Linear programming decoder for hypergraph product quantum codes

We introduce a decoder for quantum CSS codes that is based on linear programming. Our definition is a priori slightly different from the one proposed by Li and Vontobel as we have a syndrome oriented approach instead of an error oriented one, but we show that the success condition is equivalent. Although we prove that this decoder fails for quantum codes that do not have good soundness property (i.e., having large errors with syndrome of small weight) such as the toric code, we obtain good results from simulations. We run our decoder for hypergraph products of two random LDPC codes, showing that it performs better than belief propagation, even combined with the small-set-flip decoder that can provably correct a constant fraction of random errors.