Applying functions to matrices, element by element, on a quantum computer

Quantum computers can already apply polynomial functions to a matrix's eigenvalues efficiently, but doing the same thing element-by-element — a routine operation in classical computing — has been awkward or wasteful. This work builds new quantum algorithms for element-wise matrix transforms that require exponentially less memory than previous approaches, while also correcting errors in earlier constructions. The payoff spans machine learning, signal processing, and quantum simulation.