I found something in a lecture on Simon's algorithm that I do not quite understand how to interpret. There the following is said:
$$\sum_{y\in\{0,1\}^n}|y\rangle\left(\sum_{x\in\{0,1\}^n} (-1)^{x\cdot y}|f(x)\rangle\right)$$
So we assume a 1 to 1 function and $s=0$. As far as that is understandable. Then it says (that's the part I'm not quite sure of): "The probability that a measurement in String Y results is:"
$$\left|\left|\frac{1}{2^n}\sum_{x\in\{0,1\}^n} (-1)^{x\cdot y}|f(x)\rangle\right|\right|^2=\frac{1}{2^n}$$
I try to make something more understandable, which I do not quite understand. So if you have a 1 to 1 function then that means that each value occurs exactly once, say you have three bits, then there are 8 states, that is from 000 to 111. These then also occur with the same probability of 1/8 for each state. But now I do not quite understand the result of the top formula, why is $\frac{1}{2^n}$ out there, why is not it squared? What happens to the sum there? What is the exact calculation behind it?
I mean, on the one hand, I understand why $\frac{1}{2^n}$ comes out there, but on the other hand, I do not quite understand the calculation ($||...||^2$) behind it, I hope somebody can explain it a bit.