Optimal Decoding of 2D Compass Codes under Coherent Noise
Coherent noise on the physical qubits can lead to coherent error on the logical qubits in a quantum error-corrected system, even if the best recovery operator is found. An optimal decoder under coherent noise finds the minimal angle of the logical, coherent rotation for a given physical rotation angle under the uniform, coherent noise model. We show that one can optimally decode 2D compass codes without an explicit optimal decoder in polynomial complexity by simply flipping the sampled logical angle using a Majorana simulator and a sub-optimal decoder. Estimating the threshold requires tools different from MWPM decoding. As the infidelity curves don't intersect, we use finite-size scaling to evaluate optimal decoding performance by estimating the logical infidelity at the infinite distance limit. We motivate our finite size scaling ansatz based on repetition code-like compass codes for which we can exactly calculate the logical channel up to a distance of up to 1021. Under optimal decoding, our finite-size scaling methods predict a threshold of (0.274 ± 0.010)π for the surface code and a threshold of (0.252 ± 0.001)π for the m=2 Z-stacked Shor code. We compare our results to previous predictions and find a slight deviation in the case of the Z-stacked Shor code and a slight decrease in the case of the surface code.
