Photon-stimulated decay of $\chi$-matched ancillas causes unheralded bit flips in bosonic codes

Bosonic codes often read out their syndrome with a transmon ancilla whose computational levels $g$ and $f$ have equal parity and matched dispersive shifts: a decay $f\to e$ is then heralded and leaves the code intact, while the direct $f\to g$ is forbidden by parity. We show that the photons of the code open this decay at first order in the ancilla’s decay rate. Dressing by the code mode lets the charge noise that limits the ancilla’s $T_1$ drive a photon-stimulated Raman decay $\mathrm{\mid }f,n⟩\to \mathrm{\mid }g,n+1⟩$, at $(n+1)({\chi }_{ge}\mathrm{/}2g_{01}{)}^2$ times the $f\to e$ rate for a smooth noise spectrum. When the matching is set by the detuning, the decay costs $\approx {\mathrm{\Gamma }}_{fe}\mathrm{/}\alpha$ per gate and photon, whatever the coupling strength; a deliberate mismatch or matching by a drive lowers this cost, and a qubit ancilla has no such channel. The opened decay is unheralded and flips a cat with probability $\approx 1\mathrm{/}2$; being first order, it overtakes the second-order double decay $f\to e\to g$ as the ancilla improves. In a microscopic model of the Ocelot repetition cat code, it makes the bit-flip floor of the gates grow linearly with the cat size, by a factor 1.2-1.7 from four to ten photons, which would limit the noise bias to $\sim {10}^3$. The published gate data end at four photons and do not yet test this growth.