Qudit-Basis Universal Quantum Computation Using chi((2)) Interactions

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 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 or 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.