Lemniscate phase trajectories for high-fidelity GHZ state preparation in trapped-ion chains

Abstract

In trapped-ion chains, multipartite GHZ states can be prepared natively with the help of a single bichromatic laser pulse. However, higher-order terms in the expansion in the Lamb-Dicke parameter $η$ limit the GHZ state preparation infidelity for rectangular and bell-like pulses to the order of $η^4$. For tens of ions, the infidelity caused by out-of-Lamb-Dicke effects can reach several percents. We propose an amplitude and phase-modulated pulse shape, an "echoed lemniscate pulse", which cancels this contribution into error in the leading order. For the proposed pulse, the infidelity scales as $η^6$. The improved scaling is achieved because of a special phase trajectory of a collective motional mode following the figure-eight curve (lemniscate). We demonstrate that the lemniscate pulse allows achieving lower infidelity than bell-like pulses, which can be as low as $10^{-4}$ for $20$-ion chains.

Figures (6)

• Figure 1: A schematic drawing of a trapped-ion experimental setup. An ion chain in a Paul trap is illuminated by bichromatic laser field. The field of a Paul trap creates a quadratic pseudopotential confining the ions. The ions form a linear ion crystal. Two ion levels with the transition frequency $\omega_q$ form a qubit. The chain is illuminated by two beams with the frequencies $\omega_q \pm \mu$, which cause the interaction between qubit states and chain phonon modes. Two low-frequency axial phonon modes are shown: the COM mode with frequency $\omega_0$ and the stretch mode with frequency $\omega_1 = \sqrt{3}\omega_0$.
• Figure 2: Pulse shapes and phase trajectories for (a, e) rectangular pulse, (b, f) echoed rectangular pulse, (c, g) lemniscate pulse, (d, h) echoed lemniscate pulse.
• Figure 3: The GHZ state preparation infidelity as a function of field amplitude of a rectangular pulse for the number of ions from 4 to 20. Solid lines indicate the numerical results, and crosses indicate the optimal amplitude values and infidelities calculated from Eq. \ref{['eq:delta_omega_rect_pulse_theory']} and \ref{['eq:inf_eta4']}. The inset: the analytically calculated $S_x^4$ and phonon contributions into GHZ error as functions of $n$.
• Figure 4: GHZ preparation infidelity for the echoed lemniscate pulse obtained from numerical simulation at $\eta = 0.03$ for 20 ions as a function of two parameters $a$, $A$ (see Eq. \ref{['eq:lemniscate_parametrization']}). Here $\delta a = a - a_0$, $\Delta A = A - A_0$, where $a_0$ and $A_0$ are defined by \ref{['eq:lemniscate_gate_conditions']}. Along the dashed line, infidelity is of order $\sim 10^{-5}$. The inset: infidelity is plot as a function of $\delta a$ along the dashed line indicated in (a). The minimum is at $\delta a \approx 4.3\cdot 10^{-4}$, $\Delta A/A_0 \approx 0.0083$.
• Figure 5: Optimized 20-qubit GHZ state preparation (a) infidelity and (b) phonon excitation probabilities obtained from numerical simulation for rectangular, echoed rectangular, and echoed lemniscate pulse as functions of $\eta$.