Qudit-Basis Universal Quantum Computation using χ(2)\chi^{(2)} Interactions


We prove that universal quantum computation can be realized---using only linear optics and χ(2)\chi^{(2)} (three-wave mixing) interactions---in any (n+1)(n+1)-dimensional qudit basis of the nn-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)\chi^{(2)} Hamiltonians and photon-number operators generate the full u(3)\mathfrak{u}(3) Lie algebra in the two-pump-photon subspace, and showing how the qutrit controlled-ZZ gate can be implemented with only linear optics and χ(2)\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 χ(2)\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.Comment: 9 pages, 3 figure

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