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

## TL;DR

This paper demonstrates that universal quantum computation for qudits can be achieved using only linear optics and $^{(2)}$ interactions, with a novel induction method and photon injection techniques.

## Contribution

It establishes the universality of $^{(2)}$ interactions for qudit-based quantum computing and introduces a new proof technique involving coherent photon injection.

## Key findings

- Universal gate sets for qubit and qutrit bases are constructed.
- $^{(2)}$ Hamiltonians generate the full $$ Lie algebra in the two-photon subspace.
- Photon injection enables high-photon-number Fock state preparation.

## Abstract

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.