Exponentially Improved Constant in Quantum Solution Extraction

We have provided an algorithm to extract a smooth and positive definite function $\psi (x)$ encoded in quantum memory of size $2^n$ without running into the problem of exponentially suppressed sub-normalization. Through this, we remove an important bottleneck of solution information extraction, the last step, in fully solving an important class of differential equations on quantum computers. This class of problems includes solutions to the heat equation or other diffusive equations in fluid dynamics and finance.