The work proposes using SQUID couplers with relatively small Josephson energies, a departure from typical approaches that seek larger values, to minimize errors caused by “dynamical variations in mode hybridization” and resulting non-adiabatic transitions. This junction-based design maintains a strong qubit interaction while limiting undesired mixing between elements, enabling a fast, high-fidelity controlled-Z gate. The authors present a scalable tiling strategy for a miniaturized superconducting quantum processor based on merged-element transmon qubits.

Tunable SQUID Couplers Address Superconducting Qubit Challenges

SQUID couplers employing relatively small Josephson energies offer a pathway to denser superconducting circuits, addressing limitations inherent in traditional capacitive coupling schemes. This design choice actively reduces Josephson energy, typically maximized in these devices, to maintain interaction at a perturbative scale, limiting unwanted mixing between qubits and enhancing connectivity. The approach tackles a critical challenge in scaling quantum processors: balancing qubit coherence, connectivity, and packing density, all simultaneously.

Conventional linear coupling schemes introduce “dynamical variations in mode hybridization” that can trigger non-adiabatic transitions, ultimately leading to leakage errors and restricting the speed of quantum gate operations. The proposed junction-based architecture circumvents this issue by using cross-Kerr interactions controlled by external fluxes, effectively decoupling qubit interaction from problematic mode hybridization. This intrinsic control mechanism is a key distinction from earlier coupler designs, offering a more stable and predictable interaction between qubits.

Analysis of the circuit reveals that even with potential junction asymmetries and parasitic hybridization with neighboring qubits, crosstalk remains sufficiently small for practical implementation with realistic circuit parameters. This architecture anticipates SQUID couplers with even smaller junction energies and reduced loop sizes, enabled by the elimination of bulky shunt capacitors. The proposed architecture also allows for the implementation of a fast, adiabatic and high-fidelity controlled-Z gate without introducing additional modes, a component essential for complex quantum algorithms.

This gate operation is further enhanced by its robustness against junction asymmetry, a significant advantage for high-frequency qubits. The work identifies two novel forms of crosstalk mediated by the SQUID couplers, both of which are demonstrated to be sufficiently small for realistic circuit parameters.

Cross-Kerr Interactions Enable Direct Qubit Control

A junction-based coupling architecture avoids introducing additional modes that can cause leakage errors during qubit gate operations, a challenge inherent in conventional capacitive schemes. Unlike earlier designs relying on qubit nonlinearities for entanglement, this approach directly generates intrinsic cross-Kerr interactions controllable with external magnetic flux. This direct control circumvents the need for complex mode hybridization, a source of instability in previous superconducting circuits. The design’s effectiveness hinges on a specific flux $φ_{e, 1}$ + $2 φ_{e, 2}$ = 0, which prioritizes even-parity interactions like excitation hopping and cross-Kerr coupling over odd-parity interactions.

Under this condition, the interaction Hamiltonian can be rewritten using trigonometric relations, optimizing qubit connectivity. Calculations detailed in Appendix A demonstrate the impact of this approach on interaction rates. Although unconventional crosstalk may arise due to junction asymmetries and parasitic hybridization with spectator qubits, these effects are sufficiently small for realistic circuit parameters.

The architecture can propagate cross-Kerr interaction from one qubit to a non-directly coupled spectator qubit, as illustrated in a circuit diagram. The design’s impact extends beyond basic qubit coupling, offering potential for advanced applications like dissipatively-stabilized cat qubit systems. In such systems, the coupling rate, $g_{2}$ , is directly proportional to the bare zero-point fluctuations of the two modes, separating the strength of two-photon dissipation from self- and cross-Kerr induced effects.

This separation allows for greater flexibility in allocating storage and buffer mode frequencies, a significant advantage over conventional capacitively-coupled systems. The SQUID coupler facilitates engineering next-nearest-neighbor qubit interactions, demonstrated in a three-transmon circuit operating under the condition $φ_{e, 1}$ + $2 φ_{e, 2}$ = π. The resulting Hamiltonian, H_(sine-sine) = Σ $E_{J, C}^{12}$ cos(φ_(e12)/2) sin(φ_2-φ_1) + Σ $E_{J, C}^{23}$ cos(φ_(e23)/2) sin(φ_3-φ_2), illustrates the potential for complex qubit networks.

This architecture represents a departure from traditional capacitive coupling, offering a pathway towards more robust and scalable superconducting quantum processors. The ability to directly control cross-Kerr interactions without relying on mode hybridization addresses a key limitation of current designs.

SQUID Coupler Design for Fast Controlled-Z Gates

SQUID couplers establish cross-Kerr interactions controllable by external fluxes without relying on mode hybridization, a departure from earlier designs. The architecture achieves this by maintaining a perturbative scale of interaction through relatively small Josephson energies, limiting higher-order mixing between qubits while still enabling a strong cross-Kerr interaction originating from diagonal coupling elements.

Analysis of the SQUID coupler begins with a review of its Hamiltonian and external flux control parameters when mediating coupling between detuned transmon qubits, revealing a high-fidelity and fast controlled-Z gate with minimal adiabaticity overhead. A regular square lattice layout, coupled with separation of metallization layers, avoids extraneous wiring and superconducting loops, creating a gradiometric design insensitive to homogeneous magnetic flux.

The interaction rate can exceed 1 MHz for Σ $E_{J, C} / 2 π$ = 0. 8 GHz, and can be further strengthened by increasing Σ $E_{J, C}$ . The SQUID coupler also enables tunable XX interactions for implementing an adiabatic $\sqrt{i S W A P}$ gate between dual-rail transmons. “The SQUID coupler provides modeless and tunable cross-Kerr coupling, which realizes a fast and high-fidelity CZ gate between transmons with minimal adiabaticity overhead,” the paper concludes.

The SQUID coupler preserves the error hierarchy in dual-rail qubit systems, preventing the introduction of additional channels for leakage errors. Operation at the artificial sweet-spot further suppresses the impact of frequency noise. As an example of the utility of such junction-based coupling schemes, a scalable tiling strategy for a miniaturized superconducting quantum processor based on merged-element transmon qubits is presented.

Junction Asymmetry Impacts Coupler Extinction Ratio

This resilience is key because junction asymmetry can reduce the on-off extinction ratio of the coupler and introduce crosstalk with neighboring qubits, potentially undermining the fidelity of quantum operations. The design mitigates these issues, achieving interaction rates of tens of MHz while minimizing parasitic crosstalk and maintaining low infidelity despite typical levels of flux noise.

The robustness of this architecture to junction asymmetry was tested through both perturbative and numerical simulations of a controlled-Z (CZ) gate, revealing coherent errors remain below 5 x $10^{- 7}$ for gate times of 22 nanoseconds across the entire range of asymmetry studied. Detailed analysis, outlined in Appendix E, demonstrates the coupler’s ability to maintain high fidelity even when junctions are not perfectly matched, a practical advantage for large-scale fabrication.

The study also indicates that the entangling phase of the CZ gate remains stable against typical flux noise levels in superconducting circuits, further bolstering the design’s reliability. As an example of the utility of such junction-based coupling schemes, a scalable tiling strategy for a miniaturized superconducting quantum processor based on merged-element transmon qubits is presented.

Scalable Tiling Strategy for Miniaturized Quantum Processors

This design choice, detailed in the work by Kim, Butler and Painter, allows for a scalable tiling strategy aimed at creating fully miniaturized superconducting quantum processors. The approach moves beyond traditional capacitor-based coupling by employing SQUID couplers to mediate interactions between qubits, offering a pathway to increased connectivity without sacrificing qubit coherence. The researchers demonstrate that unconventional crosstalk effects are sufficiently small, a critical factor for maintaining fidelity in complex quantum computations.

👉 More information
🗞 Tunable, modeless, and hybridization-free cross-Kerr coupler for miniaturized superconducting qubits
✍️ Gihwan Kim, Andreas Butler and Oskar Painter
🧠 DOI: http://link.aps.org/doi/10.1103/cx2t-3hq7

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