Tunable Hybrid-Mode Coupler Enabling Strong Interactions between Transmons at Centimeter-Scale Distance

TL;DR

This work addresses the connectivity bottleneck in superconducting qubit architectures by introducing a tunable hybrid-mode coupler that mediates long-range interactions between transmons separated by a centimeter. The authors develop a distributed-element CPW-based model with an embedded Josephson junction, deriving a two-mode effective Hamiltonian that captures flux-dependent XX and ZZ couplings via coupler-mediated exchange, with J_{12} = ∑_m g_{1m} g_{2m} (1/Δ_{1m} + 1/Δ_{2m}). Experimentally, they extract coupler-qubit couplings (e.g., |g_{21}|/2π ≈ 160 MHz, |g_{11}|/2π ≈ 50 MHz, |g_{12}|/2π ≈ 135.18 MHz) and demonstrate XX couplings up to ~23 MHz and ZZ interactions up to ~100 MHz in optimal flux regions, in good agreement with theory. The results illuminate how coupler length, junction placement, and capacitances shape mode spectra and nonlinearities, and they discuss strategies to suppress unwanted pathways and mitigate frequency crowding, pointing to a viable path toward hardware-efficient, modular quantum processors with extended connectivity.

Abstract

The transmon, a fabrication-friendly superconducting qubit, remains a leading candidate for scalable quantum computing. Recent advances in tunable couplers have accelerated progress toward high-performance quantum processors. However, extending coherent interactions beyond millimeter scales to enhance quantum connectivity presents a critical challenge. Here, we introduce a hybrid-mode coupler exploiting resonator-transmon hybridization to simultaneously engineer the two lowest-frequency mode, enabling high-contrast coupling between centimeter-scale transmons. For a 1-cm coupler, our framework predicts flux-tunable $XX$ and $ZZ$ coupling strengths reaching 23 MHz and 100 MHz, with modulation contrasts exceeding $10^2$ and $10^4$, respectively, demonstrating quantitative agreement with an effective two-channel model. This work provides an efficient pathway to mitigate the inherent connectivity constraints imposed by short-range interactions, enabling transmon-based architectures compatible with hardware-efficient quantum tasks.

Figures (8)

• Figure 1: (a) False color sample image. Optical micrograph shows two grounded transmons ($Q_{1,2}$, blue), coupled via a flux-tunable hybrid-mode coupler (THMC, green), with an effective seperation of approximately 1 cm. The states of the qubits and coupler are read out through dispersive measurements using individual resonators ($R_{1,2,c}$, yellow). (b) Distributed-element circuit model. The THMC couples to both transmons via lumped elements characterized by per-unit-length parameters $C_0$ and $L_0$, defined in the continuum limit $\Delta x \to 0$. The embedded transmon at position $x_J$ partitions the coupler into left ($-l \leq x < x_J$) and right ($x_J < x \leq l$) segments. The embedded transmon contains a shunt capacitance $C_J$ and two Josephson junctions, with energy $E_{Jc}$, threaded by external flux $\Phi$.
• Figure 2: (a) Wave number solutions. Spatial mode characterization showing the calculated wave number $k_m$ with a geometric length $l$. The condition, amplitude at 0, corresponds to the solution of the transcendental equation Eq. \ref{['eq:transcendental']}, which determines the wavenumber. Dashed lines represent analytic predictions. (b) Flux-dependent mode envelopes. Flux envelope functions for the five lowest-frequency normal modes at a flux bias of $\Phi/\Phi_0 = 0$ (upper panel) and $\Phi/\Phi_0 = 0.5$ (lower panel). The results were obtained using the parameters provided in Table. \ref{['table:designed_params']}. The shifting of antinode values is attributed to the modulation of the Josephson inductance. (c) Experimentally measured flux-tunable spectroscopy. Transmission magnitude as a function of flux bias and driving frequency. Dashed lines correspond to the full-circuit model fits based on Eq. \ref{['eq:H_QM2']}. As $\Phi/\Phi_0$ increases from 0 to 0.5, the first mode shifts from 5.02 GHz to below 3 GHz, while the second mode shifts from 7.95 GHz to 6.622 GHz. The vertical feature near $\Phi/\Phi_0=\pm0.5$ may result from the second mode approaching the readout resonator. The horizontal resonances at 6.4 GHz and 7.2 GHz corresponds to the fundamental modes of readout resonators $R_{1,c,2}$. (d) Anharmonicity. Theoretically calculated charge energy $E_{C,m}$ as a function of flux, normalized by the mode frequency $\omega_m$ (upper panel). Theoretically calculated Kerr coefficient $\mathcal{K}_{mm} = \frac{E_{Jc}}{E_{L,m}}E_{C,m}$ (lower panel) is shown alongside the experimentally measured first-mode anharmonicity $\eta_0 = \omega_0^{21}-\omega_0^ {10}$ (dimond markers), which agrees well with the theoretical predictions. The results show that the tunable coupler exhibits weak anharmonicity in the lower-order modes, which can be adjusted by the external flux bias.
• Figure 3: Experimentally measured avoided crossings. (a) At zero coupler bias, $\Phi/\Phi_0=0$, sweeping the $Q_2$ flux reveals an avoided crossing with coupling strength $|g_{21}|/2\pi=160$ MHz. (b) Sweeping the flux bias of $Q_1$ yields a coupling strength $|g_{11}|/2\pi=50$ MHz. (c) With $Q_1$ fixed at $\omega_1/2\pi=6.504$ GHz and $\Phi/\Phi_0=0$, sweeping the coupler flux $\Phi$ reveals an avoided crossing between the coupler mode $\nu_2$ and the $Q_1$ readout resonator with a dispersive frequency 6.954 GHz, indicating $g_{r1m2}/2\pi=21.04$ MHz. Considering the dispersive shift of $10.9$ MHz, the coupling strength is inferred to be $|g_{12}|/2\pi=135.18$ MHz.
• Figure 4: (a) Schematic diagram of the coupling architecture between two transmon qubits. The 1 cm coupling length suppresses higher-order mode interference, enabling an effective two-path coupling model between the qubits. (b) Electric field profile in the coupling port as a function of flux bias $\Phi/\Phi_0$. The amplitude and phase of the field, modulated by the embedded Josephson junction, directly determine the coupling strength between the transmon and the mode. Flux engineering enables precise control over field synchronization. (c) Theoretical predictions of the mode-transmon coupling strengths by using Eq. \ref{['eq:g_jc']}. Star makers represent the result inffered from Fig. \ref{['fig:figure3']}.
• Figure 5: Experiemtally measured $XX$ interactions between qubits. (a) Chevron pattern of vacuum Rabi oscillations. The left qubit is fixed at $\omega_1/2\pi = 5.5$ GHz, while the right qubit frequency is tuned via its flux bias. Left panel: The coupler is biased at $\Phi/\Phi_0 = 0.04$, with mode frequencies $\nu_1/2\pi = 5.012$ GHz and $\nu_2/2\pi = 7.937$ GHz. Right panel: The coupler is biased at $\Phi/\Phi_0 = 0.64$, with mode frequencies $\nu_1/2\pi = 3.907$ GHz and $\nu_2/2\pi = 7.054$ GHz. The trajectory of the slowest oscillation in the Chevron pattern, plotted versus $\Phi$, is shown in (b) to clarify the distinct exchange speeds. (b) The $XX$ strength is extracted by fitting the vacuum Rabi oscillations. A strong coupling strength reaching $24.3$ MHz ($27.8$ MHz) is achieved in experiment (simulation), while the minimum coupling is $0.0$ MHz (-2.8 MHz) at $\Phi/\Phi_0 = 0.5$, yielding a switching ratio. Markers represent experimental data, the dashed line denotes calculations based on the two-mode coupling model Eq. \ref{['eq:H_eff']} in the dispersive regime, and the pink line corresponds to numerical simulations.