Efficient Quantum Error Correction for Fully Correlated Noise

TL;DR

This paper presents efficient quantum error correction schemes tailored for fully correlated noise affecting multiple qubits, demonstrating how to encode data with minimal qubits and constructing circuits for practical implementation.

Contribution

It introduces novel quantum error correction methods for fully correlated noise, detailing encoding schemes for odd and even qubit counts and providing circuit implementations.

Findings

Odd n qubits encode (n-1) data qubits

Even n qubits implement noiseless subsystems for (n-2) data qubits

Constructed quantum circuits for the proposed schemes

Abstract

We investigate an efficient quantum error correction of a fully correlated noise. Suppose the noise is characterized by a quantum channel whose error operators take fully correlated forms given by $\sigma_x^{\otimes n}$, $\sigma_y^{\otimes n}$ and $\sigma_z^{\otimes n}$, where $n>2$ is the number of qubits encoding the codeword. It is proved that (i) $n$ qubits codeword encodes $(n-1)$ data qubits when $n$ is odd and (ii) $n$ qubits codeword implements a noiseless subsystem encoding $(n-2)$ data qubits when $n$ is even. Quantum circuits implementing these schemes are constructed.