A projective algebraic curve is defined over an projective space as zeros of some homogeneous polynomial. While quantum computers almost canonically holds their states in a "ket," i.e. a vector in the projective space.
Let's say I have defined an Elliptic curve with $$f=x^3+ax+b-y^2, F=z^{\deg{f}} f(\frac{x}{z},\frac{y}{z}),$$ rational points are stored within a ket $$|Q\rangle,|P\rangle=\pmatrix{x\\y\\z}\in\mathbb{P_2(C)}, F(x,y,z)=0,$$ and what if I would like to support point arithmetic gate $+$ that somehow allows $+(|P\rangle|Q\rangle|0\rangle)=|P+Q\rangle\otimes|\psi\rangle$, this is quite likely to be non-linear so perhaps adding up some dimension would be necessary.
Is this even possible? Thanks.