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Assume I have a surface code with distance $d$ and an i.i.d error model with both single qubit depolarization and measurement errors, both with probability $p$.

In this case, one usually repeats the syndrome measurement cycles $d$ times to get the threshold. But what happens if one repeats the syndrome measurement cycles for $m*d$ times, with $2\le m$?

When I simulated this using stim, I found that the threshold decreases as $m$ increases, and the logical error rate increases with $m$. Is this a known phenomenon, and why this happens? Or maybe I have a problem with my simulations? Where can I find a discussion of this topic in the literature?

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he logical error rate increases with m

If you run for longer, there's more time for errors to occur. You need to do a conversion of the per-shot error rate to a per-round error rate or per-$d$-round (per-quop) error rate, e.g. using sinter.shot_error_rate_to_piece_error_rate, if you want to compare different numbers of rounds. Also you want enough rounds to amortize away boundary effects from the start and end of the experiment; like $3 \cdot d$ or $4 \cdot d$ rounds.

I found that the threshold decreases as m increases

You need to use larger values of $d$. If your error unit is a number of rounds linear in $d$, you will see the crossing point where different noise strengths have the same logical error rate vary a lot with $d$ when $d$ is small. For example, when using $r=2d$ the $d=5$ experiment is nearly twice as long as the $d=3$ experiment, whereas the $d=31$ experiment is basically the same length as the $d=29$ experiment.


In any case the threshold is the wrong number to estimate. The threshold is where the code doesn't work; you want to know numbers where the code does work. Behavior near threshold is qualitatively different from behavior well below threshold, where quantum computers will actually run. If you focus on near-threshold behavior, you will learn the wrong lessons about what works well and what works poorly.

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  • $\begingroup$ I am not sure that I understood. In the sentence: "You need to use larger values of d", did you mean that for large $d$, and $t=rd$, I will find the same threshold as for $t=d$, or that I will find another threshold, but I need a large $d$ to find this new value? $\endgroup$ Jul 14, 2023 at 13:01
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    $\begingroup$ @YaronJarach The threshold should be the same, but you need larger values of $d$ to approximate the limit. $\endgroup$ Jul 14, 2023 at 17:03

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