Unitary RQL Equals RQL

What carries the argument

Its core is the reweighted density-matrix doubling of the source channel. Vectorization $\mathrm{vec}(X) = \sum_{i,j} X_{ij}|i\rangle|j\rangle$ represents a mixed state as a vector on two registers, and a channel with Kraus operators $K_{t,a}$ becomes the doubled operator $\sum_a K_{t,a} \otimes K_{t,a}$. Because a reset has doubled norm $\sqrt{2}$, normalizing each step separately would lose an exponential factor; instead the right factors are reweighted by reference states $\sigma_t = (\Phi_t(\sigma_{t-1}) + \eta I/d)/\kappa$ with $\eta = 1/T$, $\kappa = 1 + \eta$, giving factors $J_{t,a} = \kappa^{-1/2}\sigma_{t-1}^{1/2} K^\dagger_{t,a} \sigma_t^{-1/2}$ and a star branch $J_{t,*} = \sqrt{\eta/(\kappa d)}\sigma_t^{-1/2}$. The identity $W^\dagger_t W_t = I$ with $W_t = \sum_a J_{t,a}|v\rangle|a\rangle + J_{t,*}|v\rangle|*\rangle$ makes each step an isometry, so each doubled contraction $C_t = \sum_a K_{t,a} \otimes J^\dagger_{t,a}$ has norm at most one and consecutive square roots cancel, yielding $C_T\cdots C_1 \mathrm{vec}(\rho_0) = \mathrm{vec}(\rho_T \sigma_T^{-1/2})\sqrt{d\kappa^T}$ and an accepting probability $q^*$ with $p^2/(ed) \le q^* \le p$. These weights are exactly the Petz recovery factors $\sigma^{1/2} L^\dagger \tau^{-1/2}$ for the reference-adjusted channel. The exactness story is separate: the left Kraus factors $K_{t,a}$ are kept exact, the right factors are approximated by the logspace-uniform unitary circuits of Lemma 3.2 built from phase estimation of two-reflection walks, a coherent failure counter composes the steps, and Grover amplification with exact reflections raises acceptance to a constant without ever creating acceptance on a no-instance. Theorem 4.1 replaces this reweighting with a backward propagation of the rejection observable and an exact history test on a certificate supplied by Merlin.