Quantum Telescoping Part X: Experimental and NISQ Limits of Telescoping Convergence

This paper provides a comprehensive analysis of the fundamental limits of quantum telescopingin experimental and noisy intermediate-scale quantum (NISQ) regimes. Building on thetheoretical telescoping framework developed in Parts I–IX, we rigorously study how physicalnoise, finite sampling, device drift, and imperfect control fundamentally cap achievable telescopingorder. We establish that in the presence of non-vanishing noise floors, all telescopingschemes satisfying our stated noise-and-measurement assumptions exhibit saturation beyond afinite refinement depth, regardless of their ideal convergence rate.We develop a complete mathematical framework incorporating:• Detailed noise models for NISQ devices with explicit error bounds• Measurement-induced lower bounds on observable telescoping increments• Statistical hypothesis testing framework for distinguishing telescoping orders• Rigorous conditions under which exponential telescoping becomes experimentally indistinguishableon NISQ devices with finite noise and polynomial resources• Device drift models and their impact on convergence verification• Quantitative trade-offs between circuit depth, sampling budget, and achievable precisionThese results formalize the unavoidable gap between asymptotic algorithmic guarantees andexperimentally realizable convergence, providing essential guidance for practical quantum algorithmdesign in the pre-fault-tolerant era.