Variational quantum circuit learning of entanglement purification in multi-degree-of-freedom

Quantum entanglement purification (EP) is a crucial technique for promising the effective function of entanglement channel in noisy large-scale quantum network. The previous EP protocols lack of a general circuit framework and become complicated to design in high-dimensional cases. In this paper, we propose a variational quantum circuit framework and demonstrate its feasibility of learning optimal protocols of EP in multi-degree-of-freedom (DoF). By innovatively introducing the additional circuit lines for representing the ancillary DoFs, e.g. space and time, the parameterized quantum circuit can effectively simulate the scalable EP process. As examples, well-known protocols in linear optics including PSBZ, HHSZ+ and etc., are learnt successfully with high fidelities and the alternative equivalent operations are discovered in low-depth quantum circuit. Our work pays the way for exploring the EP protocols with multi-DoF by quantum machine learning.