For any two states $|\phi\rangle$ and $|\psi\rangle$
Does there exist a gate $U$ such that $U|\phi\rangle = |\psi\rangle$ ?
I suppose that we know for a vector space $V$ then $\forall \quad a, b \quad \exists M$ such that $Ma = b$. Therefore the original question becomes, does the set of quantum states form a vector space?
The space of quantum states does not form a vector space. Consider a quantum state $|\phi\rangle$. If the space of quantum states is a vector space then $2|\phi\rangle$ is also a quantum state.
But $\langle2\phi,2\phi \rangle = 4\langle\phi,\phi \rangle$ by multilinearity of the inner product. And
$4\langle\phi,\phi \rangle = 4$ as $|\phi\rangle$is a quantum state so $\langle\phi,\phi \rangle$ = 1.
Therefore the space of quantum gates is not a vector space.
Then given that, could there still be a $U$ as posed?