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Given that the error mitigation matrix (meas_fitter.filter) does not vary so much for a given backend and number of qubits, then what are the advantages and disadvantages in the determination of the calibration matrix each time that we do an experiment and mitigate its error? Is any reliable error mitigation matrix available for us to use without having to calculate it ourselves? Thanks.

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Yes, if we have fixed backend, number of qubits, and noise model (e.g., Basic device noise model in https://qiskit.org/documentation/stubs/qiskit.providers.aer.noise.NoiseModel.html#qiskit.providers.aer.noise.NoiseModel), we would have a fixed calibration matrix. I think the advantage is that once we have this calibration matrix, we can use it to perform measurement error mitigation on any new experiment as long as conditions (e.g., backend, number of qubits, noise model) have not changed. In this case, we don't need to determine the calibration matrix each time we do an experiment, which I think is very convenient.

I haven't come up with any disadvantage about this method. But it is worth noting that we do need to recalculate the calibration matrix if we do the experiment on a different working environment. For example, if we choose a new backend, we might need to define a new corresponding noise model to ensure the noise on the new device can be well modeled. Subsequently, we would have a new calibration matrix for measurement error mitigation.

I can also share one tutorial on measurement error mitigation as well as a nice YouTube video.

  1. https://qiskit.org/documentation/tutorials/noise/3_measurement_error_mitigation.html
  2. https://www.youtube.com/watch?v=yuDxHJOKsVA

I hope my answer would help.

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"what are the advantages and disadvantages in the determination of the calibration matrix each time that we do an experiment and mitigate its error?"

Advantage: The noise matrix will be a more accurate description of the current noise situation. My understanding is that each day, the qubtis are cooled from 300K all the way down to about 15mK, and I can imagine that there might be tiny differences in the noise characteristics due to the qubits' slightly different local environment each day. I'm also not sure how consistent the pressure, humidity, and other properties of the surroundings of the qubits are from day to day. Experience tells us that even on the same day, two identical quantum computations done one after the other can apparently be affected quite differently by noise. Someone on the hardware team from IBM might know better about why this is, but what members of the public know is only that the noise is slightly different each day (and even each time), so it's likely that the matrix made immediately before an experiment will be the most accurate description of the noise of that experiment.

Disadvantage: Recalculating the noise matrix every time you do an experiment requires more work, and in some sense may even take away the quantum advantage. For example the noise matrix for the Melbourne chip with 15 qubits will be a $2^{15} \times 2^{15}$ matrix. The whole point of quantum computation is to avoid the $\mathcal{O}(2^N)$ scaling, so if you have to calculate a noise matrix before doing a 15-qubit experiment, maybe you might as well do the calculation on a classical computer. Ideally you would have a noise matrix prepared once, and then never have to do this $\mathcal{O}(2^N)$ procedure ever again.

The more interesting question is whether or not there is any reliable noise matrix available out there, so that people don't have to make it themselves each time. For this part I do not have an answer yet, but would be keen to know if someone else does.

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