We prove the first quantum--classical separation for a sampling problem over a continuous domain. For a class of Gibbs states $p\propto e^{-\beta E}$ on the torus $\mathbb{T}^d$ with smooth ($s$-Gevrey) potential and barrier amplitude $\alpha=e^{\beta\Delta}$, where $\Delta = \max E-\min E$, every classical algorithm---querying the value, gradient, or any higher-order derivatives of the log-density---requires $\Omega(\alpha)$ queries to sample at constant accuracy in total variation distance, while a quantum algorithm based on quantum singular value thresholding and temperature annealing samples with $\tilde{O}\left(\sqrt{\alpha}\right)$ queries to an oracle for the gradient. The advantage is quadratic in the barrier amplitude, which becomes exponential in the dimension, $e^{\Omega(d)}$, at low temperature. The classical bound is information-theoretic, holding for every classical algorithm with query access to the Gibbs potential and its derivatives at any order.
Enrico Olivucci, Mariia Sobchuk, Sehmimul Hoque et al.· 1 citation
We connect different degrees of smoothness of real-valued periodic functions to the number of qubits required for their high-precision Fourier-basis amplitude encodings as quantum states. Our resulting central observation is that the Gevrey hierarchy, which stratifies the space between smooth and analytic functions, provides a natural class for high-precision quantum algorithms. We then specialize to solving general linear partial differential equations (PDEs) with periodic boundary conditions, showing how our Fourier methods do so efficiently at varying target precisions on a quantum computer. This also demonstrates how our framework enables passage from query-complexity results to explicit elementary gate counts. As an application, we introduce a hierarchy of many-body quantum simulation pipelines that harness these high-precision algorithms to probe the linear response of atomistic systems in first quantization. Each level of the hierarchy unlocks a further polynomial-degree quantum speedup, yielding a gradual improvement in simulation efficiency as quantum computers scale.
Pooya Ronagh, Mariia Sobchuk, Xiaoran Li et al.· 0 citations
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