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The threshold theorem is valid provided that the error rate( per gate or per unit time), be under a constant threshold. But I have not read or known any proof about the constancy of theshold.

Is “the assumption of constant per gate error probability qq.” proved, practical, or valid? In other word, any quantum theory or experiments has infered or verified it?

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    $\begingroup$ It’s the other way around. There’s no viable quantum theory that is inconsistent with it. Google’s experiment shows straight-line fidelity as expected. The burden isn’t on this hypothesis, but rather on the alternate hypothesis. $\endgroup$ Commented Sep 17, 2022 at 2:58
  • $\begingroup$ @MarkS, which google's experiment? $\endgroup$ Commented Sep 17, 2022 at 6:13
  • $\begingroup$ The Sycamore random circuit sampling experiment. This showed that as you increase depth, fidelity decreases log-linearly when plotted against circuit depth. $\endgroup$ Commented Sep 17, 2022 at 13:47
  • $\begingroup$ related: cstheory.stackexchange.com/q/51851/29288 $\endgroup$
    – glS
    Commented Sep 18, 2022 at 17:04

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The quantum threshold theorem only requires the noise to be below a certain threshold. Consistency is not required. So I don’t think there’s any assumption made here.

However, an assumption that IS made, for instance in D. Aharonov and M. Ben-Or’s paper on fault-tolerance, is that noise is independent at each step:

Between the time steps, we add the noise process, which is a probabilistic process: each qubit or gate undergoes a fault with independent probability η per step, and η is referred to as the noise rate.

I’ve also heard Peter Shor mention that in one of his lectures on error correction. He continues to say that we have no reason to believe that nature “conspires” against us, and that we can assume that errors are independent (I’m paraphrasing).

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    $\begingroup$ see this "......provided $p$ is below some constant threshold, $ p<p_{\rm {th}} $, and given reasonable assumptions about the noise in the underlying hardware." , and this "....Surprisingly, the quantum threshold theorem shows that if the error to perform each gate is a small enough constant, one can perform arbitrarily long quantum computations to arbitrarily good precision, " $\endgroup$ Commented Sep 18, 2022 at 2:30
  • $\begingroup$ But I vote up your answer $\endgroup$ Commented Sep 18, 2022 at 2:31
  • $\begingroup$ Again, I think that the key here is that noise is BELOW some constant. So even if the error rate varies for certain scenarios, we just have to take some constant which is higher than all of them and we should be good. $\endgroup$ Commented Sep 18, 2022 at 5:47

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