A Closed-Form Excess-Population Latency Budget for Active Reset of Superconducting Qubits

Active reset returns a superconducting qubit to its ground state: read it out, apply a π-pulse if it reads excited. The classical decision between the two takes time, and the qubit relaxes meanwhile. Salathé and co-workers tied this cost to the ratio of feedback latency to lifetime and expected the error to scale as one minus its exponential, and Ristè and co-workers had already written the reset error to first order as readout errors plus equilibration terms linear in the latency. This work makes that accounting exact at a fixed evaluation time. A transmon with T₁ = 50 μs is prepared in its excited state and read out by an ideal terminal projection followed by a classical confusion channel with symmetric assignment error P_AE. Under that instrument the ensemble-averaged residual population after one feedback cycle is an exact weighted sum of four deterministic Lindblad evolutions, conditioned on the projected and the reported outcome. The sum is evaluated in closed form, together with the critical latency τ_crit at which the residual exceeds its zero-latency value by a chosen budget, defined throughout as an absolute increase in excitation probability of 0.01, that is one percentage point. To leading order τ_crit is the budget times T₁ divided by the pulse-firing probability p_pulse, times a prefactor of 1.019 for the parameters considered here. Discarding that prefactor while retaining the division by p_pulse costs under 2%; the further simplification to budget times T₁ carries 3.4% to 7.3% error across P_AE = 1% to 5%. Because P_AE enters only through p_pulse, the budget is weakly sensitive to assignment error over the tested range for an initially excited qubit, giving 518 to 539 ns; the readout duration enters separately, through the observation window and the post-readout population. A critical latency exists only for budgets below the maximum excess attainable within that window. The four-branch representation is exact for the stated measurement model; the finite-pulse map used to evaluate it is approximate, and is replaced here by its exact affine propagator. The closed form agrees with a master-equation solver to 3.3×10⁻⁷ and is cross-checked against stochastic trajectories.