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My questions arised basically from playing around with the STIM "getting_started" notebook, regarding the part on the rotated surface code.

There 3d syndrome extraction rounds are performed and decoded to estimate the logical error rate. In particular, this includes finding the threshold due to the different scaling of different distances.

Playing around, I changed the number of rounds to 1, expecting the threshold to vanish (as O(d) rounds are necessary to have confidence in the syndromes). But it did not, it even improved to >1%.

modified STIM code to run different distance codes with only a single round. results of the simulations showing a threshold above 1%

I suspect this is due to the "data qubit measurement" round, which is performed at the end (is this right?). As a result the plot with the correct scaling is not an argument for correct syndrome extraction (which it is not) but for the FT of the syndrome extraction circuit (the circuit is FT but the syndrome can not be trusted).

But now my question is: How would I set up an experiment to check a "valid" syndrome extraction in STIM? Meaning, an experiment that shows that actual d (or less) rounds are necessary.

I can't drop the final data qubit measurement, because then a single error after the last CX of the syndrome extraction could result in an observable flip without being detected. One could avoid placing such errors, but this seems not very elegant to me.

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    $\begingroup$ There is no perfect syndrome extraction round in the example circuits generated by stim. $\endgroup$ Commented Aug 27 at 13:42
  • $\begingroup$ Sorry, I meant the measurement of the data qubits as the last round of syndrome extraction. Of course, this is not "perfect," and I updated the question accordingly. $\endgroup$ Commented Sep 2 at 12:17
  • $\begingroup$ Regarding this question, I found the following discussion also helpful. $\endgroup$ Commented Sep 4 at 8:08

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Meaning, an experiment that shows that actual d rounds are necessary.

$d$ rounds aren't necessary.

In lattice surgery quantum computing, $d$ rounds of syndrome extraction are needed to make the circuit distance of timelike logical errors $d$. In your circuit only spacelike logical errors can occur.

See the discussion here for an explanation of timelike and spacelike errors.

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  • $\begingroup$ Do I understand correctly that this is because my observable is purely spacelike? Therefore, would even a single round of syndrome extraction be enough? $\endgroup$ Commented Sep 2 at 12:23
  • $\begingroup$ Yes correct. Yes $\endgroup$ Commented Sep 2 at 12:55
  • $\begingroup$ Thanks a lot! But then, how would I design a simple experiment to test the more complicated case where I actually need d rounds? Do I need a lattice surgery experiment (or alternatively, moving a patch), or is there a simple way to test it? $\endgroup$ Commented Sep 4 at 8:11
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    $\begingroup$ Giving the answer myself: One must do a stability experiment instead of a memory experiment. $\endgroup$ Commented Sep 4 at 9:00
  • $\begingroup$ @distort8364 Right. You can also get timelike errors in surface code Y basis memory experiments, if you accidentally mirror the topological preparation vs the topological measurement (see figure 16 and figure 14 of arxiv.org/abs/2302.07395 ). $\endgroup$ Commented Sep 4 at 15:40

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