High-Fidelity Readout of Trapped-Ion Qubits

We demonstrate single-shot qubit readout with a fidelity sufficient for fault-tolerant quantum computation. For an optical qubit stored in ${}^{40}\mathrm{Ca}^{+}$ we achieve 99.991(1)% average readout fidelity in ${10}^6$ trials, using time-resolved photon counting. An adaptive measurement technique allows 99.99% fidelity to be reached in $145\mu \mathrm{s}$ average detection time. For ${}^{43}\mathrm{Ca}^{+}$, we propose and implement an optical pumping scheme to transfer a long-lived hyperfine qubit to the optical qubit, capable of a theoretical fidelity of 99.95% in $10\mu \mathrm{s}$. We achieve 99.87(4)% transfer fidelity and 99.77(3)% net readout fidelity.

Figure 1

${}^{40}\mathrm{Ca}^{+}$ photon-count histograms with $t_b=420\mu \mathrm{s}$, for bright $S_{1/2}$ state ($\circ$) and dark $D_{5/2}$ state (•) preparations. Above-threshold events in the dark histogram give ${ϵ}_D$; below-threshold events in the bright histogram give ${ϵ}_B$. Insets show time-resolved counts for two individual 17-photon events, one from each histogram, with likelihood ratios $p_B/p_D$. A PMT dark count histogram is also shown ($▵$), normalized to the same area. Dashed curves are Poisson distributions with the same means as the histograms. The solid curve is the expected $D_{5/2}$ distribution taking into account spontaneous decay.

Figure 2

Average readout error $ϵ$ versus readout time for the ${}^{40}\mathrm{Ca}^{+}$ optical qubit, for three different analysis methods. Analyses of the experimental data are plotted as symbols; statistical uncertainty is at most the size of the symbols. The accompanying curves are simulations of ${10}^9$ trials using ideal Poisson statistics. Cusps arise from the discrete nature of photon counting. T: photon-count threshold method, where the threshold $n_c$ was optimized for each bin time. M: maximum likelihood method. A: adaptive maximum likelihood method. Inset: Simulations using method M, showing the asymptotic average error ${ϵ}_{\infty }$ and the time $t_{1.1}$ required to reach $ϵ=1.1{ϵ}_{\infty }$ as functions of the net photon collection efficiency.

Figure 3

Rate equation simulations for the ${}^{43}\mathrm{Ca}^{+}$ shelving transfer $(|\uparrow ⟩,|\downarrow ⟩)\to (|\uparrow ⟩,D_{5/2})$. The optimum average shelving transfer error ${ϵ}_T$ is shown versus total time $t_T$ allowed for the transfer, for both continuous and pulsed methods. The 393 nm intensity and (in the pulsed method) all pulse durations were numerically optimized for each $t_T$. The error increases at short $t_T$ due to higher 393 nm intensity causing off-resonant shelving of $|\uparrow ⟩$; at long $t_T$ the error increases because of decay from the $D_{5/2}$ shelf. The minimum error is $3.5\times {10}^{-4}$ at $t_T=160\mu \mathrm{s}$; for $t_T=10\mu \mathrm{s}$ an error of $4.8\times {10}^{-4}$ is available. Right: Simplified ${}^{43}\mathrm{Ca}^{+}$ level diagram; states relevant to the shelving transfer are labeled by $F$, $M_F$.