Investigation on Shor Codes as Degenerate Codes but Correct All Single Quantum Errors

In this paper, we deeply investigate the capability of the famous 9-qubit Shor codes, and reveal whether the codes can correct fully single-qubit errors or not. This paper records all syndromes in a table to easily distinguish the correctable errors. We perform a series of computer simulations for quantum communications under the depolarizing channels for messages encoded using Shor codes to evaluate the performance. We found that Shor codes are degenerate codes having three 3-pair of similar syndromes involving
$\mathrm{Z}$
errors, but the codes surprisingly can correct all single qubit errors although three 3-pair of error patterns have the same syndrome. We have successfully revealed the reason for this unique capability. Furthermore, for the given stabilizers of the Shor codes, we provide performance evaluations based on quantum word-error-rate (QWER) according to the syndrome-based error detection and error correction.