Parallel decoding of multiple logical qubits in tensor-network codes

Abstract: We consider tensor-network stabilizer codes and show that their tensor-network decoder has the property that independent logical qubits can be decoded in parallel. As long as the error rate is below threshold, we show that this parallel decoder is essentially optimal. As an application, we verify this for the max-rate holographic Steane (heptagon) code. For holographic codes this tensor-network decoder was shown to be efficient with complexity polynomial in n, the number of physical qubits. Here we show that, by using the parallel decoding scheme, the complexity is also linear in k, the number of logical qubits. Because the tensor-network contraction is computationally efficient, this allows us to exactly contract tensor networks corresponding to codes with up to half a million qubits. Finally, we calculate the bulk threshold (the threshold for logical qubits a fixed distance from the code centre) under depolarizing noise for the max-rate holographic Steane code to be 9.4%.