Quantum Codes Do Not Fix Qubit Independent Errors

The goal of this work is to analyse the performance of quantum error correction codes in regard to fixing independent errors in several qubits. 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 restrict our study to the classical 5−qubit quantum error correcting code (proposed by Laflamme and some collaborators), which is able to correct arbitrary errors in a single qubit and is also fault-tolerant. We show that this code does not fix qubit independent errors, even assuming that the correction circuit does not introduce new errors. We also prove, for qubit independent errors, that if the correction circuit of the 5−qubit quantum code detects an error, then the corrected state has central symmetry and, as a consequence, its variance is maximal. We have been able to obtain these results thanks to the high symmetry of the 5−qubit quantum code. Although the calculations needed to extend our proofs to less symmetric codes seem to be extremely complicated, we nevertheless think that the results obtained for the 5−qubit quantum code reveal a general behavior pattern of quantum error correcting codes against qubit independent errors.