  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 transversely.
However, the hardness of decoding makes the color codes not suitable as the
best candidate for experimentally feasible implementation of quantum error
correction. Here we propose an Ising model formulation that enables highly
accurate decoding of the color codes. In this formulation, 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 (4.8.8) lattice, we find an error threshold of 10.36(5)% for
bit-flip noise model, 18.47(5)% for depolarizing noise model, and 2.90(4)% for
phenomenological noise model (bit-flip error is located on each of data and
measurement qubits), all of which are higher than the thresholds of existing
efficient decoding algorithms. Furthermore, we verify that the achieved logical
error rates are almost optimal in the sense that they are almost the same as
those obtained by exact optimizations by CPLEX with smaller decoding time in
many cases. Since the decoding process has been a bottleneck for performance
analysis, the proposed decoding method is useful for further exploration of the
possibility of the topological color codes.
