Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance

We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, distance scaling as n(epsilon) for epsilon > 0 and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate k/n of 13/72 = 0.180... and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime.

Automated summary

In this study, we construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate and distance scaling. We propose efficient decoding schemes based on tessellations of closed, four-dimensional, hyperbolic manifolds. Our contribution lies in the construction of suitable manifolds using finite presentations of Coxeter groups and their linear representations over Galois fields. We establish a lower bound on the encoding rate and show its tightness for the constructed examples. Numerical simulations indicate that parallelizable decoding schemes with low computational complexity achieve high performance, even in the presence of syndrome noise.