Human-AI Co-Discovery of Quantum Sign-Embedding Algorithms

This report chronicles a collaborative human-AI discovery process that culminated in the development of sign-embedding quantum algorithms for matrix equations and functions. The project originated from a human intuition regarding the effectiveness of rational approximation for jump-type functions, such as the sign function, as a potential design principle for quantum algorithms. Rather than merely assisting in problem-solving, AI tools, including workflows later integrated into the AIM system, played a crucial role in transforming this initial intuition into a concrete research roadmap, comparing various formulations, and converging on sign embedding as the central framework. The AI system, AIM, further aided in connecting a known matrix-sign identity to broader classes of matrix equations and functions, and contributed to drafting proofs and complexity calculations. However, critical scientific judgments remained firmly within the human domain. These included selecting promising research avenues, rejecting flawed approximations, and refining implementation details. The report argues that human-AI co-discovery workflows, with AI systems acting as research partners, are most valuable for problem formation, connection discovery, derivation, and skeptical review within a human-controlled research loop, rather than as standalone theorem provers.