Quantum codes do not fix qubit independent errors

TL;DR

This paper proves that the 5-qubit quantum error correcting code cannot effectively correct qubit independent errors, showing that such errors remain unmitigated even with ideal correction circuits, due to the code's symmetry.

Contribution

The paper demonstrates that the 5-qubit quantum code does not reduce error variance for qubit independent errors, revealing limitations of this code and suggesting a general behavior pattern.

Findings

The 5-qubit code does not reduce error variance for qubit independent errors.

Error correction with the 5-qubit code results in maximum variance when errors are detected.

Symmetry of the code plays a key role in its inability to fix certain errors.

Abstract

In this work we prove that the 5-qubit quantum error correcting code does not fix qubit independent errors, even assuming that the correction circuit does not introduce new errors. We say that a quantum code does not fix a quantum computing error if its application does not reduce the variance of the error. We also prove for qubit independent errors that if the correction circuit of the 5-qubit quantum code detects an error, the corrected state has central symmetry and, as a consequence, its variance is maximum. We have been able to obtain these results thanks to the high symmetry of the 5-qubit quantum code and we believe that the necessary calculations for less symmetric codes are extremely complicated but that, despite this, the results obtained for the 5-qubit quantum code reveal a general behavior pattern of quantum error correcting codes against qubit independent errors.