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I am having trouble understanding what people mean when they say "error threshold."

I understand the answer here but it isn't code specific, it is only dependent on the number of physical qubits $n$ and the distance of the code $d$. For example, any $9$ qubit code with $d = 3$ has the same threshold as the Shor code using this method of calculation.

I realize that the threshold depends on the error model and the previous assumes tacitly that errors occur identically and independently on each physical qubit. However, I am not really sure how people in literature seem to be calculating error thresholds for specific codes, what are they doing?

I have heard many people say "just simulate your code to compute the threshold" (cf here) but I don't really understand this either because it seems like you would just reproduce what we got above?

Eric Kubischta
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Calculating the threshold as proposed in the answer you refer to gives you a worst-case estimate. Namely, in the calculation you are assuming that any error of weight $>t$ results in a logical error. Here $d=2t+1$.

One of the main reasons the surface code has good performance is that although $t+1$ errors can cause a logical error, most errors of weight $t+1$ are correctable. It's even the case that many errors with weight larger than $t+1$ are correctable. Therefore, simulations typically result in higher and more accurate thresholds.

I have heard many people say "just simulate your code to compute the threshold" (cf here) but I don't really understand this either because it seems like you would just reproduce what we got above?

So simulations don't just reproduce the threshold you would get from a calculation.

However, I am not really sure how people in literature seem to be calculating error thresholds for specific codes, what are they doing?

Usually they are doing a monte carlo simulation. They repeat the following steps many times:

  1. Take an error sample from the error model
  2. Calculate the syndrome
  3. Decode the syndrome to find a correction
  4. Check if the correction + error sample is a non-trivial (not equal to $I_L$) logical operator.
Peter-Jan
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