Efficient optical configurations for trapped-ion entangling gates

Abstract

High-fidelity and parallel realization in scalable platforms of the two-qubit entangling gates fundamental to universal quantum computing constitutes one of the largest challenges in implementing fault-tolerant quantum computation. Integrated optical addressing of trapped-ion qubits offers routes to scaling the high-fidelity optical control demonstrated to date in small systems. Here we show that in addition to scaling, capabilities practically enabled by integrated optics can substantially alleviate laser powers required for both light-shift (LS) and Molmer-Sorensen (MS) geometric phase gates acting on long-lived ground-state qubit encodings in a broad range of ion species. In the proposed gate schemes utilizing carrier nulling via ion positioning at phase-stable standing-wave (SW) nodes, our calculations suggest that suppressed spontaneous photon scattering at the SW node allows for gate drives operating at smaller Raman detunings, resulting in approximately an order-of-magnitude reduction in power requirement (and significantly larger in certain parameter regimes) for gates of a given duration and scattering-limited fidelity as compared to conventional running wave (RW)-based approaches. The SW schemes have the additional benefit of simultaneously eliminating undesired coherent couplings that typically limit gate speeds. Our work quantifies power requirements for multiple ion species and enhancements to be expected from carrier-nulled configurations practically enabled by integrated delivery, and informs experiments and systems for realization of fast and power-efficient laser-based entangling gates in scalable platforms.

Figures (11)

• Figure 1: Schematic level structures of the qubit encodings considered here. (a) We consider light shift (LS) gates performed on qubits encoded in the Zeeman sublevels of zero-nuclear-spin ($I=0$) species, $\ket{\downarrow}=\ket{{}^{2}S_{1/2}; m_J=-1/2 }$ and $\ket{\uparrow}=\ket{{}^{2}S_{1/2}; m_J=+1/2}$. Straight lines with arrows illustrate coupling of the qubit states to the sublevels in the $P$ manifolds via far-detuned Raman fields. Arrows with dashes indicate that the state $\ket{\downarrow}$ couples more weakly to the circular polarization component $E_{\sigma_-}$ of the Raman fields (since it couples via this component only to the ${}^2\mathrm{P}_{3/2}$ manifold), and vice versa for the state $\ket{\uparrow}$. Wavy lines indicate sponetaneous photon scattering (SPS) events. (b) We also consider Mø lmer-Sø rensen (MS) gates performed on qubits encoded in the "clock" transition, $\ket{\downarrow}=\ket{{}^{2}S_{1/2};F=I+1/2,m_{F}=0}$ and $\ket{\uparrow}=\ket{{}^{2}S_{1/2};F=I-1/2,m_{F}=0}$, in species with $I\neq 0$ozeri2007errors. In both cases, a small static magnetic field lifts the degeneracy of the Zeeman sublevels.
• Figure 2: Beam configurations for two-qubit gates on a radial mode using integrated optics, for qubits encoded in the ground state manifold. We assume that all beams are sourced from quasi-TE waveguide modes and are linearly polarized in the plane of the page (dashed arrows). Wave-vectors labeled have components in-plane ($k_\parallel$) and along the $z$-direction ($k_\perp$), with only the in-plane components drawn. The gray line denotes the trap axis with white dots showing the ion locations. (Left) We consider, as the most direct application of the conventional gate drive, a beam configuration that employs a pair of running wave (RW) Raman fields. The wave-vector difference $\mathbf{k_1} - \mathbf{k_2}$ lies along $\mathbf{u_y}$ maximize the drive strength for this radial direction. (Center) RW2 is a variation of the RW1 configuration where the Raman beams propagate along and transverse to the trap axis, in the plane of the page. This geometry allows for simple use of elliptical beam spots focused tightly along the radial direction to reduce power requirements, at the cost of reducing the coupling to the radial modes and introducing an unwanted coupling to the axial modes. (Right) We introduce a beam configuration where the gate is driven by a combination of a standing wave (SW) and a RW. By placing the ions at a SW intensity null, we implement a carrier-free drive that suppresses gate errors due to spontaneous photon scattering (SPS) relative to the RW configurations. In RW2 and SW different beam waists are depicted for fields 1 and 2 only for illustration; we take equal waists in our analysis.
• Figure 3: Comparison of the laser power requirement for LS gates with gate time $\tau_{\mathrm{g}}=50$$\mu$s. All beam configurations are in reference to Fig. \ref{['fig:beam_geometries_schematic']}. (a) Total laser power required given a target SPS-induced gate error $\epsilon_{\mathrm{SPS}}$, for the SW scheme (solid line) and the RW2 scheme (dashed line). Curves for the RW1 scheme have been omitted for clarity. (b) The ratios of required total laser power $P_{\mathrm{RW2}}/P_{\mathrm{SW}}$ (dashed line) and $P_{\mathrm{RW1}}/P_{\mathrm{SW}}$ (dotted line) for a given gate error quantify the advantage conferred by the SW scheme. (c) The detuning down from the $P_{1/2}$ manifold corresponding to the solutions for the SW (solid line) and RW2 (dashed line) configurations at each $\epsilon_\mathrm{SPS}$.
• Figure 4: Comparison of laser power requirement for MS gates with gate time $\tau_{\mathrm{g}}=50$$\mu$s. (a) Total laser power required given a target SPS-induced gate error $\epsilon_\mathrm{SPS}$, for the SW (solid line) and RW2 (dashed line) configurations. (b) The ratios of required total laser power $P_{\mathrm{RW1}}/P_{\mathrm{SW}}$ (dashed line) and $P_{\mathrm{RW2}}/P_{\mathrm{SW}}$ (dotted line) for a given gate error quantify the advantage conferred by the standing wave configuration. (c) The calculated detuning down from the $P_{1/2}$ manifold $\Delta$ is shown for the SW (solid line) and RW2 (dashed line) configurations.
• Figure 5: Distribution of laser power between the Raman fields in the SW scheme for the LS gate. (a) The optimal ratio of field amplitudes of the SW and the RW required to achieve a target gate error $\epsilon_{\mathrm{SPS}}$. (b) To quantify the sensitivity to the power distribution, we calculate the ratio of the gate error for equal amplitudes in the SW and RW fields and the minimum achievable gate error, for fixed total power. The detuning in each case is set by constraint \ref{['eq:Omega-gate-drive-constraint']}.