In the phenomenological noise model, we have an approximately 2% error threshold in surface codes. In the circuit-based noise model, the threshold ranges from 0.5% to 0.7%. I'm curious if there's a mapping between these two models. Is it possible to calculate, or more accurately predict, the phenomenological noise model threshold from the circuit-based noise model threshold for a given p error rate? I'm trying to understand the mapping between these two error models. To the best of my knowledge, we can calculate these threshold values numerically, but I am sure there is a mapping between them, and I do not know what it is.
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A back-of-the-envelope calculation can give some intuition: The syndrome extraction circuit of the surface code takes 5 timesteps (or more, depending on which gates are used), so there are 5 times as many error locations. If we simply divide the phenomenological threshold by 5, this would give a threshold of 2/5 = 0.4% for the surface code, which is reasonably close. Codes with high-weight stabilizers can perform very well under phenomenological noise, but will generally perform badly with circuit-level noise due to the many error locations.
As far as I'm aware there is not an exact mapping. A good approach to doing this would be to map the syndrome graph of the circuit level noise model to the syndrome graph of the phenomenological noise model. But this would be tricky to do as there are errors in the circuit level noise model that don't have the same syndrome as any error in the phenomenological noise model.
