Yongshan Ding

We construct an explicit transition matrix that exactly describes stabilizer-code dynamics under circuit-level stochastic Pauli noise. The matrix is expressed in a code-adapted basis that captures probability flow between logical and error states. For quantum error detection (QED), we incorporate post-selection by aggregating rejected outcomes into an absorbing state, so that a single transition matrix represents a full clock cycle of gates, syndrome extraction, and post-selection. Additionally, we characterize when protocol symmetries permit exact lumping to simpler, more interpretable models. The transition matrix framework enables direct application of classical stochastic-matrix techniques to analyze multi-cycle QED protocols. First, we prove that emergent logical non-Markovianity originates from QED-check imperfections at leading order, and quantify the accepted leakage injection that causes it. Second, we express leading-order QED efficiency as a function of check frequency in terms of physically meaningful parameters, and prove that less frequent checks improve efficiency in the perturbative regime. More broadly, the transition-matrix formalism provides both an analytical foundation for QED analysis and an interpretable tool for understanding code behavior under realistic noise.