  In this paper, we propose a novel bipartite entanglement purification
protocol built upon hashing and upon the guessing random additive noise
decoding (GRAND) approach recently devised for classical error correction
codes. Our protocol offers substantial advantages over existing hashing
protocols, requiring fewer qubits for purification, achieving higher
fidelities, and delivering better yields with reduced computational costs. We
provide numerical and semi-analytical results to corroborate our findings and
provide a detailed comparison with the hashing protocol of Bennet et al.
Although that pioneering work devised performance bounds, it did not offer an
explicit construction for implementation. The present work fills that gap,
offering both an explicit and more efficient purification method. We
demonstrate that our protocol is capable of purifying states with noise on the
order of 10% per Bell pair even with a small ensemble of 16 pairs. The work
explores a measurement-based implementation of the protocol to address
practical setups with noise. This work opens the path to practical and
efficient entanglement purification using hashing-based methods with feasible
computational costs. Compared to the original hashing protocol, the proposed
method can achieve some desired fidelity with a number of initial resources up
to one hundred times smaller. Therefore, the proposed method seems well-fit for
future quantum networks with a limited number of resources and entails a
relatively low computational overhead.
