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

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.

💡 Research Summary

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The paper addresses a fundamental limitation of conventional superconducting quantum processors that rely on capacitive, charge‑based linear couplers. While such couplers are simple and effective, they create a tension between three key design goals: preserving large qubit anharmonicity and coherence, achieving high qubit connectivity, and minimizing the physical footprint of the circuit. Moreover, tunable linear couplers inevitably induce time‑dependent mode hybridization, which can cause non‑adiabatic transitions, leakage errors, and ultimately limit gate speed and fidelity.

To overcome these challenges, the authors propose a junction‑based coupling architecture built around a SQUID (Superconducting Quantum Interference Device) coupler whose constituent Josephson junctions have relatively small tunneling energies (E_JC ≲ 1 GHz). The small Josephson energies keep the interaction in the perturbative regime, suppressing higher‑order mixing while still providing a sizable cross‑Kerr (ZZ) interaction that can be switched on and off with external magnetic fluxes. Crucially, the cross‑Kerr originates from diagonal (even‑parity) terms in the Hamiltonian and does not rely on hybridization of the qubit modes, thereby eliminating the extra “parasitic” modes that typically serve as leakage channels in linear‑coupler schemes.

The authors first derive the full circuit Hamiltonian for two detuned transmon qubits coupled via the SQUID. By imposing the flux operating condition φ_e1 + 2φ_e2 = 0, the interaction Hamiltonian separates into an even‑parity part (providing the desired ZZ coupling) and an odd‑parity part (responsible for longitudinal and two‑photon exchange processes). In the symmetric case (ΔE_JC = 0) only the even‑parity term survives, yielding a first‑order ZZ rate

 ζ^(1) = −ΣE_JC cos(φ_e1/2) (φ_zpf1 φ_zpf2)²,

which depends weakly on the qubit detuning and can be tuned from zero (idle) to a maximum of ≈ −27 MHz by adjusting the external flux. Because the effective linear hopping g_eff can be made vanishingly small over a wide flux interval, the second‑order hybridization‑induced ZZ term ζ^(2)_c is strongly suppressed. Numerical simulations using realistic parameters (qubit frequencies ≈ 4.5 GHz and 6.3 GHz, junction energies ≈ 0.4 GHz, coupling capacitances ≈ 0.8 fF) confirm that the average hybridization with non‑computational states stays below 0.3 % throughout the tuning range, and the qubit frequencies shift by less than 320 MHz, ensuring robustness against flux noise.

For gate implementation, the authors design an adiabatic flux pulse that moves the system from the idle point (Φ_off ≈ 0.516 Φ₀) to the “on” point (Φ_on = 0) and back. By shaping the mixing angle with a combination of a square pulse (duration β T_G) and a Slepian‑like envelope (duration (1−β) T_G), and applying a modest Gaussian low‑pass filter (σ = 0.5 ns) to mimic realistic control electronics, they achieve a controlled‑Z (CZ) gate in 22 ns with a coherent error below 3 × 10⁻⁷. The minimal adiabatic overhead (≈ 3.4 ns) is only modestly above the theoretical limit set by the on‑state ZZ rate (π/|ζ_on| ≈ 18.6 ns). Simulations that include T₁ = 1 ms relaxation predict a non‑coherent infidelity of ≈ 8 × 10⁻⁵, far below the coherent error budget.

The paper then examines robustness to junction asymmetry. Even with up to 20 % asymmetry (ΔE_JC/ΣE_JC ≤ 0.2), the odd‑parity term remains small because it scales with sin(φ_e1/2) and is further suppressed by the small zero‑point phase fluctuations (φ_zpf ≈ (2E_C/E_J)^{1/4} ≲ 0.02). Numerical results show that both the ZZ rate and the effective hopping remain essentially unchanged, and the CZ error stays below 5 × 10⁻⁷ across the full asymmetry range.

Potential crosstalk mechanisms are also investigated. Parasitic SQUID loops formed by ground connections and unintended coupling to spectator qubits can generate additional ZZ terms, but with the chosen small junction energies these contributions are on the order of 10⁻⁴ Hz, negligible for practical purposes.

Finally, the authors present a scalable architecture based on “merged‑element transmons” (mergemons) where every inter‑qubit connection is mediated by a SQUID cross‑Kerr coupler. By eliminating geometric coupling capacitors, the layout achieves a ≈ 30 % reduction in cell area, allowing dense tiling (e.g., a 7 × 7 array with 196 qubits occupying < 2 mm² per cell). The design requires only flux‑bias lines for dynamic control, dramatically simplifying wiring and reducing cross‑talk, thereby offering a clear path toward large‑scale, high‑density superconducting quantum processors.

In summary, the work demonstrates that a low‑energy SQUID coupler can provide strong, tunable, and mode‑free cross‑Kerr interactions, enabling ultra‑fast, high‑fidelity CZ gates (≈ 20 ns, error < 10⁻⁶) while maintaining robustness against fabrication asymmetries and parasitic couplings. This junction‑based, hybridization‑free coupling paradigm opens a promising route for miniaturized, highly connected quantum processors.