Parametric $\mathrm{CZ}$ Gates in a Tunable Tripod Architecture with Fixed-Frequency Superconducting Qubits

Technical Summary: Parametric CZ Gates in a Tunable Tripod Architecture with Fixed-Frequency Superconducting Qubits

Problem Statement
The implementation of quantum error correction (QEC) in superconducting quantum computers requires high gate fidelity, adequate qubit connectivity, and selective control of interactions. As QEC codes evolve toward low-density parity-check (qLDPC) structures, the requirement for qubit connectivity increases (e.g., degree-four or higher), surpassing the constraints of conventional surface codes. While sharing a single coupler among multiple qubits can enhance connectivity and reduce circuit layout complexity, it introduces significant challenges: interactions must be selectively induced for target pairs while suppressing unwanted effects on non-target "spectator" qubits. Previous architectures using fixed-frequency couplers and microwave-driven cross-resonance (CR) interactions struggle to implement controlled-phase (CZ) gates efficiently due to limited control over effective qubit-qubit interactions and resonance conditions. Furthermore, existing shared-coupler studies often focus on fixed-frequency couplers, which lack the dynamic tunability required to selectively establish resonance for specific qubit pairs without directly tuning the data qubits.

Methodology
The authors propose and numerically investigate a "tunable tripod" architecture consisting of three fixed-frequency transmon qubits connected to a single tunable coupler. The coupler is modeled as an asymmetric DC-SQUID, allowing its frequency to be adjusted via an external magnetic flux ($\Phi_{ext}$), while the data qubits remain at fixed frequencies.

• System Modeling: The system is described by a total Hamiltonian including the coupler, three qubits, and their capacitive interactions. To accurately capture leakage and non-computational transitions, the simulation truncates each circuit element to a three-level basis ($|0\rangle, |1\rangle, |2\rangle$), resulting in a full system Hilbert space of 81 levels.
• Gate Mechanism: Instead of microwave drives on individual qubits, the CZ gates are implemented via parametric resonance. The external flux applied to the coupler is modulated using a sinusoidal flat-top pulse ($\Phi_{ext}(t)$) to bring the energy levels of the target pair into resonance. Specifically, the gate relies on interactions between the computational state $|11\rangle$ and non-computational states $|20\rangle$ or $|02\rangle$.
• Optimization Protocol: The authors employ a parameter sweep followed by Nelder-Mead optimization to determine the optimal flux pulse parameters (amplitude $\Phi_{ac}$, frequency $f_d$, duration $T$, ramp ratio $\rho$, and initial phase $\phi_{init}$).
• Performance Metrics: Gate fidelity is evaluated by constructing a $4 \times 4$ computational block from the full time-evolution operator, corresponding to the target pair's basis states while assuming the coupler and spectator qubit are in their ground states. Virtual Z gates are applied to correct single-qubit phases. Fidelity is calculated against an ideal CZ gate. Leakage is defined as the population remaining outside this selected four-dimensional subspace at the end of the gate.

Key Contributions

1. Architecture Proposal: Definition and numerical validation of a tunable tripod structure where three fixed-frequency qubits share a single tunable coupler, enabling selective CZ gate implementation via flux modulation without tuning the qubits themselves.
2. Selective Gate Implementation: Demonstration of how to construct distinct CZ gates ($CZ_{01}$, $CZ_{02}$, and $CZ_{12}$) for all three possible qubit pairs within the same hardware configuration by optimizing specific flux pulse parameters for each pair.
3. Spectator Analysis: A detailed analysis of the "spectator" qubit (the non-target qubit in the pair) reveals that population preservation at the end of the gate is highly dependent on the spectator's initial state. While a spectator in $|0\rangle$ remains largely unaffected, a spectator in $|1\rangle$ can experience significant population transfer or leakage, highlighting the necessity of evaluating performance across the full computational space.

Results
The numerical simulations yielded the following results for the target-pair computational blocks (where the coupler and spectator are in the ground state):

• Gate Fidelities: The average gate fidelities ($F_{avg}$) after virtual Z correction are:
  • $CZ_{01}$: 0.9943
  • $CZ_{02}$: 0.9834
  • $CZ_{12}$: 0.9907
• Leakage: The leakage ($L$) into states outside the selected four-dimensional subspace was measured as 0.0047, 0.0139, and 0.0086 for the respective gates.
• Spectator Dynamics: For gates where the spectator qubit starts in $|1\rangle$, the probability of remaining in the computational basis is significantly lower than when it starts in $|0\rangle$. For instance, in the $CZ_{01}$ gate with a spectator in $|1\rangle$, the system exhibits leakage and state mixing involving states like $|101\rangle$ and $|011\rangle$.
• Preliminary Multi-Qubit Analysis: A preliminary analysis of a CCZ gate using the same parameter sweep data yielded a corrected fidelity of approximately 0.8415, with a leakage of ~11.51% relative to the eight-dimensional computational subspace. The authors note this is a preliminary search, not an optimized result.

Significance and Claims
The paper claims to demonstrate the feasibility of constructing CZ operations for each qubit pair in a shared-coupler architecture through flux modulation, while maintaining fixed data-qubit frequencies. This approach offers a potential pathway to increased connectivity required for advanced QEC codes without the complexity of individual flux-tuning for every qubit.

However, the authors explicitly state that these results highlight a critical requirement: performance evaluation and control optimization must extend beyond the target pair to include the full computational space, particularly the spectator qubit. The dependence of spectator population preservation on the initial state suggests that while pairwise gates are feasible, the dynamics of the full system must be carefully managed. The authors modestly frame the CCZ results as a preliminary indication that extending this architecture to multi-qubit gates requires further optimization of pulse shapes and control over unwanted conditional phases. They conclude that future work must address relaxation, dephasing, and flux noise to assess experimental viability and explore further connectivity extensions.