Abstract
In this paper, new conditions on parameters in difference sets are derived to satisfy symplectic inner product, and new constructions of quantum stabilizer codes are proposed from the conditions. The conversion of the difference sets into parity-check matrices is first explained. Then, the proposed code construction is composed of three steps, which are to choose the generators of quantum stabilizer code, to determine the quantum stabilizer groups, and to determine subspace codewords with large minimum distance. The quantum stabilizer codes with various length are also presented to explain the practicality of the code construction. The proposed design can be applied to quantum stabilizer code construction based on combinatorial design.
Highlights
Quantum theory gives the probability of the possible outcomes for a measurement on a physical system [1]
Quantum computers which are based on quantum theory give us the possibility deal on the various tasks such as factoring the large integer number that shows the substantial speed-up in polynomial time over the best classical algorithm [2,3]
After establishment in 1997 [7], quantum stabilizer codes have played a prominent role in quantum error correcting codes (QECCs)
Summary
Quantum theory gives the probability of the possible outcomes for a measurement on a physical system [1]. The modified circulant matrix has been proposed to construct the parity-check matrix and the results for entanglement-assisted quantum error correction codes are explained [8]. In [9], two sub-matrices are proposed to satisfy the constraint of parity-check matrix for quantum stabilizer codes with length seven. Innovative designs of the parity-check matrix have been proposed for LDPC codes with better performances or with easy implementation. The application of combinatoric design on LDPC codes was proposed to increase the girth of the parity-check matrices [12]. New constructions of quantum stabilizer codes based on DSs are proposed. From the suitable DSs, the circulant matrices are designed and used to construct the parity-check matrix.
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