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 $\mathfrak{u}(3)$ 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 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.

Figure 1

Schematic for constructing the ${\mathrm{\Lambda }}_2[Z]$ gate in the logical-qubit basis [Eq. (4)] using ${\chi }^{(2)}$ interactions and linear optics. QFC1 and QFC2: Quantum-state frequency conversions. DM: Dichroic mirror. ${\mathrm{SFG}}_{\pi }$: Generalized sum-frequency generation (3) with $\theta =\pi$.

Figure 2

Schematic for constructing the ${\mathrm{\Lambda }}_3[Z]$ gate in the logical-qutrit basis (5) using ${\chi }^{(2)}$ interactions and linear optics. SHG: Second-harmonic generation. ${\mathrm{\Lambda }}_2[Z]$: The optical circuit from Fig. (1) with modifications described in the text. SPDC: Type-I phase-matched spontaneous parametric down-conversion.