A fast renormalization-group tensor-network decoder for planar quantum low-density parity-check codes

We present a fast and Bayes-optimal-approximating tensor network decoder for planar quantum

LDPC codes based on the tensor renormalization group algorithm, originally proposed

by Levin, and Nave. By precomputing the renormalization group flow for the null syndrome,

we need only recompute tensor contractions in the causal cone of the measured syndrome at the

time of decoding. This allows us to achieve an overall runtime complexity of O(pnχ6) where p is

the depolarizing noise rate, and χ is the cutoff value used to control singular value decomposition

approximations used in the algorithm. We apply our decoder to the surface code in the code capacity

noise model and compare its performance to the original matrix product state (MPS) tensor network

decoder introduced by Bravyi, Suchara, and Vargo. The MPS decoder has a p-independent runtime

complexity of O(nχ3) resulting in significantly slower decoding times compared to our algorithm in

the low-p regime.