Experimental implementation of non-Clifford interleaved randomized benchmarking with a controlled-$S$ gate

Hardware-efficient transpilation of quantum circuits to a quantum device native gate set is essential for the execution of quantum algorithms on noisy quantum computers. Typical quantum devices utilize a gate set with a single two-qubit Clifford entangling gate per pair of coupled qubits; however, in some applications access to a non-Clifford two-qubit gate can result in more optimal circuit decompositions and also allows more flexibility in optimizing over noise. We demonstrate calibration of a low-error non-Clifford controlled-$\frac{\pi }{2}$ phase (cs) gate on a cloud-based IBM Quantum system using the Qiskit Pulse framework. To measure the gate error of the calibrated cs gate we perform non-Clifford cnot-dihedral interleaved randomized benchmarking. We are able to obtain a gate error of $5.9\left(7\right)\times {10}^{-3}$ at a gate length 263 ns, which is close to the coherence limit of the associated qubits, and lower error than the back-end standard calibrated cnot gate.

Figure 1

The cs gate realized with a closed-loop calibration. (a) Pulse schedule with the flat-top width ${\tau }_{\mathrm{sq}}$ = 21.3 ns. The schedule consists of two CR pulses $C{R}_{-}$ and $C{R}_{+}$ on the ControlChannel u0 with echo pulses $X\left(\pi \right)$ applied on DriveChannel d0 of the control qubit. Local gates in Eq. (10) are also applied to the DriveChannel d1 of the target qubit. Pulse instructions in d0 and d1 are played in the rotating frame of the control and the target qubits, respectively. The ControlChannel u0 is physically connected to the control qubit, whereas pulses are played in the rotating frame of the target qubit to drive CR interaction. A circular arrow of $VZ\left(\theta \right)$ represents the virtual-$Z$ rotations with rotation angle $\theta$. (b) cnot-dihedral interleaved RB. Dotted lines show fit curves of the ground state population measured by standard RBs in $|00\rangle$ and $|++\rangle$ basis, while solid lines show fits of interleaved RB. Triangle and cross symbols show raw experiment data of 10 different random circuits.

Figure 2

Average gate errors as a function of the flat-top width of the CR pulse ${\tau }_{\text{sq}}$ estimated by different benchmark techniques. The corresponding total gate time ${\tau }_{\mathrm{CS}}$ is shown in the top axis. Blue circles and red triangles represent $r_g^{\mathrm{rb}}$ and $r_g^{\mathrm{qpt}}$, respectively. The green dotted line shows the theoretical lower bound of the average gate error calculated by the total gate time ${\tau }_{\mathrm{CS}}$ and the average $T_1$ and $T_2$ values of the qubits during the experiment. The filled area represents the coherence limit with $T_1$ and $T_2$ values with variance of $1\sigma$. See text for a detailed discussion.

Figure 3

Distribution of single-qubit and two-qubit gate errors of ibmq_paris at the time of experiment. Single-qubit gate errors measured by the Hadamard operation are shown in nodes of the qubit coupling map, while two-qubit gate errors measured by cnot operation are shown in graph edges. Error values are represented by color maps shown in the bottom.

Figure 4

Typical experimental results for calibration experiments. Measured population is converted into the expectation value of Pauli operators. (a) Rough amplitude calibration. The blue and black lines show the cosinusoidal fit for the experimental results and the optimal amplitude $A_0$. (b) Rough phase calibration. The blue and red lines show the cosinusoidal fit for the experimental result of ${\mathcal{S}}_{\varphi g}^{\mathrm{scan}}\left(\varphi \right)$ and ${\mathcal{S}}_{\varphi e}^{\mathrm{scan}}\left(\varphi \right)$, respectively. The black line shows the optimal phase ${\varphi }_0$. (c) Rough amplitude calibration. The solid and dotted lines show the fit for the result of initial ($A=0.236$) and final experiment ($A=0.237$) within the closed-loop calibration. The cosinusoidal function is used for the fit with $N$-dependent decay and baseline $F\left(N\right)=e^{-\alpha N}cos[4(\pi /4+{\delta }_A)N+\pi /2]+aN+b$. Here $\alpha ,a$, and $b$ are additional fit parameters introduced empirically. The residual error per gate after the final experiment is $-1.25\times {10}^{-3}$ rad, which is lower than the threshold of ${10}^{-3}\pi$. (d) Compensation tone calibration. The blue and red lines show the result of ${\mathcal{S}}^{xy4}$ without and with the calibrated compensation tone, respectively. The cosinusoidal fit with decay for those curves yields crosstalk amplitude of 176.2 kHz and 6.7 kHz. All data in (a)–(c) are measured with ${\tau }_{\text{sq}}=21.3$ ns, while (d) is measured with ${\tau }_{\text{sq}}=0$ ns.