Quantum Telescoping Part VIII: Learned Quantum Telescoping

This paper introduces learned quantum telescoping, a framework in which the refinementsteps of a quantum telescoping scheme are selected using data-driven or learning-based methods.Building on the channel-level telescoping theory developed in Parts I–VII, and complementingexisting adaptive product-formula methods 21, 22, we formalize learning as an oracle thatproposes refinement channels based on training data, prior simulations, or experimental measurements.We prove that learning can optimize telescoping constants and enable data-drivenrefinement strategies that improve typical-case performance in structured regimes, while remainingsubject to the same fundamental lower bounds on telescoping order and query complexity.Our results clarify one precise role of machine learning in quantum simulation: learning canguide which quantum channels to use, but cannot overcome information-theoretic limits inherentto the target dynamics. We provide explicit constructions, sample complexity bounds, andconnections (with explicit metric caveats) to quantum signal processing, variational algorithms,and shadow tomography. This work develops a principled framework for integrating machinelearning into quantum simulation while respecting quantum information-theoretic constraints.