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

TL;DR

This paper presents optimized quantum error correction schemes for fully correlated quantum channels, improving implementation efficiency and protecting both quantum and classical information with fewer gates.

Contribution

It introduces new error correction schemes for fully correlated channels that require fewer gates and can protect quantum and classical data simultaneously.

Findings

Scheme for odd n uses 3k CNOT gates for quantum data

Scheme for even n protects quantum and classical bits with minimal gates

Implementation demonstrated on multiple quantum computing platforms

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 2 classical bits. The encoding operation $\sigma \otimes \rho \mapsto \Phi(\sigma \otimes \rho)$ can be done by $3k+2$ CNOT gates and a single quibt 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 $2$-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.