Recursive expansion of Tanner graph: a method to construct stabilizer codes with high coding rate

Abstract

Quantum stabilizer codes face the problem of low coding rate. In this article, following the idea of recursively expanding Tanner graph proposed in our previous work, we try to construct new stabilizer codes with high coding rate, and propose XZ-type Tanner-graph-recursive-expansion (XZ-TGRE) code and Tanner-graph-recursive-expansion hypergraph product (TGRE-HP) code. XZ-TGRE code have zero asymptotic coding rate, but its coding rate tends to zero extremely slowly with the growth of code length. Under the same code length, its coding rate is much higher than that of surface code. The coding rate of TGRE-HP is the constant 0.2, which is the highest constant coding rate of stabilizer codes to our best knowledge. We prove that the code distance of XZ-TGRE code scales as $O(log(N))$, and that of TGRE-HP code scales as $O(\log \sqrt{N})$, where $N$ is the code length. Moreover, the code capacity noise threshold of XZ-TGRE code is around 0.078, and that of TGRE-HP code is around 0.096. This articles shows that the idea of recursively expanding Tanner graph might have potential to construct quantum codes with good performance.