High-fidelity transmon-coupler-activated $CCZ$ gate on fluxonium qubits

The Toffoli gate takes a special place in the quantum information theory. It opens up a path for efficient implementation of complex quantum algorithms. Despite tremendous progress of the quantum processors based on the superconducting qubits, realization of a high-fidelity three-qubit operation is still a challenging problem. Here, we propose a way to perform a high-fidelity $CCZ$ gate on fluxoniums capacitively connected via a transmon qubit, activated by a microwave pulse on the coupler. The main advantages of the approach are relative quickness, simplicity of calibration and significant suppression of the unwanted longitudinal $ZZ$ interaction. We provide numerical simulation of a 95-ns-long gate of higher than 99.99% fidelity with realistic circuit parameters in the noiseless model and estimate an error of about 0.25% under the conventional decoherence rates.

Figure 1

$CCZ$ gate concept. (a) Circuit layout of the proposed device consisting of three computational fluxoniums $F_1$, $F_2$, $F_3$ (purple) capacitively coupled by a transmon $T$ (orange). (b) The state-dependent spectrum of the coupler $0$-$1$ transition. After a single Rabi oscillation of the coupler population, caused by an external drive at the coupler main transition associated with the $|111⟩$ computational state, the state $|111⟩$ accumulates extra phase $\pi$. The rest of the computational states remains undisturbed, that altogether corresponds to the action of a $CCZ$ operation.

Figure 2

The schematic representation of the interaction between one of the three fluxonium qubits and transmon coupler. The left purple box represents the lowest allowed energy transitions from the $|0⟩$ and $|1⟩$ fluxonium states. In the right orange box, the dashed line shows undisturbed transition frequency of the $0$-$1$ transmon transition. Due to the qubit interaction, illustrated by the arrows, the fluxonium transitions $1$-$2$ and $0$-$3$ push apart the transmon transition associated with fluxoniums states $|0⟩$ and $|1⟩$. As a result, the coupler transition frequency is different for the different computational states. By similar arguments, considering the transmon coupler simultaneously interacting with three fluxonium qubits, we obtain a state-dependent spectrum of the coupler shown in Fig. 1.

Figure 3

The detuning dependence of the coupler behavior, approximated by a two-level model, after a 78-ns-long Gaussian $2\pi$ pulse. The solid green line shows the system population (left axis) and the orange dashed curve presents the common phase acquired on both states (right axis). The vertical dotted lines denote the location of the coupler $0$-$1$ transition associated with the eight computational states. The enlarged area shows the behavior of the population and the phase, accumulated on the qubit states, far from the resonance that is essential for the gate.

Figure 4

$CCZ$ gate performance as a function of the Gaussian pulse amplitude. For each amplitude value we optimize gate fidelity as a function of the external pulse duration $\tau$ (c) and frequency $f$ (d), see Eqs. (4) and (5). The resulting optimal fidelity and the corresponding average population leakage from the computational subspace associated only with the transmon ground state are shown in (a),(b). The bottom plot (d) demonstrates that with the decreasing pulse amplitude, the frequency becomes closer to the resonance transition $|1110⟩$-$|1111⟩$ frequency, indicated by a dashed line.

Figure 5

Quantum circuit of the two-pulse $CCZ$ operation. CCPhase and CCPhase* gates are supposed to be implemented via driving the $|1110⟩$-$|1111⟩$ and $|0000⟩$-$|0001⟩$ transitions. The microwave $2\pi$ pulses on the transmon coupler are shown above the corresponding gates. The change of the computational basis allows one to cancel the unwanted longitudinal $ZZ$ interaction with different phase accumulation signs during the CCPhase-like gates.

Figure 6

The cumulative distribution of the parasitic longitudinal interactions ${\zeta }_{ZZ}$, ${\zeta }_{ZZZ}$ and the characteristic value $\mathrm{\Delta }$ of the state-dependence coupler spectrum calculated for different error spreads $\epsilon$. The vertical line correspond to the designed parameters: ${\zeta }_{ZZ}=5$ kHz, ${\zeta }_{ZZZ}=15.8$ Hz, and $\mathrm{\Delta }=116$ MHz.