A Tunable, Modeless, and Hybridization-free Cross-Kerr Coupler for Miniaturized Superconducting Qubits

TL;DR

This work proposes a junction-based SQUID coupler with small Josephson energies to realize intrinsic cross-Kerr interactions between superconducting qubits, avoiding mode hybridization and extra modes that cause leakage. By operating under the condition phi_e1 + 2 phi_e2 = 0, the scheme isolates even-parity interactions and yields a tunable ZZ rate zeta that can be switched off at an idle point Phi_off and switched on at Phi_on, enabling fast CZ gates. Full-Hamiltonian simulations show CZ gates with durations around 22 ns achieving coherent errors below 3e-7, with robust performance against realistic junction asymmetries and flux-noise levels; a gradiometric layout further mitigates sensitivity to external flux. A scalable tiling strategy for a fully miniaturized mergemon processor demonstrates how all interactions can be mediated by junction-based couplers, achieving high density without bulky capacitors and enabling potential phonon-engineering strategies to further improve coherence. Overall, the work provides a practical path to compact, high-fidelity quantum processors using all-junction, tunable cross-Kerr couplers.

Abstract

Superconducting quantum circuits typically use capacitive charge-based linear coupling schemes to control interactions between elements such as qubits. While simple and effective, this coupling scheme makes it difficult to satisfy competing circuit design requirements such as maintaining large qubit anharmonicity and coherence along with a high degree of qubit connectivity and packing density. Moreover, tunable interactions using linear coupling elements produce dynamical variations in mode hybridization, which can induce non-adiabatic transitions, resulting in leakage errors and limiting gate speeds. In this work we attempt to address these challenges by proposing a junction-based coupling architecture based on SQUID (superconducting quantum interference device) couplers with relatively small Josephson energies. SQUID couplers provide intrinsic cross-Kerr interactions that can be controlled by external fluxes and that do not rely on mode hybridization. The small Josephson energies of the coupler maintain the interaction at a perturbative scale, which limits undesired higher-order mixing between coupled elements while achieving a sufficiently strong cross-Kerr interaction originating from diagonal coupling elements. Based on these properties, we show that a SQUID coupler can be used to implement a fast, adiabatic, and high-fidelity controlled-Z gate without introducing extra modes, and the operation is robust against junction asymmetry for high-frequency qubits. Although unconventional crosstalk may arise due to junction asymmetries and parasitic hybridization with spectator qubits, we show that these effects are sufficiently small for realistic circuit parameters. As an example of the utility of such junction-based coupling schemes, we present a scalable tiling strategy for a miniaturized superconducting quantum processor based on merged-element transmon qubits.

Figures (14)

• Figure 1: Tunable cross-Kerr coupling and CZ gate via a SQUID coupler.a, Circuit schematic of two transmons coupled via a SQUID coupler. b, ZZ interaction rate $\zeta$ and average hybridization as functions of $\Phi_{e,1}$. Red (green) dashed line indicates where $\zeta$ is zero (maximal). Black dashed line represents $\zeta$ calculated from perturbation theory. c, Eigenfrequencies of the first 5 excited states as functions of $\Phi_{e,1}$ under the operating condition $\Phi_{e,2} = -\Phi_{e,1}/2$. d, Flux waveform $\Phi_{e,1}(t)$ and the corresponding $\zeta(t)$, used for a 22 ns-long CZ gate. e, Coherent error of CZ gate for various gate durations $T_\text{G}$ (red circles). Yellow dashed line represents the contribution from non-adiabatic state transitions.
• Figure 2: Robustness against junction asymmetry.a, ZZ interaction rate as a function of $\Phi_{e,1}$ and junction asymmetry. Junction capacitance asymmetry is set to be the same as the Josephson energy asymmetry. Red triangles indicate $\Phi_\text{off}$ obtained from circuit quantization, and black dashed line shows $\Phi_\text{off}$ estimated from perturbation theory. b, Coherent error of the 22 ns-long CZ gate as a function of junction asymmetry.
• Figure 3: Sensitivity to flux noise.a, SQUID coupler circuit with physical external fluxes and realistic inductive network. Capacitors are omitted for brevity. b, Echo dephasing times $T_{\phi, \text{echo}}^{1/f, \ket{10}}$ and $T_{\phi, \text{echo}}^{1/f, \ket{01}}$ calculated for $1/f$ flux noise with amplitudes $A_{\Phi_{e,i}}=10^{-6}\Phi_0$ and $A_{\Phi_{e,o}}=A_{\Phi_{e,o}'}=5\times10^{-6}\Phi_0$. Here we assume worst-case scenario in which the outer SQUID flux noises are anti-correlated. c, Calculated echo dephasing time and minimum achievable gate time ($\pi/\zeta_\text{on}$) as a function of $\Sigma{E_{J,C}}$ for $1/f$ flux noise with the same flux noise amplitudes as in b. d, Coherent error of the 22 ns-long CZ gate as a function of external flux offsets, $\delta\Phi_{e,1}$ and $\delta\Phi_{e,2}$. Green and gray dashed lines represent the range of flux offsets corresponding to the RMS deviation over a 1 hr drift period for $1/f$ flux noise with amplitudes $A_{\Phi_{e,i}}=A_{\Phi_{e,o}}=A_{\Phi_{e,o}'}=5\times 10^{-6}\Phi_0$, assuming an anti-correlation in the outer SQUID flux noises.
• Figure 4: Unconventional crosstalk mediated by SQUID couplers.a, Circuit schematic of a chain of three transmons with nearest-neighbor connection via SQUID couplers. Coupler capacitances are omitted for brevity. b, Energy level diagram of a with transition matrix elements (solid orange lines) due to longitudinal interaction provided in eq. (\ref{['eq:Hlongitudinal']}). c, ZZ interaction rate $\zeta_{13}$ calculated from circuit quantization numerically as a function of $\Delta E_{J,C}$ (purple circles). Black dashed line indicates $\zeta_{13}$ obtained from perturbation theory. d, Circuit schematic of two transmons coupled via a SQUID coupler and a spectator transmon coupled to transmon 2 over a parasitic capacitance $C_\text{para}$. e, Spectator ZZ interaction rates $\zeta_{1S}$ and $\zeta_{2S}$ as functions of spectator 0-1 transition frequency $\omega_S$, with $C_\text{para} = 30 \ \text{aF}$. f, $\zeta_{1S}$ and $\zeta_{2S}$ as functions of $C_\text{para}$, estimated at $\omega_S/2\pi = \omega_{2}/2\pi - 60\ \text{MHz}$ (black dashed line in e).
• Figure 5: Mergemons on a square lattice for a miniaturized quantum processor. Orange (yellow) shape corresponds to high voltage side metalization for low frequency (high frequency) qubits, where an overlap with low voltage side metalization (purple) forms a mergemon. High voltage side metalization of each mergemon extends to their nearest-neighbors, forming SQUID couplers. Low voltage side metalization is connected galvanically to ground metalization (blue), forming the outer SQUID loops. Each mergemon is connected galvanically to its lumped-element readout resonator (green) via a Josephson junction for junction readout. For convenience in illustration, elements including Josephson junctions are intentionally displayed bigger than their realistic relative length scales. Scale bar represents the anticipated lattice constant provided in the main text.