Leakage-Tolerant High-Fidelity Logical X Gate Tailored for Erasure Qubits in a Transmon-Mechanical Phononic System

We present a high-fidelity single-qubit logical X gate realized in a frequency-tunable transmon qubit dispersively coupled to a mechanical phonon mode. The logical qubit is encoded in a binomial code using dressed eigenstates of the hybrid system, while the gate is implemented by optimized dual-tone microwave controls applied to the mechanical mode. Using bandwidth-limited Fourier gradient-ascent pulse engineering (GRAPE) optimization with the Adam algorithm, we obtain an ideal (dissipation-free) average logical fidelity of 99.99% with a gate time of 0.8 μs. Because both the static coupling and the two-phonon drive conserve the total excitation parity, transient mid-gate population borrowed into higher Fock states coherently returns to the logical subspace at the end of the gate, leaving a residual leakage of 0.001% in the ideal case, against a maximum transient value of 2.9% during the gate. Full Lindblad master equation simulations in the dressed eigenbasis, including realistic decoherence channels (T₁^q = 30 μs, T_ϕ^q = 30 μs, T₁^c = 100 μs, T_ϕ^c = 300 μs), yield an average logical-state fidelity of 96.58%, dominated by phonon energy relaxation. A decoherence channel decomposition shows that the four channels contribute 1.53%, 1.13%, 0.26% and 0.73% for phonon T₁, phonon dephasing, transmon T₁ and transmon dephasing respectively, the latter being strongly suppressed relative to its bare value by dispersive dressing. The parity-changing phonon-loss events (≈ 1.8% by the end of the gate) populate the orthogonal odd-parity subspace and are therefore in principle detectable as heralded erasures. Robustness analysis with recalibration of the dressed basis and carrier frequencies shows fidelity above 99.9% for amplitude deviations of about -2.1% to +2.8% and anharmonicity deviations of about -0.13% to +0.15%. These results identify a promising control strategy for erasure-compatible bosonic quantum computing, subject to a complete error-conversion and fault-tolerance analysis.