Evolving graph codes with improved error thresholds for Pauli channels

Computing the quantum capacity of a quantum channel is challenging due to the phenomenon of superadditivity of coherent information. For a one-parameter family of quantum channels, such as the depolarizing channel, a closely related problem is that of determining the error threshold, defined as the largest noise level up to which reliable quantum error correction against i.i.d. noise remains possible. Graph states, which form a subclass of stabilizer states, have previously been shown to yield quantum codes with thresholds surpassing those of repetition codes for Pauli channels. In this work we propose the use of a genetic algorithm to discover new families of graph states that have a higher threshold than repetition codes for Pauli channels. Our findings include, for the depolarizing channel, a graph state exhibiting a threshold higher than the 5-qubit repetition code with only 10 channel copies. Additionally, we present novel families of graph-state codes for the BB84 channel and the $2$-Pauli channel achieving higher thresholds than repetition codes. These results expand the set of graph-structured quantum codes that have positive coherent information beyond the hashing bound for special classes of Pauli channels.