Time-Integrated Leakage as a Dynamical Benchmark for Multi-Mode Superconducting-Qubit Reset

Reset performance in superconducting qubits is often summarized by the population that remains at the end of the pulse. This is an important number, but it does not describe what happens during the reset itself. A protocol may reach a small final residual while keeping the qubit in a leakage state for a comparatively long part of the trajectory. In this work, we therefore use time-integrated leakage, together with the usual endpoint population, to study a multi-mode dissipative reset model under flux-frequency control. We examine finite thermal occupation, parameter sweeps, frequency disorder, an auxiliary-chain ablation, and a literature-based rate benchmark. At the representative ablation point, adding the auxiliary chain reduces Lf from 8.56 to 0.78 ns and shifts the sustained Pf less-than- 10-2 crossing from 39.5 to 9.8 ns. Interestingly, the configuration without the chain still reaches the smaller long-time residual. The two metrics can therefore favor different reset configurations. For the same 300 ns analysis window, an effective rate model built from the f-ket and e-ket decay rates reported by Zhou et al. gives Lf is 108.0 ns, while the complete simulated configuration gives 0.783 ns. We use this only as a kinetic reference and not as a reproduction of the experiment. In the wider set of simulations, thermal occupation is the main source of degradation, while the frequency disorder considered here has a much weaker effect. The results suggest that time-integrated leakage is a useful quantity to report alongside endpoint residuals, especially when reset is part of repeated quantum-error-correction cycles.