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.