Control the qubit-qubit coupling with double superconducting resonators

1. Problem Statement

In large-scale superconducting quantum processors, tunable couplers are essential for enabling switchable interactions between qubits, facilitating fast two-qubit gates, and minimizing residual $ZZ$ coupling and idling errors. While transmon-based couplers are currently mainstream, they often require complex fabrication, large frequency tuning ranges, and additional flux lines, which introduce flux noise and consume valuable space and cabling resources in dilution refrigerators.

The authors address the challenge of achieving high-fidelity, switchable qubit-qubit coupling with a simpler architecture that minimizes flux noise and hardware overhead. Specifically, they investigate the efficacy of a double-resonator coupler architecture, where two fixed-frequency resonators mediate the interaction between two tunable qubits, aiming to demonstrate the ability to switch the coupling from "off" (zero) to "on" (gate strength) with minimal frequency shifts.

2. Methodology

The study employs a combination of theoretical modeling and experimental verification on a custom superconducting quantum chip.

• Device Architecture:

  • The sample consists of two tunable Xmon qubits coupled to two common, fixed-frequency superconducting resonators.
  • The resonators have fixed frequencies ($\omega_a \approx 4.47$ GHz and $\omega_b \approx 4.80$ GHz).
  • The qubits ($\omega_1, \omega_2$) are independently tunable via Z-control lines (DC-bias currents and flux pulses).
  • Interactions are primarily capacitive.

• Theoretical Framework:

  • The system is modeled using a second-quantization Hamiltonian including direct qubit-qubit coupling ($g_{12}$), direct resonator-resonator coupling ($g_{ab}$), and qubit-resonator couplings ($g_{\lambda\beta}$).
  • The effective qubit-qubit coupling strength ($g_{eff}$) is derived as a function of detuning ($\Delta_{\lambda\beta}$). The theory predicts that by tuning qubit frequencies between the two resonator frequencies, the contributions from the two resonators can cancel each other out (destructive interference), effectively switching off the coupling even if direct coupling exists.

• Experimental Techniques:

  • Frequency Domain: Two-tone spectroscopy was performed using a network analyzer and microwave sources to map the energy spectrum. By sweeping the DC bias of one qubit while fixing the other, the authors measured the anti-crossing gaps, which directly reflect the effective coupling strength.
  • Time Domain: Vacuum Rabi oscillation experiments were conducted. A pulse sequence was used to excite one qubit, tune both qubits to near-resonance via flux pulses for a specific interaction time ($\tau$), and then measure the population exchange. This allowed the observation of the coupling envelope and oscillation periods.
  • Conditions: Measurements were performed at a base temperature above 25 mK without a Josephson parametric amplifier (JPA), resulting in a lower Signal-to-Noise Ratio (SNR) compared to state-of-the-art setups, yet sufficient to observe the coupling dynamics.

3. Key Contributions

• Demonstration of Switch-Off Capability: The paper experimentally validates that a double-resonator coupler can completely switch off the effective qubit-qubit coupling ($g_{eff} \approx 0$) by tuning the qubit frequencies to a specific point where the indirect interactions mediated by the two resonators cancel out.
• Minimal Frequency Tuning: The study shows that the transition from the "switching off" point to a functional "two-qubit gate" point (where $g_{eff} > 5$ MHz) requires a frequency shift of only ~50 MHz. This is a significant reduction compared to other architectures requiring larger tuning ranges.
• Hardware Efficiency: The architecture eliminates the need for additional flux lines dedicated to the coupler (since the resonators are fixed frequency), thereby reducing flux noise, minimizing the number of cables in the dilution refrigerator, and allowing for a more compact chip design.

4. Key Results

• Frequency Domain Measurements:

  • Spectroscopy revealed anti-crossing gaps between qubits and resonators.
  • By fixing Qubit-2 and sweeping Qubit-1, the anti-crossing gap (representing $g_{eff}$) was observed to shrink from ~10 MHz to < 2 MHz (near the switching-off point) as the qubit frequencies approached the cancellation condition.
  • At the optimal switching-off point (Qubit-2 $\approx$ 4.37 GHz), the gap became almost invisible, confirming $g_{eff} \approx 0$.
  • Shifting the frequency by 50 MHz from this point restored the coupling to **5 MHz**, suitable for two-qubit gates.

• Time Domain Measurements:

  • Vacuum Rabi oscillations were observed in the time domain.
  • Despite the low SNR (due to the absence of a JPA and higher base temperature), the envelope of the oscillation clearly varied with the flux pulse amplitude.
  • At the switching-off point (flux amplitude $\approx$ -6.0 mV), the oscillation envelope was flat (no energy exchange).
  • As the flux amplitude changed (increasing detuning or moving away from the null point), clear oscillatory behavior emerged, confirming the tunability of the coupling strength.
  • Simulation using the Lindblad master equation matched the experimental trends, accounting for decoherence ($T_1, T_2$).

5. Significance and Future Outlook

• Scalability: The double-resonator coupler offers a promising path for scaling superconducting quantum processors. Its simple fabrication and lack of dedicated flux lines for the coupler reduce the "wiring bottleneck" in dilution refrigerators.
• Noise Suppression: By operating near the switching-off point with minimal frequency shifts, the system is less susceptible to flux noise, potentially improving gate fidelities and reducing residual $ZZ$ errors.
• Compactness: Future designs can utilize narrower coplanar waveguide resonators to further reduce the physical footprint of the coupler on the chip.
• Potential Improvements: The authors note that future iterations could benefit from the integration of Josephson parametric amplifiers (JPAs) and Purcell filters to enhance SNR and coherence times, as well as better calibration of Z-pulse distortions.

In summary, this work provides a robust experimental proof-of-concept for a double-resonator coupler that achieves precise, switchable qubit-qubit interactions with minimal hardware overhead, addressing critical bottlenecks in the scaling of superconducting quantum computers.