Li injection in the surface code

Question

I am trying to reproduce Li-injection method in the surface code (from here). Specifically, I start with d1=7 surface code, perform the injection + two rounds of stabilizer measurements, and check the "acceptance rate". If there were no errors, I continue by d2=7, so I just measure all data qubits and check if there was a logical error.

Specifically, I am using STIM so I actually inject S gate instead of T (my circuit can be found in crumble). I only induce two-qubit gate depolarizing error with probability 0.1%, to match for one point from the paper. Nevertheless, I am not able to reproduce the results.

Going over all possible errors Li mentioned, for T state he counts 6 two-qubits errors from the CNOT of the stabilizer measurement. In the case of the S gate, I should actually have only 5 (because Y on the injected qubit is not an error), so I should have a logical error of 5p/15 = p/3.

Now, my problem is that Li states that these errors are originated solely in the first round of measurement, and these are the only possible errors that leads to a logical error (for first order in p). But why? If, for example, I have an XI error on the second data qubit in the second round - it is not detectable, and leads to a logical error. Indeed, if I assume NO errors in the second round, I get exactly p/3 logical error rate, but acceptance rate of ~ 75% which is much higher than in the paper; on the other hand, if I DO assume errors in the second round (as should be), I get the correct acceptance rate of ~ 59%, but much higher logical error rate ~ 1.8p.

So, my question is two-fold: First, why does Li ignore the possibility of errors in the second round, like the example I gave above? Second, If Li is correct, what do I miss and how do I incorporate it in my simulations?

Answer 1

It looks to me like your ending measurement isn't fault tolerant, and that's the source of the issue not the injection. You seem to be trying to measure the Y operator transversally, but that doesn't work.

If you include 10 rounds instead of 2, it will be much clearer the issue is coming from the end of the circuit instead of the start.

For an easy implementation just insert noiseless MPP instructions for every stabilizer, with a detector declared for each. For a planar fault tolerant Y measurement that would run one a noisy machine, see "Inplace Access to the Surface Code Y Basis". It's for a rotated surface code and you're using unrotated, but the rotated surface code state occurs halfway through the unrotated surface code cycle so you can just trigger it then instead of at the normal cycle boundary.