A numerically exact, non-Markovian, non-Gaussian noise model for open quantum system dynamics

We develop a multichannel random-telegraph-noise hierarchical-equations (RTN-HE) framework for interacting multiqubit systems to describe non-Markovian, intrinsically non-Gaussian open quantum system dynamics driven by classical stochastic processes. Starting from the stochastic Liouville equation, we derive a systematic multi-channel generalization of the Shapiro-Loginov identity and construct a closed hierarchy of mixed system-noise moments. Owing to the dichotomic algebra of telegraph processes, the hierarchy terminates exactly, embedding the non-Markovian reduced dynamics into a finite-dimensional linear system of ordinary differential equations without stochastic trajectory sampling. An equivalent formulation in the noise-configuration basis yields a tensorized Liouvillian with explicit Kronecker-product structure, enabling sparse and efficient propagation on an enlarged state space. We further show that thermal detailed balance can be enforced consistently through a constant counterterm, without modifying the homogeneous generator, establishing thermodynamically controlled long-time behavior within the RTN-HE formalism. The framework is illustrated in two distinct settings: excitation energy transfer in a reduced Fenna-Matthews-Olson complex, where RTN-HE provides a compact stochastic surrogate to hierarchical equations of motion dynamics, and Bell-state storage in a quantum teleportation protocol, where the model enables a direct comparison between non-Gaussian telegraph noise and Gaussian processes matched at the level of second-order correlations. These results position RTN-HE as a controlled theoretical approach for isolating and quantifying the dynamical consequences of finite memory and higher-order moment noise statistics in multi-site open quantum systems.