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 (√kappa) 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 (Lambda), a smoothed-score query at noise level (τ) gives access to the resolvent ((Lambda+τ-1I)-1). Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with q=O!left(bigl(logkappa+log(e√ d/δrm TV)bigr)log(e√ d/δrm TV)right) smoothed-score queries for total variation error (δrm TV), improving the condition-number dependence from (√kappa) 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 (kappa); in particular, for fixed dimension and accuracy, the bit complexity is (O(log2kappa)). 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 (Omega(logkappa)) 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.