Quantum algorithms for continuous-variable problems, from partial differential equations (PDEs) to statistical sampling, must encode real-valued periodic functions as quantum states to a target precision. We show that Gevrey regularity, stratifying functions between merely smooth and analytic, is the natural class controlling this cost: the Gevrey order sets the Fourier-tail decay and hence the discretization needed for a given precision. Exploiting this, we give high-precision quantum PDE solvers whose cost is polynomial in dimension, breaking the curse of dimensionality for general linear PDEs, with explicit gate complexity and applications to many-body linear-response simulation. We then apply the same Gevrey control to build an efficient quantum Gibbs sampler for potentials with periodic boundary conditions, via quantum singular value filtering and temperature annealing. Crucially, Gevrey order s > 1 is also exactly where an adversary can hide probability mass from any classical algorithm, giving a matching unconditional lower bound: the first provable quantum-classical separation for a sampling problem over a continuous domain, quadratic in the energy barrier and exponential in dimension at low temperature.