Correcting Coherent errors with Surface Codes

Question

Following this article: 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!

Answer 1

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 $\theta$.