Can you give an example of how to use Shor's algorithm to break ECC? I have a friend that is a professor in QC department in one of the famous Chinese universities, he is unable to answer this question
Shor I can. Sorry.
The maths is I think well established and universally accepted. I am by no means an expert, but section 2 of this paper guides you through it.
ECC security is reliant on the effective impossibility of solving the Discrete Logarithm Problem; it being implausibly difficult to reverse elliptic curve point multiplication using "normal" computers.
Shor's algorithm is famous for solving prime factorisation for any given integer. This can be applied to discrete logarithms (see: https://en.wikipedia.org/wiki/Shor%27s_algorithm#Discrete_logarithms), because the algorithm is equivalent to the hidden subgroup problem for finite Abelian groups. I'll not go into it further because as I say I'm no expert and the maths gets beyond me at this point.
Here's the relevant bit from the paper I mentioned above. I'll make a second post shortly answering your second question about how QCs work.
