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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2\mathrm{D}$</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(0.274\pm 0.010)\pi$</tex> for the surface code and a threshold of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(0.252\pm 0.001)\pi$</tex> 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.