I am following this introduction paper for the surface code theory.
The question:
I am struggling to understand the derivation of the equation (12) of this paper. This equation gives the probability of having an $X_L$ (logical $X$) error for a protected logical qubit after one surface code cycle.
The result is:
$$P_L^s=d \frac{d!}{(d_e-1)!d_e!}p_e^{d_e}.\tag{12}$$
Where $d$ is the array distance, and $d_e=(d+1)/2$.
What I understand:
Let me start by what I understand. In the paper they provide an example of what can cause a logical error. Below is the image the discussion is based on.
As illustrated in (a), the first and third $Z$ measurement qubits are detecting an error. It could be due to an $X$ error on the second and third data qubits (on (b), I call this error $X_2 X_3$), or to an $X$ error to the first and two last data qubits as represented on (c) (I call this error $X_1 X_4 X_5$). Conceptually we cannot know.
However, the situation in (b) is more likely to occur and hence we will assume it is the case. In this case we can apply the $X$ operator on the second and third data qubit and it will fix the error. Now, if actually the situation in (c) occured, such correction would induce a logical $X$ error. Indeed we would apply $X_2 X_3$ (to correct what we believed occured) where an error $X_1 X_4 X_5$ is the one that actually occured. In the end it means that the "net" effect is $X_1 X_2 X_3 X_4 X_5$ which is a logical $X$ operator: hence it introduces a logical error.
I am fine with this example.
Where I struggle to understand:
My struggle start on the precise counting, and generalization of this example on a full surface. The probability to have a logical error is the probability to face an uncorrectable event (such as the example (c) before). Then we have to answer two questions:
- What characterizes an uncorrectable event?
- How much of those events can occur?
I am struggling to understand the answer to those two questions. In the paper they answer the following (I reformulate what I believe they say in my own words).
- For arrays with distance $d$, an uncorrectable event will occur when $(d+1)/2$ qubits errors are mis-identified as $(d-1)/2$ qubits error. Actually there might have more possibilities than that, but such possibilities are the one giving the highest probability of error (we neglect the rest).
- The precise counting of these possibilities gives for a 2D array: $d \frac{d!}{(d_e-1)!d_e!}$.
In the particular example of the 1D array here, I agree that $(d+1)/2$ qubits errors were misidentified as $(d-1)/2$ qubits error. But why would it be a general result? How can I be sure of that?
Assuming the point 1. is correct, I don't understand how we end up with the factor $d \frac{d!}{(d_e-1)!d_e!}$