We prove that universal quantum computation can be realized---using only
linear optics and χ(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 χ(2) Hamiltonians and photon-number operators generate the
full 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 χ(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 χ(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.Comment: 9 pages, 3 figure