Parallel decoding of multiple logical qubits in tensor-network codes

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. Holographic codes are interesting examples of tensor-network codes, so we first test out this parallel decoding strategy on the max-rate holographic Steane (heptagon) code. For holographic codes with a constant number of logical qubits, the 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. We then calculate the bulk threshold (the threshold for logical qubits a fixed distance from the code center) under depolarizing noise for the max-rate holographic Steane code to be 9.4%. We also introduce some further holographic error-correcting codes and calculate their thresholds under depolarizing noise. One example is based on an 11-qubit code due to Gottesman, which has asymptotic rate 0.114 and threshold 13.8%. Another code we consider is an asymptotically zero-rate holographic code, which performs extremely well under dephasing noise in the X, Y, or Z basis with a threshold of 50%.