High-Fidelity Universal Gate Set for ${{}^9\mathrm{Be}}^{+}$ Ion Qubits

We report high-fidelity laser-beam-induced quantum logic gates on magnetic-field-insensitive qubits comprised of hyperfine states in ${{}^9\mathrm{Be}}^{+}$ ions with a memory coherence time of more than 1 s. We demonstrate single-qubit gates with an error per gate of $3.8(1)\times {10}^{-5}$. By creating a Bell state with a deterministic two-qubit gate, we deduce a gate error of $8(4)\times {10}^{-4}$. We characterize the errors in our implementation and discuss methods to further reduce imperfections towards values that are compatible with fault-tolerant processing at realistic overhead.

Figure 1

Schematic of the ion trap, formed with two gold-coated, stacked wafers [14]. Top view of the trap (on the right) showing the load zone $\mathcal{L}$ and experiment zone $\mathcal{E}$. Ions are transported from $\mathcal{L}$ to $\mathcal{E}$ with time-varying potentials applied to the segmented control electrodes (colored orange hues). The positions of RF and control electrodes are exchanged in the lower layer (cross section in inset). Coherent manipulations are implemented on ions confined in $\mathcal{E}$.

Figure 2

Relevant energy level structure for ${{}^9\mathrm{Be}}^{+}$ ions (not to scale). Transitions to the electronic excited states are used for Doppler cooling, repumping, and qubit state measurement as described in the text.

Figure 3

Laser beam geometry for stimulated-Raman transitions. Copropagating beams $2a$ and $2b$ are used to implement high-fidelity single qubit gates; two-qubit entangling gates use all three beams as described in the text.

Figure 4

Average fidelity for single-qubit-gate randomized benchmarking sequences, plotted as a function of sequence length. We determine the average error per computational gate to be $3.8(1)\times {10}^{-5}$ and state preparation and measurement error to be $2.0(3)\times {10}^{-3}$ for these data sets. Error bars show the standard error of the mean for each point.

Figure 5

ML Bell-state error (red dots), plotted as a function of $-2\pi /\mathrm{\Delta }$ where $\mathrm{\Delta }$ is the Raman detuning, for a constant gate duration of approximately $30\mu \mathrm{s}$. The simulated contributions to the Bell state error from Raman and Rayleigh scattering (Supplemental Material [18]) are shown with the blue and purple dashed lines, respectively. For large $|\mathrm{\Delta }|$ the Raman scattering error approaches zero, however, the Rayleigh scattering error remains approximately constant at $1.7\times {10}^{-4}$. The black line is the sum of the Raman and Rayleigh scattering errors and the error due to the composite microwave pulses used for qubit state preparation and detection (uncertainty indicated by the gray band). Error bars for the measured Bell state fidelity are determined from parametric bootstrap resampling [42] of the data and represent a $1\sigma$ statistical confidence interval.

Figure 6

ML Bell-state error (red dots) as a function of gate duration $t_{\mathrm{gate}}$ for a constant Raman beam detuning $\mathrm{\Delta }\simeq -2\pi \times 730\mathrm{GHz}$. The black line shows the separately determined error and uncertainty (gray shade) due to the microwave pulses used for $|2,2⟩\leftrightarrow |\uparrow ⟩$ state transfer. The three dashed lines show the sum of the expected gate errors including photon scattering and mode frequency fluctuations (which are slow compared to gate durations shown) for three different rms magnitudes of mode frequency fluctuations (Supplemental Material [18]). The gate error increases quadratically with increasing $t_{\mathrm{gate}}$ due to such frequency fluctuations; however, for $t_{\mathrm{gate}}=30\mu \mathrm{s}$ the error due to these frequency fluctuations is approximately $1\times {10}^{-4}$.