Neil Thompson

Quantum machine learning can be fast when linear-algebra subroutines are paired with the right data and access models. This paper organizes QML’s leading algorithms mostly under an HHL framework, assembling a side-by-side runtime table and making explicit the conditions that govern real gains: efficient state preparation, sparsity, condition numbers, and readout. We pinpoint where exponential advantages plausibly survive (e.g., quantum-assisted Gaussian process regression with polylog-sparse, well-conditioned kernels, and topological data analysis on engineered complexes), and where advantages shrink to strong polynomial factors, especially under full-vector recovery or dense, ill-conditioned matrices. We also align these claims with quantum-inspired ''dequantized'' methods that narrow several gaps, and we frame payoffs using a quantum-economic lens, highlighting near-term, high-yield workloads and longer-term hardware/data access needs.