Efficient unitarity randomized benchmarking of few-qubit Clifford gates

Unitarity randomized benchmarking (URB) is an experimental procedure for estimating the coherence of implemented quantum gates independently of state preparation and measurement errors. These estimates of the coherence are measured by the unitarity. A central problem in this experiment is relating the number of data points to rigorous confidence intervals. In this work we provide a bound on the required number of data points for Clifford URB as a function of confidence and experimental parameters. This bound has favorable scaling in the regime of near-unitary noise and is asymptotically independent of the length of the gate sequences used. We also show that, in contrast to standard randomized benchmarking, a nontrivial number of data points is always required to overcome the randomness introduced by state preparation and measurement errors even in the limit of perfect gates. Our bound is sufficiently sharp to benchmark small-dimensional systems in realistic parameter regimes using a modest number of data points. For example, we show that the unitarity of single-qubit Clifford gates can be rigorously estimated using few hundred data points under the assumption of gate-independent noise. This is a reduction of orders of magnitude compared to previously known bounds.

Figure 1

Schematic difference between the single-copy implementation (a) and the two-copy implementation (b) of the unitarity randomized benchmarking protocol. Each line represents a system on the base Hilbert space $\mathcal{H}$. In the single-copy implementation, the expected value of the measurement $\mathrm{Tr}\left[E_{\mathcal{H}}{\mathcal{G}}_{\mathbf{j}}\left({\overline{\rho }}_{\mathcal{H}}\right)\right]$ needs to be squared to obtain $q_{\mathbf{j}}$, whereas in the two-copy implementation $q_{\mathbf{j}}=\mathrm{Tr}\left[E{\mathcal{G}}_{\mathbf{j}}^{\otimes 2}\left(\overline{\rho }\right)\right]$ yields the direct outcome.

Figure 2

Comparison of the $99\%$ confidence intervals around the average sequence purity ${\overline{q}}_m$ calculated with and without our variance bound at several different sequence lengths. The plot is based on a simulated URB experiment of the single-qubit Clifford group with $N=250$ samples per sequence length $m$. The empirical average sequence purity ${\overline{q}}_m$ (marked with a cross) is plotted versus the sequence length $m$ on a semilogarithmic scale. The larger (blue) bars indicate the $99\%$ confidence interval without variance Eq. (28) and the smaller (red) bars indicate the $99\%$ confidence interval of Eq. (18) based on our sharp variance bound Eq. (19). Here we used a priori estimates of the unitarity and SPAM parameters of $u=0.98$ and $\parallel {\overline{\rho }}_{\mathrm{err}}{\parallel }_1^2={\parallel E_{\mathrm{err}}\parallel }_{\infty }^2=0.02$, respectively. Then Eq. (27) yields $L=1.02+0.2\sqrt{2}$. For completeness, a least-squares fit according to the model ${\overline{q}}_m=B{u}^{m-1}$ [see Eq. (7)] is shown in the yellow solid line. This yields $u\approx 0.987$.

Figure 3

Number of sequences $N$ versus the sequence length $m$ for various values of the unitarity $u$ when benchmarking the single-qubit Clifford group ($d=2$). Confidence parameters are $\epsilon =0.02$ and $\delta =0.01$. The SPAM parameters are $\parallel {\overline{\rho }}_{\mathrm{err}}{\parallel }_1^2={\parallel E_{\mathrm{err}}\parallel }_{\infty }^2=0$. By Eq. (27) then $L=1$ is used. The number of sequences is asymptotically independent of the sequence length. This is consistent with our variance bound Eq. (19).

Figure 4

Semilogarithmic plot of the variance bound ${\sigma }^2$ as a function of the unitarity $u$ for various magnitudes of SPAM errors in the large sequence limit for single-qubit Clifford URB ($d=2$). The black dash-dotted line is a reference line plotting ${\sigma }^2={(1-u)}^2$. The differently colored solid lines indicate the various magnitudes of SPAM errors, where $\parallel {\overline{\rho }}_{\mathrm{err}}{\parallel }_1^2={\parallel {\overline{E}}_{\mathrm{err}}\parallel }_{\infty }^2=\eta$. There are two regimes. For small SPAM errors and small $u$, the variance scales as ${(1-u)}^2$, whereas for nonzero SPAM errors and large $u$, the variance approaches a constant.

Figure 5

Color plot of the number of sequences $N$ as a function of the SPAM parameters $\parallel {\overline{\rho }}_{\mathrm{err}}{\parallel }_1^2$ and $\parallel E_{\mathrm{err}}{\parallel }_{\infty }^2$ in the large sequence length limit for single-qubit Clifford URB ($d=2$). The parameters $u=0.99$ and $\epsilon =0.02, \delta =0.01$ were used. This plot illustrates the sensitivity of our result to SPAM errors. In particular, the number of sequences increases most significantly when both state preparation and measurement errors are large.

Figure 6

Number of sequences $N$ as a function of the number of qubits $q$ comprising the system for different values of the SPAM parameters. A fixed unitarity $u=0.99$ and the large sequence length limit are used. The interval bound $L$ is computed using Eq. (27) as a function of the SPAM quantities (see legend). The confidence parameters $\epsilon =0.02, \delta =0.01$ were used. The dashed line indicates the first-order bound Eq. (28) corresponding to $\parallel {\overline{\rho }}_{\mathrm{err}}{\parallel }_1^2={\parallel {\overline{E}}_{\mathrm{err}}\parallel }_{\infty }^2=0.02$. For the given confidence and SPAM parameters, our bound gives an improvement of the required number of sequences up to five-qubit systems.