{ "abstract": "The quantum baker map is a standard model of quantum chaos, but rigorous analysis of how it spreads random perturbations remains difficult. Following the repeated ``baker plus refreshed diagonal perturbation'' architecture of Schack and Caves, we study a tractable strong-noise model based on the Balazs--Voros quantization: at each time step, its unitary propagator is followed by a diagonal unitary whose entries are independent and uniform on the unit circle. The resulting random propagator is a structured random walk on $\\U(N)$, $N=2^k$. We quantify how quickly the baker dynamics spreads this basis-local randomness at three levels: ensemble means, two-copy statistics, and the full law of the accumulated propagator. The walk becomes an exact unitary $1$-design at time $k+1$, while its fixed-accuracy $2$-design mixing time is $Θ(\\log N)$. Thus the induced state ensembles reproduce Haar means and two-copy statistics on the Ehrenfest scale. The full law mixes much more slowly. A phase-adapted path coupling gives normalized Wasserstein mixing in $O_\\varepsilon(N)$ steps, whereas metric-entropy and Haar small-ball estimates give an $Ω_\\varepsilon(N/\\log N)$ lower bound and asymptotically maximal distance at every $t=o(N/\\log N)$. The law remains singular with respect to Haar measure for every $t