Hi this question is stem from What is meant with "shot" in quantum computation?.
If we just look for the distribution of the result, is there any theoretical difference between method 1 and method 2 in qiskit simulator?
Method 1. "shots = 1000" circuit result.
Method 2. Iterate the "shots = 1" circuit 1000 times and accumulate every result.
The methods are different on physical devices because "under the hood" there are control operations that perform various actions at the beginning and end of an iteration. For example, there are reset operations to put all states back to their initial values, lasers must be turned off or on, and calibration steps may be taken. On a simulator it is unlikely that any of these operations are taken into account and so the results should not be different.
However, and this is a big however, it depends how the data are processed to give you a result from the simulator. If you are given the raw values (or a probability distribution) then you don't have to worry - you can count your 1000 values in any manner you want. But if you are given corrected values, perhaps due to some smoothing function that is supposed to correct for experimental noise, then that correction will happen after every iteration, not after every shot, and so the resulting data will be different. This is the subject of a recent question where qiskit returns quasiprobabilities after an iteration as opposed to probabilities.
Overall: in theory there is no difference, but when run on a physical device there are practical differences both in terms of how many opportunities there are for the device to change between trials and in terms of how the data may be processed before being given to you.
