Highly accurate decoder for topological color codes with simulated annealing

Quantum error correction is an essential ingredient for reliable quantum computation for theoretically provable quantum speedup. Topological color codes, one of the quantum error-correction codes, have an advantage against the surface codes in that all Clifford gates can be implemented transversally. However, hardness of decoding makes the color codes not suitable as the best candidate for experimentally feasible implementation of quantum error correction. Here we propose a highly accurate decoding scheme for the color codes using simulated annealing. In this scheme, we map stabilizer operators to classical spin variables to represent an error satisfying the syndrome. Then we construct an Ising Hamiltonian that counts the number of errors and formulate the decoding problem as an energy minimization problem of an Ising Hamiltonian, which is solved by simulated annealing. In numerical simulations on the square-octagon lattice, we find an error threshold of 10.5% for bit flip noises and 18.3% for depolarizing noises, both of which are higher than the thresholds of existing decoding algorithms. Furthermore, we verify that the achieved logical error probabilities are almost optimal in the sense that they are almost the same as those obtained by exact optimizations by CPLEX with much smaller decoding time. Since the decoding process has been a bottleneck for performance analysis, the proposed decoding method is useful for further exploration of possibility of the topological color codes.