Process verification of two-qubit quantum gates by randomized benchmarking

We implement a complete randomized benchmarking protocol on a system of two superconducting qubits. The protocol consists of randomizing over gates in the Clifford group, which experimentally are generated via an improved two-qubit cross-resonance gate implementation and single-qubit unitaries. From this we extract an optimal average error per Clifford operation of $0.0936$. We also perform an interleaved experiment, alternating our optimal two-qubit gate with random two-qubit Clifford gates, to obtain a two-qubit gate error of $0.0653$. We compare these values with a two-qubit gate error of $\sim 0.12$ obtained from quantum process tomography, which is likely limited by state preparation and measurement errors.

Figure 1

Homodyne signal as a function of the cross-resonance tone length and pulse sequences for the standard cross-resonance gate (a) and for the $Z{X}_{-\pi /2}$ (b). All tones are applied to the control qubit driving line (Q1), with the $\pi$ pulses and cross-resonance tones applied at the control and target qubit frequencies, respectively. The system is in its ground state at the beginning of each sequence. The $\pi$ pulse between the two cross-resonance tones in the $Z{X}_{-\pi /2}$ gate, along with the opposite sign of the second cross-resonance pulse, echo away the fast $IX$ component of the driving Hamiltonian seen in (a) and leave only the $ZX$ interaction. The dashed vertical lines mark the half-gate length used for the RB experiments for the $Z{X}_{-\pi /2}$ gate and the equivalent length for the standard cross-resonance gate.

Figure 2

QPT of the $Z{X}_{-\pi /2}$ gate with ${\tau }_2=178$ ns. (a) Experimentally extracted Pauli transfer matrix. The gate fidelity is $F_g=0.8830$ raw and $F_{\mathrm{mle}}=0.8799$ after applying a maximum likelihood algorithm. (b) Ideal Pauli transfer matrix representation of the $Z{X}_{-\pi /2}$ gate.

Figure 3

RB experiments on a two-qubit system. (a) Two-qubit pulse sequence for 5 Cl gates randomly selected from the two-qubit Cl group. Tall Gaussians represent $\pi$ rotations, whereas short ones represent $\pi /2$ rotations. A final Cl gate is added to make the sequence self-inverting. (b) Fidelity decay for $|00\rangle$ in the standard (circles) and in the interleaved (triangles) RB protocols. The decays are fitted to an exponential model to extract an average error per Cl operation of $r=0.0936\pm 0.0058$ (standard protocol) and a $Z{X}_{-\pi /2}$ error of $r_{\mathcal{C}}=0.0653\pm 0.0014$. The arrow shows the truncation that would correspond to the pulse sequence shown in (a) and which has an average $|00\rangle$ state fidelity of over 50$\%$.

Figure 4

Average error per Cl operation as a function of the $Z{X}_{-\pi /2}$ half-gate length ${\tau }_2$. An estimated error from the measured $T_2$ echo is shown as dashed lines, with an interval of confidence of $1\sigma$ represented by the shaded area. The estimated error for a coherence time equal to twice the qubit relaxation times (dash-dotted) is also plotted. Error bars in the data represent $1\sigma$ confidence intervals. These results suggest that our average gate error is currently limited by qubit coherence.