If I would design library for quantum computation I would naively consider a sequences of entangled qudits with unit length as a building blocks. I.e., unit length elements from $$\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes\cdots\otimes\mathbb{C}^{d_k}$$ and denote this space by $S^{d_1d_2\dots d_k}$, because it is in fact a unit sphere.
Then any circuit would be just a unitary operator $U$ acting on $\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes\cdots\otimes\mathbb{C}^{d_k}.$
If we would like to consider mixed states then that would be taking an arbitrary probability measure $P$ on $S^{d_1d_2\dots d_k}.$ It would be convenient because the probabilities on the output of $U$ would be modeled by pushforward measure $U_*(P)$, i.e. $P\circ U^{-1}.$ Even more generally if we would have an arbitrary measurable function $f:S^{d_1d_2\dots d_k}\to\Omega$ then the previous would still hold, thus we can define quantum measurement in this manner as well (and any quantum operation as well).
If we have any quantum hardware then the question boils down whether we can set up any pure state and mixed states and whether an abstract unitary operation/function $(U/f)$ have a realization in hardware. To test if a hardware works we just gather statistics.
Does the formalism that I describe constitutes of all possible operations that quantum computer can do? If so, is there any quantum computation library which is based on the formalism that I sketched above?
