Abstract: The quantum capacity is a fundamental bound on the rate of a quantum error correcting code: it gives the maximal number of logical qubits k that can be encoded in n noisy physical qubits, subject to random depolarizing or erasure errors. A major achievement of classical coding theory is the construction of LDPC error correcting codes approaching the classical capacity of symmetric binary channels. In 2017, Lloydt, Shor, Thompson have constructed an asymptotic family of QLDPC stabilizer codes based on graph states that achieve the capacity of the random erasure channel, at the prize of polylog stabilizer weights. On the quantum depolarizing channel, however, things are more complicated and the quantum capacity is not even known explicitly, although the hashing bound provides a strict lower-bound. Here, based on [LST17], we build a family of linear rate log-sparse QLDPC error correcting codes that achieve the hashing bound on the quantum depolarizing channel: R = 1 - h(p) - p log_2(3). Moreover, we show that these codes can be decoded to high threshold using a pure belief propagation algorithm without additional postprocessing. Our two contributions are based on a new graphical model for representing such codes, computing their threshold and performing local decoding.
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