# Modeling building blocks for quantum computation

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?

• it seems like you are looking for a more abstract way to state the basic building blocks of quantum mechanics in general, I don't see much here that is specifically related to quantum computation. By the way, taking "unit elements" in some $\mathbb C^n$ isn't enough: you want complex projective spaces $\mathbb{CP}^n$ to also take care of the global phase. Moreover, the probability measures in QM are not arbitrary.
– glS
May 4, 2021 at 23:37
• I believe the unit vectors are precisely states (en.wikipedia.org/wiki/State_space_(physics)), thus $S^n$ models the state space. The $\mathbb{CP}^n$ space is useful when we already have fixed state and we want to measure probabilities in some basis. However, I can imagine that there is a mapping $\phi:S^n\to S^n$ which do not factorize to $\bar\phi:\mathbb{CP}^n\to\mathbb{CP}^n$, i.e. such that do not map complex lines to complex lines in $\mathbb{C}^n$. May 5, 2021 at 11:36
• I believe any probability measure on borel space of $S^n$ would work. In particular the discrete one which we have in the case of mixed states. May 5, 2021 at 11:37
• if you define states via density matrices/linear functionals defined on a set of events (the functional analysis approach), then yes these are "unit vectors". But then these are not just elements of some $\mathbb C^n$, but rather positive linear functionals on a C* algebra. Regardless, I still don't understand the connection between this and quantum computation. This formalism is equivalent to the standard one in terms of bra-ket etc, which is likely more convenient to actually perform calculations in practice
– glS
May 5, 2021 at 12:30