  The error threshold of a one-parameter family of quantum channels is defined
as the largest noise level such that the quantum capacity of the channel
remains positive. This in turn guarantees the existence of a quantum error
correction code for noise modeled by that channel. Discretizing the
single-qubit errors leads to the important family of Pauli quantum channels;
curiously, multipartite entangled states can increase the threshold of these
channels beyond the so-called hashing bound, an effect termed superadditivity
of coherent information. In this work, we divide the simplex of Pauli channels
into one-parameter families and compute numerical lower bounds on their error
thresholds. We find substantial increases of error thresholds relative to the
hashing bound for large regions in the Pauli simplex corresponding to biased
noise, which is a realistic noise model in promising quantum computing
architectures. The error thresholds are computed on the family of graph states,
a special type of stabilizer state. In order to determine the coherent
information of a graph state, we devise an algorithm that exploits the
symmetries of the underlying graph, resulting in a substantial computational
speed-up. This algorithm uses tools from computational group theory and allows
us to consider symmetric graph states on a large number of vertices. Our
algorithm works particularly well for repetition codes and concatenated
repetition codes (or cat codes), for which our results provide the first
comprehensive study of superadditivity for arbitrary Pauli channels. In
addition, we identify a novel family of quantum codes based on tree graphs. The
error thresholds of these tree graph states outperform repetition and cat codes
in large regions of the Pauli simplex, and hence form a new code family with
desirable error correction properties.
