Testing the robustness of robust phase estimation

The robust phase estimation (RPE) protocol was designed to be an efficient and robust way to calibrate quantum operations. The robustness of RPE refers to its ability to estimate a single parameter, usually gate amplitude, even when other parameters are poorly calibrated or when the gate experiences significant errors. Here we demonstrate the robustness of RPE to errors that affect initialization, measurement, and gates. In each case, the error threshold at which RPE begins to fail matches quantitatively with theoretical bounds. We conclude that RPE is an effective and reliable tool for calibration of one-qubit rotations and that it is particularly useful for automated calibration routines and sensor tasks.

Figure 1

(a) ${}^{171}\mathrm{Yb}^{+}$ energy level diagram. The required 369-nm-laser transitions are shown for Doppler cooling (C), initial state preparation (SP), and detection (D). (b) State labels in ${}^2{S}_{1/2}$ hyperfine manifold. The qubit levels are $|0\rangle$ and $|1\rangle$. (c) System overview. Laser beam geometry and microwave horn for qubit operations are overlaid on an image of the trap chip.

Figure 2

Relative distribution of observed photon counts for putative $|0\rangle$ (dark) and $|1\rangle$ (bright) states. Each experiment is repeated 2500 times. The dashed line shows the optimal detection threshold of two photons. Larger detection errors are introduced by choosing a different threshold. At the optimal threshold, the measurement error is 1.2% (sum of the left-most two red bars).

Figure 3

State preparation error as a function of operation duration. We typically use a 50-$\mu \mathrm{s}$ optical pumping duration to ensure high fidelity initialization in $|0\rangle$; intentionally using a shorter duration injects error. Over the plotted range, the exponential fit to the infidelity $E$ as a function of initialization duration $t$ is $E\left(t\right)=0.95e^{-1.5t}+0.025$. At short duration, the infidelity limit is due to asymmetric state populations after Doppler cooling. At long duration, the apparent initialization infidelity is limited by measurement error. The $1/\sqrt{8}$ limit for additive error in RPE corresponds to a preparation time of about 700 ns.

Figure 4

Observed RPE failure rate as a function of the detection threshold (circles). The solid black curve shows the maximum detection error probability, ${\delta }_{\mathrm{meas}}$, defined in Sec. 4a and computed from the histograms in Fig. 2. We observe a sharp onset of RPE failure when the threshold is set at 17 counts, in good agreement with when ${\delta }_{\mathrm{meas}}>{\delta }_{\mathrm{bound}}$. Similarly, RPE also fails when enough dark detections are mislabeled as bright, a situation which only occurs for a threshold of zero.

Figure 5

Observed RPE failure rate as a function of qubit preparation duration (bottom axis) and number of samples (left axis). The background shading also indicates the observed failure rate. The top axis shows the maximum additive error ${\delta }_{\mathrm{prep}}$ corresponding to the qubit preparation duration. We estimate that ${\delta }_{\mathrm{prep}}$ exceeds ${\delta }_{\mathrm{bound}}=1/\sqrt{8}$ beyond 700 ns (vertical red line). The boundary of the onset of failure shifts as a function of the number of samples as expected.

Figure 6

Histograms of rotation angle estimates from RPE with and without additional phase damping errors. The bounds for protocol success are shown as vertical dashed lines. We injected 369-nm (measurement) light at four intensities to drive phase damping. RPE only fails at the strongest intensity ($-20$ dB), although even here the majority of trials succeed. At lower intensities, we observe that the distribution of estimates is significantly tighter than the provable bounds reported by the protocol.