Scalable quantum metrology via recursive optimization

Achieving higher precision in quantum metrology demands the optimization of both entangled-state preparation and readout processes. However, direct machine optimization using variational dynamics becomes impractical and time consuming, particularly with a large particle number. Here, we develop a heuristic recursive screening-breeding algorithm designed to efficiently optimize variational dynamics for scalable quantum metrology. Leveraging the similarity between optimal variables for adjacent particle numbers, the algorithm recursively updates variables from few to many particles, significantly reducing computational overhead. This enables the identification of optimized dynamics for entangled state preparation and readout, even with a large particle number. As a proof of concept, we apply the algorithm to efficiently optimize twist-and-turn dynamics for spin cat state preparation, interaction-based readout, and time-reversal metrology. Notably, in well-recursive cases, optimized variables can be approximated via fitting without recursion. Our recursive optimization algorithm establishes an efficient framework for scalable quantum metrology protocols, providing a practical avenue for quantum sensing with many-body systems.