High-Contrast $ZZ$ Interaction Using Superconducting Qubits with Opposite-Sign Anharmonicity

For building a scalable quantum processor with superconducting qubits, $ZZ$ interaction is of great concern because its residual has a crucial impact to two-qubit gate fidelity. Two-qubit gates with fidelity meeting the criterion of fault-tolerant quantum computation have been demonstrated using $ZZ$ interaction. However, as the performance of quantum processors improves, the residual static $ZZ$ can become a performance-limiting factor for quantum gate operation and quantum error correction. Here, we introduce a superconducting architecture using qubits with opposite-sign anharmonicity, a transmon qubit, and a $C$-shunt flux qubit, to address this issue. We theoretically demonstrate that by coupling the two types of qubits, the high-contrast $ZZ$ interaction can be realized. Thus, we can control the interaction with a high on-off ratio to implement two-qubit controlled-$Z$ gates, or suppress it during two-qubit gate operation using $XY$ interaction (e.g., an iSWAP gate). The proposed architecture can also be scaled up to multiqubit cases. In a fixed coupled system, $ZZ$ crosstalk related to neighboring spectator qubits could also be heavily suppressed.

Figure 1

(a) Layout of a two-dimensional nearest-neighbor lattice, where circles at the vertices denote qubits, and gray lines indicate couplers between adjacent qubits. The lattice consists of two-type qubits arranged in an $\text{−}A\text{−}B\text{−}A\text{−}B$- pattern in each row and column. The circles with $A$ and $B$ are qubits with opposite-sign anharmonicities, and each one can be treated as three-level anharmonic oscillators (b). Typically, transmon qubits and $C$-shunt flux qubits can be modeled as anharmonic oscillators with negative and positive anharmonicity, respectively. Qubits can be coupled to each other (c) directly via a capacitor or (d) indirectly using a resonator.

Figure 2

Numerical calculation of the energy levels of the coupled qubit system as a function of the qubit detuning $\mathrm{\Delta }={\omega }_a-{\omega }_b$ for qubit frequency ${\omega }_b/(2\pi )=5.5\mathrm{GHz}$, anharmonicity $\alpha /2\pi =250\mathrm{MHz}$ [8, 26], and coupling strength $\mathrm{g}/2\pi =15\mathrm{MHz}$. (a) Qubits with same-sign anharmonicity ${\alpha }_{a(b)}=-\alpha$; (b) qubits with opposite-sign anharmonicity ${\alpha }_{a(b)}=\mp \alpha$. The inset highlights the triple degeneracy point in the two-excitation manifold $\{|02⟩,|20⟩,|11⟩\}$.

Figure 3

Numerical results for $ZZ$ coupling strength $|\zeta |$. (a) $|\zeta |$ versus qubit detuning $\mathrm{\Delta }$ for anharmonicity asymmetry ${\delta }_{\alpha }=|{\alpha }_b|-|{\alpha }_a|=0$, where the dashed line shows results for the $AA$ type. (b) $|\zeta |$ versus ${\delta }_{\alpha }$ for $\mathrm{\Delta }/2\pi =-150\mathrm{MHz}$, where the dashed line shows perturbational result. The inset shows that $|\zeta |$ could be suppressed below 60 KHz with the typical anharmonicity asymmetry ($-20–20\mathrm{MHz}$). (c) $|\zeta |$ versus $\mathrm{\Delta }$ and ${\delta }_{\alpha }$. Horizontal (vertical) cut through (c) denotes the result plotted in (b). (d) $ZZ$ coupling in a system comprising two qubits that are coupled via a resonator [26].

Figure 4

Numerical results for CZ gate and iSWAP gate implementation in our architecture. The system parameters used are same as in Fig. 2. (a),(d) Typical pulses with small overshoots for realizing CZ gates and iSWAP gates, where the full width at half maximum is defined as hold time. (b),(e) Leakage ${ϵ}_{\mathrm{laek}}$ and swap error ${ϵ}_{\mathrm{swap}}$ versus hold time for system with anharmonicity asymmetry ${\delta }_{\alpha }=0$ and optimal overshoots [26]. (c),(f) Gate error $1-F$ versus typical anharmonicity asymmetry. The phase error ${\delta }_{\theta }$, ${\delta }_{\varphi }$ with respect to the ideal phase parameters (0, $\pi$ for CZ gate, and $\pi /2$, 0 for iSWAP gate) and leakage error ${ϵ}_{\mathrm{leak}}$ are also presented for identifying the major source of error.