Quantum punctured Reed-Muller codes

Quantum technology is a candidate for the sixth generation of telecommunications (6G) in 2030. This paper proposes a new quantum coding scheme in the class of Calderbank-Shor-Steane (CSS) codes based on the Reed-Muller codes, since Reed-Muller codes are one of provably classical Shannon Capacity achieving codes. This paper keeps the proposed new codes simple by making dual-containing CSS codes having Hx = Hz, such that no further search is required for Hx or Hz. The proposed quantum Reed-Muller codes are designed from the classical Reed-Muller codes with puncturing to obtain new codes with block-length of n = 7 qubits capable of error correcting t = 1 qubit error and quantum information of k = 1 qubit, of which the extension to other block-lengths are expected to be straightforward. This paper performs a series of computer simulations to evaluate the quantum word error rate (QWER) performances under the depolarizing channels. The proposed quantum Reed-Muller codes satisfy the stabilizer codes requirement, where the diagonal element of symplectic inner product (SIP) is kept zero to guarantee a successful error correction. The proposed codes have a structure with a length close to that of the perfect quantum codes. Our results confirm that the stabilizers of the proposed codes can extract the syndrome and distinguish all error patterns of any single qubit error. This paper also confirm that the simulated performances are agreed by the theoretical QWER performances indicating that the both coding design and simulation environment are correct. These results confirm that the classical Reed-Muller codes can be used directly as QECC by puncturing the parity check matrix H. The results of this paper are expected to open new ground on the invention of new quantum codes derived from the classical codes.