Improving threshold for fault-tolerant color code quantum computing by flagged weight optimization

Quantum error correction (QEC) is an essential component for realizing fault-tolerant quantum computation. Color codes are the promising QEC codes because they have an advantage against surface codes in that all Clifford gates can be implemented transversally. However, the threshold of color codes under circuit-level noise is relatively low mainly because the measurement of high-weight stabilizer generators causes an increase in the circuit depth, and thus substantial errors are introduced. This makes color codes not the best candidate. Here, we propose a method to suppress the impact of such errors by optimizing weights of decoders using flag qubits and reducing the circuit depth using cat states. We set the weights of the decoder based on the conditional error probability conditioned on the information of flag qubits. In numerical simulations, we improve the threshold of the (4.8.8) color code under standard circuit-level noise from 0.14% to around 0.3% using the integer programming decoder. Moreover, the achieved logical error rates at low physical error rates are almost one order of magnitude lower than those of the (4.8.8) color codes that employ the existing QEC schemes. This method can also be applied to other weight-based decoders, making the color codes more promising for the candidate of experimental implementation of QEC. Furthermore, one can utilize this approach to improve the threshold of wider classes of QEC codes, such as high-rate quantum LDPC codes.