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I have been learning about quantum error correction using the zero-noise extrapolation method from this paper and have been pleased with the results. This method takes advantage of the fact that the quantum gate noise is predictable as seen, for example, in the plot below. Here, I have taken a simple 2-qubit circuit given by qc.x(0), qc.h(0), qc.cx(0,1), qc.x(0) and added noise to it in the form of pairs of CNOT gates (a noisey identity matrix). I am plotting the $|01\rangle$ solution for this circuit as a function of the number of CNOT pairs that I used. This is done using the qiskit noise model with the IBMQ-montreal gate error rates. Note that the real solution is 0.5.

enter image description here

Clearly, the noise follows a very predictable relationship which to me is unexpected. Instead, what I expected was the noise to be scattered evenly across the real solution (0.5) with the amount of scatter being proportional to the number of gates, as shown in the made up plot below.

enter image description here

My question is, why does the noise follow the predictable curve instead of a random scattering about the real solution? And a follow up question, for an arbitrary circuit, how can I predict what the curve will look like?

Update: I ran my noisy circuits on the real IBMQ-athens machine to see if it is anywhere close to the simulated results. The plot is below is meant to be the same as the first plot (except the maximum depth is a less because I could not exceed 900).

enter image description here

Clearly the noise is no where near as organized as in the simulated version. What I don't know is if that was a failure on my part to simulate the noise correctly, or if it is a problem of the qiskit noise simulator itself.

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    $\begingroup$ Is the first plot coming from actual hardware or is it from using a noise model? $\endgroup$
    – KAJ226
    Nov 7, 2020 at 23:47
  • $\begingroup$ The first plot is from the qiskit noise model where I used the function NoiseModel.from_backend. $\endgroup$
    – thespaceman
    Nov 8, 2020 at 0:33
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    $\begingroup$ I think that although the noise model does try to mimic the behavior of the hardware, it is still kinda fixed... what I mean by that is that on real hardware, the noise fluctuates from run to run somewhat. The gate fidelity changes from run to run rather than stay fixed like it would have with the noise model. $\endgroup$
    – KAJ226
    Nov 8, 2020 at 0:37
  • $\begingroup$ Follow up question, why is the noise model so different compared with the noise from a real QC (plot 1 with my plot 3 respectively). $\endgroup$
    – thespaceman
    Nov 9, 2020 at 18:05
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    $\begingroup$ The noise on the hardware fluctuates randomly within certain range from run to run. The noise model is only meant to represent the hardware noise model in a simple sense. Now, what you can do is to do more shots... This will smoothen out the curve you have more. IBM has max number of shots of 8192 set on their device to make sure people don't take up all the time, but you can submit many jobs and average out the results. $\endgroup$
    – KAJ226
    Nov 9, 2020 at 19:22

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The curve you are observing seems intuitive to me, since it shows how decoherence works. As you add more gates the state of the qubits tend to the ground state asymptotically .

enter image description here

You can probably use T1 and T2 to predict the curve.

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    $\begingroup$ I'm far from being an expert, so converting this answer in a community wiki to allow others to chip in. $\endgroup$
    – luciano
    Nov 8, 2020 at 0:04
  • $\begingroup$ Interesting, so the asymptotic relationship is given by decoherence and not gate noise. Would I be correct in saying that if I isolate the gate noise, that it should look something like my second plot? $\endgroup$
    – thespaceman
    Nov 8, 2020 at 0:39
  • $\begingroup$ Just a note, if you can edit your response to answer my follow up question then I can accept yours as the official answer. Thanks! $\endgroup$
    – thespaceman
    Nov 10, 2020 at 18:26
  • $\begingroup$ well.. I'm not fully sure what you mean with "gate noise" in this context. Hopefully, somebody else more knowledgeable might jump in. $\endgroup$
    – luciano
    Nov 10, 2020 at 21:46
  • $\begingroup$ Okay, understandable and I appreciate your help. The noise I am referring to is the noise due only to gate failure without accounting for decoherence in anyway. $\endgroup$
    – thespaceman
    Nov 10, 2020 at 23:16

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