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
κ \sqrtκ κ dependence on the condition number. We show that this barrier disappears when the sampler is allowed to query \emph{smoothed scores}, namely gradients of the logarithms of the Gaussian-convolved densities. For a Gaussian target with precision matrix
Λ Λ Λ , a smoothed-score query at noise level
τ τ τ gives access to the resolvent
( Λ + τ − 1 I ) − 1 (Λ+τ^{-1}I)^{-1} ( Λ + τ − 1 I ) − 1 . Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with
q = O ( ( log κ + log ( e d / δ T V ) ) log ( e d / δ T V ) ) q=O\!\left(\bigl(\logκ+\log(e\sqrt d/δ_{\rm TV})\bigr)\log(e\sqrt d/δ_{\rm TV})\right) q = O ( ( log κ + log ( e d / δ TV ) ) log ( e d / δ TV ) ) smoothed-score queries for total variation error
δ T V δ_{\rm TV} δ TV , improving the condition-number dependence from
κ \sqrtκ κ 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 κ κ ; in particular, for fixed dimension and accuracy, the bit complexity is
O ( log 2 κ ) O(\log^2κ) O ( log 2 κ ) . 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 ( log κ ) Ω(\logκ) Ω ( log κ ) 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.