高维量子低密度奇偶校验码纠缠度

Entanglement degree of high-dimensional quantum low-density parity check codes

量子纠错与量子计算是量子信息科学坚实的基础和重要的组成部分.在实际应用中,如大气传输中的量子通信,将需要多种数学运算,其中包括量子纠错码.量子纠错码可以抵抗噪声,但由于构造量子纠错码依赖于量子纠缠,因此被认为是困难的.利用图态解决码字纠缠度是一个很有前途的解决方案,但高维图态构造起来仍有诸多困难,上述困难可以巧妙地通过码字纠缠的上界和下界来解决.本文根据稳定子码循环差集的特性和经典低密度奇偶校验(low-density parity check,LDPC)码的U和B组合,构造了高维量子低密度奇偶校验(quantum low-density parity check,QLDPC)码.通过计算新码元的非Z型生成元并求出其最小数目得到新码元的纠缠上界;再计算新码校验矩阵的秩作为纠缠下界.当码字纠缠上界和下界不同时,利用机器学习中的学习向量量化(learning vector quantization,LVQ)算法可同时求得码字纠缠度和编码复杂度,以此推得它们之间的关系.在计算运行速度方面,对比拉格朗日乘数法中的迭代算法,LVQ算法运行速度提高了37:68%,而且在稳定性和精度方面,LVQ算法的性能优于拉格朗日乘数法中的迭代算法.本文在量子码字纠缠度的测量中迈出了重要的一步,为设计具有更高译码效率的量子纠错码提供了帮助.

The important part of quantum information science is quantum error-correction and quantum computation.In the practical application,some methods of quantum communication,such as the quantum communication based on the atmosphere transmission,require a mix of mathematics-potentially to solve the problem.The quantum error-correction code is an efficient method to calculate quantum communication.However,since the construction of quantum error-correction codes depends on the degree of quantum entanglement,it is difficult to be widely used in practice.At present,the graph state is a promising solution to reduce the degree of codewords entanglement.But there are still many difficulties in constructing the high-dimensional graph state.More importantly,the above difficulties can be skillfully solved by the upper and lower bounds of codewords entanglement.Based on the characteristics of the cyclic difference set of stabilizer codes and the combination of U and B of classical low-density parity check(LDPC)codes,a high-dimensional quantum low-density parity check(QLDPC)code is constructed creatively.By calculating the characteristics of the non-Z-type generators of the new codewords,the minimum number is obtained and the entanglement upper bound of the new code is acquired.And then,the rank of the check matrix of the new code is calculated as the lower bound of entanglement.When the upper bound and lower bound of entanglement are different,the learning vector quantization(LVQ)algorithm in machine learning can be used to simultaneously obtain the codewords entanglement and the coding complexity of the code,in which the relationship between them can be derived.In terms of calculating the running speed,compared with the iterative algorithm in the Lagrange multiplier method,the running speed of LVQ algorithm is increased by 37:68%.In this paper,a measurement of quantum codewords entanglement is proposed,which supports a helpful role in the design of quantum error-correction codes with higher decoding efficiency.