Benchmarking Coherent Errors in Controlled-Phase Gates due to Spectator Qubits

A major challenge in operating multiqubit quantum processors is to mitigate multiqubit coherent errors. For superconducting circuits, besides crosstalk originating from imperfect isolation of control lines, dispersive coupling between qubits is a major source of multiqubit coherent errors. We benchmark phase errors in a controlled-phase gate due to dispersive coupling of either of the qubits involved in the gate to one or more spectator qubits. We measure the associated gate infidelity using quantum-process tomography. We point out that, due to coupling of the gate qubits to a noncomputational state during the gate, two-qubit conditional-phase errors are enhanced. Our work is important for understanding limits to the fidelity of two-qubit gates with finite on-off ratio in multiqubit settings.

Figure 1

(a) Spectator qubits S1 and S2 are coupled to gate qubits G1 and G2, between which we perform a controlled-phase gate. (b) Energy-level diagram of the states $|G1G2⟩=|11⟩$ and $|02⟩$, which are shifted due to dispersive interaction with the spectator qubits. (c) Dispersive coupling strengths ${\zeta }_1$, ${\zeta }_2$, and ${\zeta }_{12}$ as a function of detuning $\mathrm{\Delta }$ between the gate qubits. (d) Qubit connectivity of the device studied. Numbers next to the qubits indicate qubit idling frequencies in gigahertz.

Figure 2

Conditional-phase and leakage errors due to spectator qubits. (a) Pulse sequence for the conditional-phase measurement. The rectangular pulses are flux pulses, short rf bursts represent $\pi /2$ and $\pi$ pulses, and long rf bursts at the end represent readout pulses. Dashed pulses indicate that measurements are performed with and without that pulse. (b) Example of a conditional-phase measurement with the spectator qubit in the $|0⟩$ state (dark-blue and dark-green data) and in the $|1⟩$ state (light-blue and light-green data). The corresponding sinusoidal fits are shown as solid lines. (c) Conditional-phase error $\delta {\mathrm{\Phi }}_c$ between (c) $\mathrm{Q}4$ and $\mathrm{Q}2$ due to spectator qubit $\mathrm{Q}1$ and between (d) $\mathrm{Q}1$ and $\mathrm{Q}4$ due to spectator qubit $\mathrm{Q}3$ as a function of the detuning between the spectator qubit and its neighboring gate qubit. (e),(f) Corresponding leakage errors. (g) Conditional-phase error $\delta {\mathrm{\Phi }}_c$ from (c),(d) as a function of detuning between the gate qubit states $|11⟩$ and $|02⟩$ during the gate. The solid lines in (c),(d),(g) are calculated with Eq. (3) and those on in (e),(f) are calculated with Eq. (5).

Figure 3

Measured conditional-phase error $\delta {\mathrm{\Phi }}_c$ (orange circles) in the controlled-phase gate between $\mathrm{Q}4$ and $\mathrm{Q}2$ as a function of spectator-qubit configuration $|\mathrm{Q}1\mathrm{Q}6\mathrm{Q}7⟩$. Each data point represents the mean of six measurements, and error bars indicate one standard deviation. The open squares are calculated values (see the main text).

Figure 4

Increase in controlled-phase-gate error $\delta {\epsilon }_{\mathrm{CZ}}$ in the presence of multiple spectator qubits as a function of the total dispersive shift of the gate qubit $\mathrm{Q}4$ (orange circles). The top axis indicates the states of the three spectator qubits during each measurement. The solid line is a calculation of the gate error in the presence of phase errors only. Error bars are derived from a bootstrapping method.

Figure 5

Dynamical-phase errors in the presence of multiple spectator qubits. Dynamical-phase error $\delta {\mathrm{\Phi }}_d$ (orange circles) on Q4 during the controlled-phase gate between $\mathrm{Q}4$ and $\mathrm{Q}2$ as a function of spectator-qubit configuration $|\mathrm{Q}1\mathrm{Q}6\mathrm{Q}7⟩$. Each data point represents the mean of two measurements and error bars indicate one standard deviation. The open squares are calculated values (see the text).