# Convert Coherent Noise to Clifford Errors with Probability on Surface Codes

Following my question about the equivalence of coherent and no coherent error, in surface codes.

Now I understand, it is not equivalent. I tried to read some articles about it, and I couldn't find a straightforward recipe, to convert certain coherent errors in surface code, to Pauli errors that happen in probability (in order to boost simulation time) which will give the same logical errors as it was if the coherent error was simulated.

The thing that I am looking for, is an answer like:

1. If you have a coherent error that happens in your $$n$$ qubits
2. Assume your coherent error is of the form $$E=E_1\otimes...\otimes E_n$$
3. Which can be shown also as decompose to Pauli like $$E=({a_1 I+b_1 X + c_1 Y + d_1 Z})\otimes...\otimes ({a_n I+b_n X + c_n Y + d_n Z})$$
4. You can build a circuit, which will have the same logical error probability if you will use Pauli errors in the probabilities $$p$$ to every Pauli combination, where $$p(Some Pauli Combination)$$ is a function of {$$a_i ,b_i, c_i , d_i$$}

So, how can I choose the proper $$p$$ that will give me a good approximation to simulate the same logical error rate?

• Since these aren’t equivalent error models, I don’t think there is a straightforward way to go from one to the other. Something that is almost what you want is “randomized compiling,” which is a way to convert arbitrary coherent/incoherent errors in your qubits into just Pauli errors Feb 20, 2022 at 16:43
• Feb 20, 2022 at 16:44
• If randomized compiling exist, how is it that there is no equivalent to my model? Feb 20, 2022 at 16:54
• You could randomly compile your surface code circuit. Whatever noise is in the circuit would then be mapped to Pauli noise that you can calculate from the original noise. But this results in a totally different threshold (potentially) from the version that is not randomly compiled. Feb 20, 2022 at 23:07