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Suppose I have a quantum circuit with a few measurements (say $N$) on a single qubit. Before all the measurements, I generated calibration circuits and the 2 by 2 calibration matrix for that qubit of interest. However, I wonder after the first mid-circuit measurement, how can I determine if this matrix is still the same? In other words, will one mid-circuit measurement affect the error on the qubit, therefore influencing the following measurement errors (in mid-circuit measurement)?

I wonder if there's a way to test that, and/or generate the calibration circuits and the matrix (with $2^N$ dimensions) in this case.

Please feel free to let me know if anything is unclear in the question. Thanks for the help!

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    $\begingroup$ What do you mean with "the matrix" here? Are you referring to the unitary matrix representing the circuit? But if there are measurements, the overall evolution is not unitary, so you cannot describe it by such a matrix. $\endgroup$
    – glS
    Jan 11 at 8:44
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    $\begingroup$ @glS Thanks for the comment! By that I meant the calibration matrix of the qubit of interest. I don't know if we perform measurements on the qubit multiple times will influence the error rate on that qubit. $\endgroup$
    – ZR-
    Jan 13 at 5:47
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    $\begingroup$ It sounds like you have a specific piece of hardware in mind, but aren't saying what it is. It seems pretty relevant to the question, e.g. maybe your qubits are actually qudits and the measurement process can heat the qubits out of the computational space causing leakage errors. $\endgroup$ Jan 15 at 20:04
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    $\begingroup$ @Craig Gidney Thanks for the comment, I was using the simulator with noise model imported from ibmq casablanca. $\endgroup$
    – ZR-
    Jan 17 at 17:54
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"I wonder after the first mid-circuit measurement, how can I determine if this matrix is still the same?"

I had one of my students do a daily benchmarking of the various IBM chips and part of this project was to monitor the calibration matrix as a function of time.

Unfortunately, the calibration matrix is not a constant with respect to time. It changes from day to day, and from hour to hour, and even on shorter time intervals than that. However, people use it anyway, and from measurement to measurement within the same run of a circuit, you can use a single calibration matrix for the whole process. It's not perfect, but it's a valid and most people don't do this very rigorously anyway (for good reasons, because the quantum error mitigation technique works fairly well anyway, and it was never meant to be perfect anyway).

If you did want to test this rigorously, you could implement a very simple circuit in which you know what the outcome is supposed to be. Make a calibration matrix and then test how well the quantum error mitigation works with that calibration matrix, in terms of giving you the correct output state at the end. Then try the same thing again, but have the circuit contain a mid-circuit measurement in addition to the last one (and as much else being equal as possible). How much worse is the final output state after quantum error mitigation, with that calibration matrix, than for the previous circuit? Then try again with two mid-circuit measurements. Then with three, and eventually with $N$. Does the quantum error mitigation get worse and worse in terms of giving the desired output state at the end? In our experience, we could never come up with any solid trends or results because of how noisy the data would be. If you find things to be similar to what we found, then it's further evidence that you shouldn't try to be overly rigorous with this already approximate (and reasonably well-working) scheme.

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