Localized charges, reset noise, and boundary memory in matrix-product-conserving quantum chains

Abstract

We construct boundary observables that retain memory for a quantified time window in quantum chains whose basis configurations carry a conserved ordered product of matrix labels. Under strong-irreducibility and proximality hypotheses on the label matrices, projective contraction of random matrix products localizes a conserved charge in the root-mean-square over uniform basis configurations. A finite-time inequality then converts readout overlap and reset leakage into a correlation bound: for a conserved reference observable $H$, a readout $F$, accumulated squared leakage $\mathcal B$, and the normalized Hilbert-Schmidt inner product, $\langle F,Φ(F)\rangle\ge 2\langle H,F\rangle^2/(\|H\|_2^2+\mathcal B)-\|F\|_2^2$. Here $Φ$ is a sequence of conserving channels and partial resets. The bound holds for each such circuit, without averaging its gates. It yields an exponential memory window in the distance from the noisy region, with quantitative corrections for spatially distributed noise and imperfect conservation. For a thirteen-state elementary-matrix model, exact polynomial certificates give root-mean-square localization error $e^{-r/1400}$ between the charge and its restriction to the last $r$ sites, and limiting variance at least $1/19$ for a rational charge. A related normalized-Gram charge gives an eight-site certificate retaining more than half the specified readout correlation through two boundary-reset rounds. Uniform depolarization imposes an inverse-noise-rate lifetime ceiling. A two-qubit IBM experiment illustrates the general reset inequality; it does not realize the matrix-product chain. We provide proofs, exact certificate programs, and an independent recount of the archived experimental outcomes.