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I am completely new to statistical analytics. What I am trying to do is to implement some appropriate tools of statistical analytics in quantum computation. I try to explain the situation clearly. We have a qubit which is in the superposition state and then we measure it. Therefore it will collapse into state 0 or state 1. After measurement the probability of finding the qubit in state Zero is 50% and finding the qubit in state 1 is also 50%. When we measure the qubit for n times, the results are not exactly 50%-50% since the process is random based. So how to determine the distribution type of the measurement result?

Coder
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    There are some puzzling inconsistencies among your questions. As a matter of terminology, you have a *sample* from a *single* population: that sample consists of a series of 0 and 1 observations. You cannot generally hope that they will have the same counts, no matter how many observations you make, so (3) is unanswerable. (5) is a *Binomial* distribution: search our site. It seems you are in a situation identical to the classic textbook question of estimating the chance that a coin will land heads. How do you see your situation differing from that one? – whuber Nov 27 '19 at 20:55
  • Thanks for your comment. This situation is very similar to the coin situation however the state of the qubit before measurement is completely unknown to us. Can you please explain more about the number of populations that why we have just one single population? – Coder Nov 27 '19 at 21:05
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    The state of the coin before measurement is completely unknown to the observer, too--otherwise it wouldn't be modeled as a random variable that is independent of all other variables. The *population* in both cases is the set $\{0,1\}$ and both cases can be viewed as sampling from this population with replacement. – whuber Nov 27 '19 at 21:32
  • Thanks a lot, your explanation was very helpful, I also edited the question according to your explanation. – Coder Nov 27 '19 at 23:16
  • @whuber Following P(k out of n) = (n! / k!(n-k)!)(p^k)(1-p)^(n-k) in binomial distribution, if n is a large number like 10000 and k = 1000 then running a computer program to calculate P based on the factorials of n and k will be a very costly job, can you please recommend any alternative solutions? – Coder Nov 28 '19 at 19:14
  • "Very costly" only if the algorithm is poor: there are efficient ways to compute the log factorials and one can approximate the Binomial distribution with extreme accuracy (using Normal, Chi-squared, or Poisson approximations, depending on the values of $n,$ $k,$ and $p$). For instance, the built-in double-precision calculation for base `R` (called `bdinom`) will compute these probabilities for $n=10^7$ and $k=0,1,\ldots,10^7$ in a couple of seconds: `system.time(dbinom(0:1e7, 1e7, 1/2))`. – whuber Nov 29 '19 at 17:10

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When we measure the qubit for n times, the results are not exactly 50%-50% since the process is random based. So how to determine the distribution type of the measurement result?

A variable that has 50% probability to be '0' and 50 % probability to be '1' follows a Bernoulli distribution (which more generally allows other probabilities for the two states as well).

The counts of the occurrence of '1' or '0' among a specific number of repetitions of measurements of a Bernoulli variable, follows a Binomial distribution.

Sextus Empiricus
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