Efficient Measurement of Quantum Gate Error by Interleaved Randomized Benchmarking

We describe a scalable experimental protocol for estimating the average error of individual quantum computational gates. This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits. We apply this protocol to a superconducting qubit system and find a bounded average error of 0.003 [0,0.016] for the single-qubit gates $X_{\pi /2}$ and $Y_{\pi /2}$. These bounded values provide better estimates of the average error than those extracted via quantum process tomography.

Figure 1

Randomized benchmarking protocols. (a)–(b) Schemes for the standard and interleaved benchmarking protocols. The target gate, $\mathcal{C}$ (green) is interleaved with random gates ${\mathcal{C}}_i$ (orange) chosen from ${\mathrm{Clif}}_n$. A final gate ${\mathcal{C}}_{m+1}$ (red) is performed to make the total sequence equal to the identity operation.

Figure 2

Experimental implementation of interleaved RB. (a) Measurement of average fidelity over 32 random sequences each of lengths between 2 and 96. The blue data (circles) show the result of the standard RB protocol, while red (triangles) and orange (diamonds) data correspond to interleaved sequences for $X_{\pi /2}$ and $Y_{\pi /2}$ gates, respectively. All data are well described by the first model of Eq. (2), with $p=0.984\pm 0.004$ (standard RB), $p_{\overline{\mathcal{C}}}=0.978\pm 0.005$ (interleaved $X_{\pi /2}$) and $p_{\overline{\mathcal{C}}}=0.979\pm 0.001$ (interleaved $Y_{\pi /2}$) to 95% confidence. Error bars are the standard error of the mean of each point. (b) Pauli transfer maps from process tomography of the $X_{\pi /2}$ and $Y_{\pi /2}$ gates with corresponding gate errors of ${0.011}_{-0.009}^{+0.011}$ and ${0.020}_{-0.008}^{+0.009}$, respectively.