Tensor-Network Codes

We introduce tensor-network stabilizer codes which come with a natural tensor-network decoder. These codes can correspond to any geometry, but, as a special case, we generalize holographic codes beyond those constructed from perfect or block-perfect isometries, and we give an example that corresponds to neither. Using the tensor-network decoder, we find a threshold of 18.8% for this code under depolarizing noise. We show that, for holographic codes, the exact tensor-network decoder (with no bond-dimension truncation) has polynomial complexity in the number of physical qubits, even for locally correlated noise, making this the first efficient decoder for holographic codes against Pauli noise and, also, a rare example of a decoder that is both efficient and exact.