# How to know exactly error threshold of surface code in terms of number of physical qubit

I asked some question about surface code properties and I recommended following research paper from what I asked before. Research paper : https://journals.aps.org/pra/pdf/10.1103/PhysRevA.86.032324

In section VII in this paper, they mentioned about error correction by using Edmond's minimum-weight perfect-matching algorithm. Also they mentioned that it worked perfectly for sufficiently sparse errors, but begins to fail as the error density increases and as the length of the error chains increases. In this sentences, I really wants to know how much degree (numerically) of each of them should be to make this surface code as defected code that cannot be used anymore even error correction is worked.

1. sparse errors

2. error density increase

3. length of error chain

• There are multiple questions here and not all of them are clear. I think it's clearer to open a different question for each individual question. Jan 18, 2023 at 8:48
• The question in the title is a bit confusing. The unit of a codes threshold is the probability that an error occures. Jan 18, 2023 at 8:50
• I will modify it and open question for you. Thank you for your comments
– 김동민
Jan 18, 2023 at 9:41

In order for you to know the dependence of the probability threshold with respect to the number of qubits for the planar code, which is the one considered in Fowler's review, which I will be refering in this answer, it is important to understand the concept of distance $$d$$.
As seen in FIG. 3, a number of $$X$$ operators in a horizontal line moving through the entire surface code yield a logical $$X$$ operator in the surface code. For the case of the figure, the minimum number of $$X$$ and $$Z$$ operators required for such an operation is 5, thus the distance of the code is 5. A distance of a code defines that it can corrects all errors of weight $$(d-1)/2$$ or lower. For the case of the MWPM decoding method, an example is provided in FIG. 5, where for a distance-4 repetition code, a syndrome can correspond to two weight two physical errors.