# Correcting Coherent errors with Surface Codes

I wonder about the modeling of coherent error and its effect on syndrome after measurement. They say that applying U on the initial logical state, and stabilizing using syndrome measurement, will collapse the state to $$exp(i\theta_sZ)$$ where $$\theta_s$$ is a new angle.

I feel that a few steps were skipped. I would be happy to see a detailed mathematical explanation that shows how the angle was changed from arbitrary, to one that depends on the syndrome measurement.

Thank you!

• I'll try looking into this - super busy period because it's pre-march meeting
– Lior
Feb 23, 2022 at 13:41

Here is a small example where tried to answer myself.

Consider Bit Flip 3-Repetition code, that protects 1 X error:

$$|0_L>=|000> ; |1_L>=|111>$$

Since it protects $$X$$, it protect also linear combination $$aI+bX$$.

Therefore also $$R_x(\theta)=cos(\theta/2)I-i*sin(\theta/2)X$$

Now let's imagine the following flow:

1. Initialize $$|000>$$
2. Error on qubit #2 of small $$\theta: cos(\theta/2)|000>-i*sin(\theta/2)|010>$$
3. Stabilize by syndrome measurement 2 possibilities:
• measure no error syndrome in probability $$cos(O/2)^2 -> |000>$$
• measure error X syndrome in probability $$sin(O/2)^2 -> |010>$$ and correction to $$|000>$$

Therefore X error with a probability of $$sin(\theta/2)^2$$ is equivalent to a certain error $$(probability=1)$$ of small rotation in a degree of $$\theta$$.