Entanglement-assisted quantum error-correcting codes based on the circulant permutation matrix

Entanglement-assisted quantum error-correcting (EAQEC) codes, a wider class of stabilizer codes, allow us to construct quantum error-correcting codes from arbitrary classical codes with the help of ebits, i.e. maximally entangled quantum states. However, in the construction of EAQEC codes, it is desirable to use as small number of ebits as possible because of the high cost in the preparation of those entangled states. In addition, it is also desired that the error-correcting capability of classical codes from which EAQEC codes are constructed is as high as possible. In order to solve those problems, we consider in this paper a class of EAQEC codes that are based on classical LDPC codes whose parity check matrices are designed by the circulant permutation matrix. Then we give some sufficient conditions on those parity check matrices so that the obtained EAQEC codes need only one ebits. Moreover, we also clarify the condition on which parity check matrices satisfying above sufficient conditions have no 4-cycle, which is one of the important characteristics for the sum-product decoding to exhibit high error-correcting capability.