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In Black Mirror S04 E04 named Hang the DJ, How did the real-life app particularly choose two specific persons to run 1000 simulations for them? Does the app run 1000 simulations for every two possible match? If we assume the app has 1 million members, then it roughly must run 5*10^14 simulations!
Plus, we see many other couples with different numbers over their heads. I couldn't identify if they're Amy and Frank, or other couples.
Firstly, yes, they're all Amy and Frank - each is one of the simulated couples who climbed the wall.
The couple we are focused on is simulation 998, which matches the number of rebellions logged...
We have to assume that this is couple 1000 & that the two simulations who did not rebel are somehow still inside the simulation. Their fate is not mentioned, AFAIK. Presumably once the final results are in, all the simulations are terminated & the occupants are never aware. We do see all the rebellious couples 'dissolving'.
As regards the math, or processing power needed to perform these simulations, you always need to be aware that Black Mirror never concerns itself with the science behind the technology. It simply asks you to believe that at this point in time, technology is sufficiently advanced to be able to achieve it.
How is rarely important to the plot.
The show is more usually concerned with the human implications of the technology - how we interact with it, what it makes us do to ourselves and each other as a result of it.
If we assume the app uses proximity of signed-up members to generate 'quick' matches, then either it tested everybody in the bar, very rapidly, or only Amy and Frank were signed up. It really isn't important which, merely that the results echoed the claim from the beginning of the simulation that the chances of eventually making the correct match are 99.8%.
The final scene IMHO just gives some "aww" factor, when we see that in real life Amy and Frank are about to meet for the first time... And we, the audience, are pretty certain they will 'live happily ever after' - which in itself is quite rare for Black Mirror.
therwise there would be difficulty explaining how the 2 'fails' have been qualified as 'not rebelling' if they could potentially live another 10 years before they did rebel It seems a fair assumption that the "no rebellion" cutoff happens when either Frank or Amy decides to go with their appointed soulmate (or escape with someone completely different, I guess...), which means the simulation can be marked as failed when the date of the "true soulmate reveal" has passed. It makes sense, Frank and Amy match when they even refuse to accept what a perfect algorithm tells them.
The question has been answered, but a small part of your question wasn't addressed:
If we assume the app has 1 million members, then it roughly must run 5*10^14 simulations!
You assume that any member is interested in any other member.
Filtering for only single people who meet each other's criteria (presumably you tell the app preferred age range, gender, etc.), that's considerably less than 5*1014.
Additionally, the fact that Amy and Frank run into each other exactly when the result is presented strongly suggests that the simulations are only run for people you come into contact with, which again significantly reduces the amount of pairs that need to be checked.
