Quantum Telescoping Part II: Channels and Hamiltonian Simulation

This paper develops the theory of quantum telescoping for quantum channels, with a particularfocus on Hamiltonian simulation. Building on the axiomatic framework introduced in QuantumTelescoping Part I [11], we formalize channel-level telescoping schemes using the diamondnorm, analyze convergence and error propagation under composition, and establish rigorous connectionsbetween telescoping order and simulation complexity. We show that product-formulamethods exhibit power-law telescoping with polynomial increment decay, while quantum signalprocessing induces exponential telescoping with increment decay e−ad, yielding the optimalO(t+log(1/ε)) complexity. LCU implementations of truncated series achieve super-polynomial(factorial) decay. This perspective yields transparent complexity bounds, clarifies the role of incrementalrefinement in quantum simulation, and provides a unifying framework for comparingclassical and quantum simulation strategies.