I'm working on an implementation of the algorithms described Brassard et al. in the following paper: arXiv:quant-ph/0005055v1.
I managed to make the amplitude amplification cases working but I'm stuck with the amplitude estimation because in the paper there is the definition of a "special" operator in a way I don't understand how to realize it. It is the $\Lambda_M(U^ĵ)$ defined at the bottom of page 15 as
$$|j\rangle|y\rangle\mapsto|j\rangle(|U^j |j\rangle)$$ for $0 \le j \le M$
For a generic integer M and a unitary operator $U$ of size N (!= M), and where the exponent j is its repetition j times. Any idea on how to realize it in practice, as a product of matrices (not necessarily universal), or build element by element?