When does a three-level readout help? Leakage flags, erasure qubits and the qg T1-aware decoder

Transmon qubits leak to their second excited state, and a three-level readout can flag it. We ask whether this helps the qg T1-aware decoder, a minimum-weight matching decoder whose weights follow the final data readout and which a register-mean polar-bias witness switches on only when relaxation dominates [1], [2]. Six simulated tests on a circuit-level surface-code memory, each with its criterion fixed before the reported run, give a consistent answer. (i) When a leaked qubit keeps its value, erasing flagged qubits makes every decoder worse, and the whole qg gain (1.68) comes from the qubit-level reweighting. (ii) When leakage scrambles the value, the flag and the reweighting combine better than either alone: 1.68 over the best leakage-aware decoder at distance 3. (iii) A Bayesian weight for the flag, a flip probability with the leak asymmetry between and , contains both cases, is never worse than ignoring leakage, and is not uniformly better than erasure (one of four predictions fails). The combined gain grows with distance, from 1.48 to 1.77. (iv) A coherent three-level simulation of a diabatic CZ gives and a leaked qubit that keeps its value and still flips its ancilla with probability 0.97. For such transmons the flag is useless, erasure hurts, and the qg gain (1.81 at , 2.02 at ) needs no qutrit readout. (v) On erasure qubits, which herald T1 decays as they happen, a naive combination with qg is harmful when heralding is good (0.46 at heralding efficiency ); scaling the prior of unheralded decays by , a rule then tested with its own pre-registration, removes the loss and keeps a gain of 1.41–1.68 at . All results are simulations, reproduced by scripts and pinned by regression tests.