  Machine learning techniques are increasingly being used in fundamental
research to solve various challenging problems. Here we explore one such
technique to address an important problem in quantum communication scenario.
While transferring quantum information through a noisy quantum channel, the
utility of the channel is characterized by its quantum capacity. Quantum
channels, however, display an intriguing property called super-additivity of
coherent information. This makes the calculation of quantum capacity a hard
computational problem involving optimization over an exponentially increasing
search space. In this work, we first utilize a neural network ansatz to
represent quantum states and then apply an evolutionary optimization scheme to
address this problem. We find regions in the three-parameter space of qubit
Pauli channels where coherent information exhibits this super-additivity
feature. We characterised the quantum codes that achieves high coherent
information, finding several non-trivial quantum codes that outperforms the
repetition codes for some Pauli channels. For some Pauli channels, these codes
displays very high super-additivity of the order of 0.01, much higher than the
observed values in other well studied quantum channels. We further compared the
learning performance of the Neural Network ansatz with the raw ansatz to find
that in the three-shot case, the neural network ansatz outperforms the raw
representation in finding quantum codes of high coherent information. We also
compared the learning performance of the evolutionary algorithm with a simple
Particle Swarm Optimisation scheme and show empirical results indicating
comparable performance, suggesting that the Neural Network ansatz coupled with
the evolutionary scheme is indeed a promising approach to finding non-trivial
quantum codes of high coherent information.
