Topological quantum error correction in fractal dimensions II: decoding and threshold estimation

We investigate quantum error correction with topological codes on 3D fractal lattices with Hausdorff dimension 3-δ. We show concrete decoding schemes for correcting the bit-flip and phase errors. For the phase errors, we have developed both a minimum-weight-perfect-matching decoder and a cluster decoder on the 3D fractal lattice. For the bit-flip errors, we have developed a particular type of local cellular-automaton decoder, the sweep decoder, for the 3D fractal code. We also show that bit-flip errors can be corrected in a single shot. For all these decoders, we have numerically estimated the corresponding error threshold using Monte Carlo simulations.