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

Abstract

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 $\sigma \otimes \rho \mapsto \Phi (\sigma \otimes \rho )$ only requires 3k CNOT gates (each with one control bit and one target bit). After the encoded state $\Phi (\sigma \otimes \rho )$ goes through the channel, we can apply the inverse operation $\Phi ^{-1}$ 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 $\sigma \in \{|ij{\rangle }{\langle }ij|: i, j \in \{0,1\}\}$ to protect a 2k-qubit state $\rho$ and two classical bits. The encoding operation $\sigma \otimes \rho \mapsto \Phi (\sigma \otimes \rho )$ can be done by $3k+2$ CNOT gates and a single-qubit Hadamard gate. After the encoded state $\Phi (\sigma \otimes \rho )$ goes through the channel, we can apply the inverse operation $\Phi ^{-1}$ 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 two-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.