Smoothed Score Queries and the Complexity of Sampling

We study the query complexity of sampling from high-dimensional Gaussian distributions using gradient information. In the standard oracle model, exact gradients expose only matrix-vector products with the precision matrix, leading to polynomial approximation barriers and a characteristic \sqrtκ dependence on the condition number. We show that this barrier disappears when the sampler is allowed to query smoothed scores, namely gradients of the logarithms of the Gaussian-convolved densities. For a Gaussian target with precision matrix Λ, a smoothed-score query at noise level τ gives access to the resolvent (Λ+τ^{-1}I)^{-1}. Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with q=O!\left(\bigl(\logκ+\log(e\sqrt d/δ_{\rm TV})\bigr)\log(e\sqrt d/δ_{\rm TV})\right) smoothed-score queries for total variation error δ_{\rm TV}, improving the condition-number dependence from \sqrtκ to logarithmic. We also study finite-bit gradient oracles. Using coordinatewise quantization of the transformed smoothed-score answers and a final dithering step, we obtain a sampling scheme whose total communicated gradient information is polylogarithmic in κ; in particular, for fixed dimension and accuracy, the bit complexity is O(\log^2κ). To complement these upper bounds, we introduce a channel-synthesis, or reverse-Shannon, converse technique for sampling lower bounds. This converts total-variation simulation guarantees into communication requirements and yields an Ω(\logκ) lower bound on the required gradient information. Together, these results identify smoothed scores as a provably more informative oracle for sampling and give nearly matching upper and lower bounds for its finite-bit complexity.