^{-1}$ to produce $\tilde σ\otimes ρ$ so that a partial trace operation can recover $ρ$. When $n$ is even and equals $2k+2$, we describe a hybrid quantum error correction scheme using any one of the two classical bits $σ\in \{|ij\rangle \langle ij|: i, j \in \{0,1\}\}$ to protect a $2k$-qubit state $ρ$ and 2 classical bits. The encoding operation $σ\otimes ρ\mapsto Φ(σ\otimes ρ)$ can be done by $3k+2$ CNOT gates and a single quibt Hadamard gate. After the encoded state $Φ(σ\otimes ρ)$ goes through the channel, we can apply the inverse operation $Φ^{-1}$ to produce $σ\otimes ρ$ so that a perfect protection of the two classical bits $σ$ and the $2k$-qubit state is achieved. If one uses an arbitrary $2$-qubit state $σ$, the same scheme will protect $2k$-qubit states. The scheme was implemented using Matlab, Mathematica, Python, and the IBM's quantum computing framework qiskit.