Method For Constructing Quantum Code Based On Algebraic Curves

In the process of quantum communication,the quantum bits carrying information are inevitably affected by the external environment,which affects the coherence of the quantum states,causing information errors.In order to avoid the influence of quantum noise and quantum decoherence in the process of quantum communication,the emergence of quantum error correction code is particularly important.A common construction method for quantum error correcting codes is constructed based on classical error correcting codes and their dual codes.Since the code on the algebraic curves has good asymptotic behavior,this paper will use the algebraic curves to construct the quantum error correction code.For a first type one-point algebraic geometry code,the dual code is still the first type of algebraic geometry code,but this feature does not hold for the two-point algebraic geometry code.We need to calculate the parameters of the dual code separately.The specific research content of this paper is as follows:1.A method for constructing symmetric quantum error correcting codes using two-point codes on Hermitian curves is proposed.Firstly,by using the Weierstrass semigroup of Hermitian curve to analyze its Riemann-Roch space structure,the construction method and performance parameters of the classical Hermitian two-point code and its dual code are determined.Then the corresponding quantum Hermitian two-point code is constructed by CSS construction method,and the calculation method of its performance parameters is given.The calculation and verification of the parameters are carried out by way of example.Finally,comparing the quantum Hermitian two-point code with the corresponding one-point code,it is verified that the quantum Hermitian two-point code has better performance than the quantum Hermitian one-point code.2.A method for constructing symmetric quantum error correcting codes using two-point codes on Suzuki curves is proposed.Firstly,by using the Weierstrass semigroup of Suzuki curve to analyze its Riemann-Roch space structure,the construction method and performance parameters of the classical Suzuki two-point code and its dual code are determined.Then the corresponding quantum Suzuki two-point code is constructed by CSS construction method,and the calculation method of its performance parameters is given.The calculation and verification of theparameters are carried out by way of example.Finally,comparing the quantum Suzuki two-point code with the corresponding one-point code,it is verified that the quantum Suzuki two-point code has better performance than the quantum Suzuki one-point code.3.A method for constructing asymmetric quantum error correcting codes using two-point codes on Hermitian curves and Suzuki curves is proposed.The corresponding quantum error correction code is constructed by asymmetric CSS construction method.Then the simulation analysis of the constructed asymmetric quantum code is carried out.The simulation results show that the performance of the constructed asymmetric quantum error correction code will change better with the increase of the asymmetry value.