Nonlinear Phase Gates Beyond the Lamb-Dicke Regime

Abstract

Nonlinear phase gates are essential to achieve the universality of continuous-variable quantum processing and its applications. We present a deterministic protocol for generating nonlinear phase gates in trapped ion systems using simultaneous two-tone sideband drives beyond the Lamb-Dicke regime. Our approach harnesses higher-order interaction terms typically neglected or suppressed to construct nonlinear phase gates. This methodology enables high-fidelity gate engineering with a near three-fold reduction in control pulses compared to state-of-the-art theoretical proposals.

Figures (15)

• Figure 1: Quantum circuit for the cubic phase gate protocol in Eq. (\ref{['eq:protocol']}). The protocol consists of repeated application of the composite gate sequence $U_1$, $U_3$, $U_1$, $U_2$ applied $N$ times, optimized over all $t_k^{(l)}$ for fixed $\eta$ and $\Omega_{k}$. While each unitary $U_k$ is realized via multi-tone sideband pulses beyond the Lamb-Dicke regime, $U_1$ and $U_2$ can be alternatively implemented as high-fidelity laser-free operations: displacement $D(\beta)$ via resonant classical electric fields LoNature2015SpinDependentforce and squeezing $S(r)$ via parametric trap modulation burd2019quantumburd2024squeezing. Since the qubit occupies the $|+\rangle_y$ eigenstate of the of $\hat{\sigma}_y$ and interaction Hamiltonians, it remains stationary throughout the evolution.
• Figure 2: Wigner function representations of (a) the target cubic gate $\mathcal{U}^{(3)}(\zeta_3)$ with cubicity $\zeta_3= 1$ on ground states as in (\ref{['eq:cubicstate']}), and (b) the generated state obtained using the optimized $N=3$, $C_{N=3}^{(3)}$ (parameters in Table \ref{['tab:params_3g']}). The protocol achieves fidelity $\mathcal{F} = 0.99986$ and closely reproduces the non-classical features, with similar total negativity volumes $V_{-}(\rho) = \frac{1}{2} \int |W_\rho(\mathbf{r})| - W_\rho(\mathbf{r}) d\mathbf{r}$ of $V_{-}(\text{target}) = 0.226$ and $V_{-}(\text{generated}) = 0.2229$.
• Figure 3: Wigner functions for the cubic gate acting on coherent state inputs. Top row: target states $\mathcal{U}^{(3)}(\zeta_3=1)\,|\alpha\rangle$ for $\alpha\in\{-1,+1,-i,+i\}$. Bottom row: states generated by the optimized $N=3$ gate $C_{N=3}^{(3)}$ (same gate parameters as in Table \ref{['tab:params_3g']}). Fidelities: $\mathcal{F}(\alpha=1)=0.99937$, $\mathcal{F}(\alpha=-1)=0.99965$, and $\mathcal{F}(\alpha{=}\pm i)=0.969$. These examples complement Fig. \ref{['fig:repetition-fidelity']}, where the infidelity trend is reported across purely imaginary $\alpha$.
• Figure 4: Comparison of Wigner function negativity along the position axis, i.e. $p=0$ for the target cubic phase gate on the ground state in Fig. \ref{['fig:wignercomp']} and protocols with increasing gate complexity ($N=1, 2, 3$). The $N=3$ protocol achieves high fidelity of 0.99986 with the target state.
• Figure 5: Wigner-negativity cut from the Figure \ref{['fig:wignercoherent']} for the target cubic state and the state generated from the coherent state input $\ket{\alpha=-i}$ by optimized sequence (\ref{['eq:protocol']}). The optimal cut follows the line of maximal negativity $p = -0.0687\,q - 1.9703$ between phase space points $(q,p)=(-12.50,-1.11)$ and $(-6.22,-1.54)$.