Randomized Benchmarking of Multiqubit Gates

We describe an extension of single-qubit gate randomized benchmarking that measures the error of multiqubit gates in a quantum information processor. This platform-independent protocol evaluates the performance of Clifford unitaries, which form a basis of fault-tolerant quantum computing. We implemented the benchmarking protocol with trapped ions and found an error per random two-qubit Clifford unitary of $0.162\pm 0.008$, thus setting the first benchmark for such unitaries. By implementing a second set of sequences with an extra two-qubit phase gate inserted after each step, we extracted an error per phase gate of $0.069\pm 0.017$. We conducted these experiments with transported, sympathetically cooled ions in a multizone Paul trap—a system that can in principle be scaled to larger numbers of ions.

Figure 1

The red circles show one minus the average probability of measuring an error at the end of sequences of random Clifford unitaries $\overline{E}(l)$ as a function of the sequence length $l$ in the two-qubit benchmarking experiment. By fitting the data to the expression in Eq. (1) (red, upper line), we find an error per random Clifford unitary ${\epsilon }_g=0.162(8)$. The preparation/measurement error, ${\epsilon }_m$, is 0.086(22) (recall that measurement error includes the error for an additional inverting unitary before detection). Blue squares show the results for running random sequences with an additional $\widehat{G}$ inserted after each Clifford unitary. Fitting this data to the same functional form (blue, lower line) and using Eq. (2) yields an error of ${\epsilon }_{\widehat{G}}=0.069(17)$ and ${\epsilon }_m=0.132(26)$. The error bars in the plot represent the standard deviation of the mean of the sequences’ frequency of correct measurement outcome. Error bars for inferred parameters are based on bootstrap resampling [22, 38].

Figure 2

Black circles and blue squares show 1 minus the average probability of error for each qubit in the single-qubit benchmarking experiment. The solid lines are the best fits of the data to Eq. (1) with $n=1$ [5]. The best-fit values for ${\epsilon }_g$ give the error per computational gate (as defined in the text for the single-qubit benchmark), which we find to be 0.010(2) and 0.007(2), respectively.