Experimental Demonstration of a Cheap and Accurate Phase Estimation

We demonstrate an experimental implementation of robust phase estimation (RPE) to learn the phase of a single-qubit rotation on a trapped ${\mathrm{Yb}}^{+}$ ion qubit. We show this phase can be estimated with an uncertainty below $4\times {10}^{-4}\mathrm{rad}$ using as few as 176 total experimental samples, and our estimates exhibit Heisenberg scaling. Unlike standard phase estimation protocols, RPE neither assumes perfect state preparation and measurement, nor requires access to ancillae. We crossvalidate the results of RPE with the more resource-intensive protocol of gate set tomography.

Figure 1

(a) RPE and (b) GST experimental sequences. Each sequence starts with the state $\rho$ and ends with the two-outcome measurement $M$. (a) An RPE sequence consists of repeating the gate in question either $L$ or $L+1$ times. (b) In GST, a gate sequence $F_i$ is applied to simulate a state preparation potentially different from $\rho$. This is followed by $\lfloor L/|g_k|\rfloor$ applications of a germ—a short gate sequence $g_k$ of length $|g_k|$. Finally, a sequence $F_j$ is applied to simulate a measurement potentially different from $M$.

Figure 2

Analytic upper bounds on the RMSE of the RPE phase estimate. Because RPE is potentially biased, the RMSE does not go to zero in the limit of infinite $N$, but instead, approaches a floor of $\pi /(2L_{\max })$. A larger additive error $\delta$ produces a larger bias, and thus requires a larger $N$ and larger $L_{\max }$ to achieve a small RMSE. For example, $N=16$ is not large enough to reach the floor for $L_{\max }=1024$, but increasing $N$ to 370 we easily saturate the bound for most values of $\delta$.

Figure 3

RMSE versus $L_{\max }$ for RPE estimates of $\alpha$ from 100 subsampled data sets of size $N=8$ and $N=32$. While analytic bounds are at best $\pi /(2L_{\max })$, we see this can be pessimistic. When the additive errors, which can bias the RPE estimate, are sufficiently small, increasing $N$ improves accuracy.

Figure 4

Scaling of the RMSE of estimates of $\alpha$ as a function of total samples $S(N,L_{\max })$, with $L_{\max }=1024$. For the analytic curves, we resample with $N\in \{8,16,\dots ,256\}$, while for the experimental curves we restrict ourselves to $N\in \{8,16,32\}$ (as higher resampling rates would introduce nontrivial correlations between data sets). For each curve, $N$ increases incrementally from left to right. Analytic bounds are derived using the techniques of Fig. 2. Experimental data points take the RMSE of 100 subsampled data sets for both RPE and GST. While the analytic bounds converge to $\pi /2048$, we see standard quantum limit scaling (i.e., error scaling $\propto 1/\sqrt{S}$) of RPE experimental estimates. As discussed in the text, this is because our experimental device has very low additive error, and thus the RPE estimates are essentially unbiased, and can achieve greater accuracy with increasing number of samples. GST estimates also exhibit standard quantum limit scaling.

Figure 5

RMSE for RPE and GST estimates of $\alpha$ as function of total number of total samples $S(N,L_{\max })$ using 100 subsampled data sets. As opposed to Fig. 4, here we fix $N=8$ and vary $L_{\max }$. Each sequential data point corresponds to setting $L_{\max }\in \{1,2,3,\dots ,1024\}$. RPE achieves the same level of accuracy as GST using far fewer resources.