Universal hybrid quantum computing in trapped ions

Using discrete and continuous variable subsystems, hybrid approaches to quantum information could enable more quantum computational power for the same physical resources. Here, we propose a hybrid scheme that can be used to generate the necessary Gaussian and non-Gaussian operations for universal continuous variable quantum computing in trapped ions. This scheme utilizes two linear spin-motion interactions to generate a broad set of nonlinear effective spin-motion interactions including one- and two-mode squeezing, beam splitter, and trisqueezing operations in trapped ion systems. We discuss possible experimental implementations using laser-based and laser-free approaches.

Figure 1

One-mode squeezing operation generated by Eq. (2). We plot the infidelity $1-\mathcal{F}$ versus normalized gate duration $t_f\mathrm{\Omega }$, while keeping ${\mathrm{\Omega }}_2{t}_f$ constant. This dependence is shown for squeezing parameters $2{\mathrm{\Omega }}_2{t}_f\equiv r_s$ of 1.5 (blue dashed), 1.0 (purple dotted), and 0.5 (solid light blue). For large values of $t_f$, $1-\mathcal{F}\to 0$, as described in the text. The inset shows the probability $\mathcal{P}$ of phonon state $|n_j\rangle$ for a squeezed state with $r_s=1.5$, generated by integrating Eq. (2) for $t_f\mathrm{\Omega }/2\pi \simeq 2.2$.

Figure 2

Two-mode squeezing operation generated by Eq. (2). Here we show the infidelity $1-\mathcal{F}$ versus the normalized gate duration $t_f\mathrm{\Omega }/2\pi$. This dependence is shown for squeezing parameters $r_{2s}$ of 0.75 (red dashed), 0.5 (pink dotted), and 0.25 (orange solid). For larger values of $t_f$, all three calculations converge to that of ${\widehat{H}}_{\text{eff}}$. The inset shows the probabilities $\mathcal{P}$ of occupying a state with $n_j$ phonons in the $j$ mode and $n_{j^{\prime }}$ phonons in the $j^{\prime }$ mode for $r_{2s}=0.75$ and $t_f\mathrm{\Omega }/2\pi \simeq 1.7$.

Figure 3

Beam-splitter operation generated by Eq. (2). Here we show the infidelity $1-\mathcal{F}$ versus normalized gate duration $t_f\mathrm{\Omega }/2\pi$. This dependence is shown for two squeezing parameters $r_{bs}=\pi /4$ (solid turquoise) and $r_{bs}=\pi /2$ (dashed green), for a system initialized to the state $|\downarrow \rangle |n_j=1,n_{j^{\prime }}=0\rangle$. The inset shows the probabilities of the phonon subspace being in the state $|10\rangle$ and $|01\rangle$ versus time, for a $r_{bs}=\pi /2$ interaction such that $t_f\mathrm{\Omega }/2\pi \simeq 2.5$.

Figure 4

Trisqueezing operation generated by Eq. (2). We show the infidelity $1-\mathcal{F}$ versus normalized gate duration $t_f\mathrm{\Omega }/2\pi$ for three squeezing parameters $r_{3s}=0.12$ (brown dashed), $r_{3s}=0.09$ (light brown dotted), and $r_{3s}=0.06$ (tan solid). The inset shows the probabilities $\mathcal{P}$ of each phonon state $n_j$ for $r_{3s}=0.12$ and $t_f\mathrm{\Omega }/2\pi \simeq 2.9$ generated by Eq. (2), showing that only states with phonon numbers that are multiples of 3 are populated.