On The Design of Non-CSS Quantum Error Correction Codes with High Quantum Information

In this paper, we propose quantum [[10, 10, 2]] stabilizer codes constructed from the existing quantum [[5, 5, 1]] Hamming codes. The proposed quantum [[10, 10, 2]] codes satisfy the symplectic inner product (SIP) to ensure stabilizer generators commute each other for successful detections and corrections. A comprehensive end-to-end construction from parity-check matrix (PCM) of classical codes is provided to ensure quantum error correction codes (QECC) with higher information. We perform a series of computer simulations to evaluate proposed quantum [[10, 10, 2]] codes in terms of quantum word error rate (QWER) performances under the depolarizing channels. The results confirm that the proposed quantum [[10, 10, 2]] codes can successfully detect and correct all single quantum bit (qubit) errors, which is validated by the curve of QWER performances that coincide with the theoretical QWER performances. The results are expected to contribute to the development of QECC in quantum computing and communication systems.