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