Experimental Demonstration of a Resonator-Induced Phase Gate in a Multiqubit Circuit-QED System

The resonator-induced phase (RIP) gate is an all-microwave multiqubit entangling gate that allows a high degree of flexibility in qubit frequencies, making it attractive for quantum operations in large-scale architectures. We experimentally realize the RIP gate with four superconducting qubits in a three-dimensional circuit-QED architecture, demonstrating high-fidelity controlled-z (cz) gates between all possible pairs of qubits from two different 4-qubit devices in pair subspaces. These qubits are arranged within a wide range of frequency detunings, up to as large as 1.8 GHz. We further show a dynamical multiqubit refocusing scheme in order to isolate out 2-qubit interactions, and combine them to generate a 4-qubit Greenberger-Horne-Zeilinger state.

Figure 1

(a) Picture of our superconducting 4-qubit 3D cQED system with 5 cavities. Four single-qubit chips (false colored: red, blue, green, and pink) are mounted to couple to the bus cavity (center pocket) and individual readout cavities (outer pockets). (b) Close-up photograph of a 3D qubit chip. (c) Diagram of the 4-qubit 3D cQED system. (d) The microwave drive for the RIP gate (cyan arrow) is blue detuned by frequency $\mathrm{\Delta }$ from the dressed cavity resonance ${\omega }_{\mathrm{gggg}}$. Here, $g$ and $e$ indicate the ground and excited states of a qubit.

Figure 2

(a) Excited state population ($P_{\uparrow }$) of the qubit $A2$ (see Table 1) versus single RIP pulse gate time $\tilde{\tau }$ and detuning $\mathrm{\Delta }/2\pi$, from Ramsey experiments for $Z_2{Z}_3$ shown in (f). The red dashed line indicates a threshold time below which no coherent oscillation is observed. (b) Theoretical prediction of $ZZ$ oscillations at ${\tilde{\epsilon }}_R/2\pi =315\mathrm{MHz}$. Inset: Simulated residual photons $⟨n(\tilde{\tau })⟩$ vs $\tilde{\tau }$ and $\mathrm{\Delta }/2\pi$. The red region indicates $⟨n(\tilde{\tau })⟩>$ 0.01; $⟨n(\tilde{\tau })⟩$ drops sharply after the threshold time. (c) $P_{\uparrow }$ of $A1$ for $Z_1{Z}_2{Z}_3$ in (f), showing $ZZZ$ interactions among qubits $A1$, $A2$, and $A3$ as a function of $\tilde{\tau }$. (d) Theoretical calculation of $ZZZ$ oscillations at ${\tilde{\epsilon }}_R/2\pi =200\mathrm{MHz}$. (e) $P_{\uparrow }$ of $A2$ vs $\tilde{\tau }$ at three detuning points. $ZZ$ oscillations from the experiment (blue circles) and theory (red curves) show good agreement. The theoretical drive amplitude is fine-tuned at ${\tilde{\epsilon }}_R/2\pi =315\pm 15\mathrm{MHz}$. (f) Illustration of the pulse sequences for the Ramsey experiment (top) and 4-qubit refocused RIP gate schemes for $Z_2{Z}_3$ (middle) or $Z_1{Z}_2{Z}_3$ (bottom).

Figure 3

(a) Pulse sequence of the 2-qubit refocused RIP gate scheme and $P_{\uparrow }$ of a target qubit from a tune-up procedure using a Ramsey experiment when a control qubit is in the ground state (blue) and in the excited state (red). (b) Population of $|0⟩$ versus the number of Cliffords from the 2-qubit RB on $A2\text{−}A3$ at $\mathrm{\Delta }/2\pi =20\mathrm{MHz}$ with 40 randomization sequences (colored dots). (c) $Z{Z}_{\pi /2}$ fidelity between $A2\text{−}A3$ vs $\mathrm{\Delta }/2\pi$ (red dots). The error bars are from the 2-qubit RB fit. Theoretical upper and lower limits on the gate fidelity without (black dashed line) and with the measurement-induced dephasing (green region, dashed curves) are calculated from minimum and maximum coherence times in Table 1.

Figure 4

(a) Experimentally reconstructed density matrix of a 4-qubit GHZ state from the quantum state tomography [30] performed on device A. The 4-qubit refocused RIP gate scheme is used to create the GHZ state. (b) Theoretically reconstructed ideal density matrix of a 4-qubit GHZ state using the exact gate sequences from the experiment in (a). (c) Theoretical density matrix with static $ZZ$ interactions.