Quantum Telescoping Part VII: Telescoping for Open Quantum Systems (Lindbladian Dynamics)

This paper extends the theory of quantum telescoping to open quantum systems governedby Lindblad generators. We develop a comprehensive channel-level telescoping framework forcompletely positive trace-preserving (CPTP) semigroups and conduct a rigorous analysis ofachievable convergence rates in Lindbladian simulation. We show that dissipative dynamicswith non-normal generators admit at best power-law telescoping under physically reasonableoracle models, in sharp contrast with the exponential telescoping possible for Hamiltonian evolution.We identify structural conditions—such as normality, commutativity, or unitary dilation—under which faster convergence may occur, and we establish that the absence of efficientquantum signal processing mechanisms obstructs exponential telescoping for typical non-normalLindbladians. We provide detailed convergence analysis, explicit error bounds, computationalcomplexity characterizations, and concrete examples including a non-normal toy model illustratingpseudospectral obstructions. These results establish fundamental limits on open-systemsimulation and clarify the distinction between coherent and dissipative quantum dynamics froma telescoping perspective, with implications for quantum algorithm design, noise modeling, andquantum thermodynamics.