How do you implement the gates for Grover's algorithm for more than 4 elements?

Question

I am currently working intensively on the Grover algorithm and have understood the individual "building blocks" of the algorithm so far. There are also references in the literature to Nielsen, e.g. an implementation example, but only for 4 elements.

My question is, of a more general nature. How would you actually implement a gate for more elements than just 4? Is there a general approach? The Grover operator G is composed of the n-fold Hadamard transformation and of the conditional phase change and a renewed H transformation. If I have a unitary matrix, eg. the one for the phase shift in the Grover operator, how can I convict this common unitary matrix?

I summarize briefly:

1. How do you convert the gates in the Grover algorithm for N bits? Is there a general derivation somewhere?

2. How can a unitary matrix e.g. the phase shift $ 2 |0\rangle \langle 0| -I $ realize as a gate? Are there any general phrases that might somehow express that?

3. What does a general construction procedure look like to create a gate from a unitary matrix?

Answer 1

Consider this paper where an implementation of Grover's algorithm is given for three qubits.

The only difficulty when extending to $n$ qubits is that you need a $C^n\operatorname{NOT}$ gate (bit flip oracle) or a $C^{n-1}Z$ gate (phase oracle). The phase oracle is made trivially by combining a $C^{n-1}\operatorname{NOT}$ gate with two Hadamard gates. The multiple controlled $\operatorname{NOT}$ gate is constructed from different Toffoli-gates.