  We prove that universal quantum computation can be realized---using only
linear optics and $\chi^{(2)}$ (three-wave mixing) interactions---in any
$(n+1)$-dimensional qudit basis of the $n$-pump-photon subspace. First, we
exhibit a strictly universal gate set for the qubit basis in the
one-pump-photon subspace. Next, we demonstrate qutrit-basis universality by
proving that $\chi^{(2)}$ Hamiltonians and photon-number operators generate the
full $\mathfrak{u}(3)$ Lie algebra in the two-pump-photon subspace, and showing
how the qutrit controlled-$Z$ gate can be implemented with only linear optics
and $\chi^{(2)}$ interactions. We then use proof by induction to obtain our
general qudit result. Our induction proof relies on coherent photon
injection/subtraction, a technique enabled by $\chi^{(2)}$ interaction between
the encoding modes and ancillary modes. Finally, we show that coherent photon
injection is more than a conceptual tool in that it offers a route to preparing
high-photon-number Fock states from single-photon Fock states.
