Short answer:So what a quantum computer does it give you the optimal result without the time delay of calculating results which aren't optimal.
Long answer: It's advantage is ability to from deterministic processing to non-deterministic processing, to solve the optimal answer to a problem.
Our computer are deterministic, which means that for a problem it has to through and check all possibilities and find which one is the best (hence optimal). The problem is that some problems (like
the Travelling Salesman Problem are so complex to solve that there is no efficient way to sale them optimally. Trying to solve it with a large data set could take years because it has test all combinations of routes. (As a side note there are quick ways to 'solve' it, but these are not optimal.) So it has to try 2
n combinations (where n is the number of cities). Each one of those combinations takes a certain amount of time to solve, so even with a small set of cities it can a while.)
An non-deterministic computer (quantum computer if you will) will respond in one of a few ways.
First it will 'know' the answer and spit it straight out.
Second if it finds a wrong combination it suffers no time penalty, meaning that in a 2
n problem 2
n - 1 results take no time at all to calculate (assuming there is only one optimal answer.) To our perception these two results are same.
So what a quantum computer does it give you the optimal result without the time delay of calculating results which aren't optimal.
Also see P Vs NP, Turing Machines, and Time Complexities (Big O notation)