Function extraction via Chebyshev interpolation of quantum integral estimates

We present an algorithm for extracting a smooth, positive-definite function ψ(x) encoded in a quantum state vector of size 2^{n}, while avoiding exponentially suppressed subnormalization of a naive extraction step. The method addresses a known limitation in quantum algorithms for differential equations, namely, the efficient retrieval of classical information from quantum-encoded solutions. It applies to settings where the target solution is smooth and positive definite, such as certain instances of the heat equation and related diffusive models arising in fluid dynamics and finance.