Quantum Telescoping Part V: Fault-Tolerant Telescoping and Error-Corrected Simulation

This paper studies quantum telescoping in the presence of physical noise and fault-toleranterror correction. Building on the telescoping framework and lower bounds developed in Parts I–IV, we analyze how simulation error and hardware noise interact under fault-tolerant execution.We show that telescoping order alone does not determine end-to-end accuracy: instead, theachievable refinement depth is limited by the error-correction overhead and logical noise floorwhen resources are bounded. We prove a general noise dominance condition under which exponentialtelescoping is preserved, and establish resource-bounded cutoffs for power-law schemes.These results clarify which simulation advantages persist under fault tolerance and explain whyQuantum Signal Processing (QSP)-based methods remain optimal even after accounting forerror correction. We provide extensive examples, detailed proofs, numerical analysis, concreteresource estimates, and visual phase diagrams that bridge ideal algorithmic analysis with realisticquantum hardware constraints.