We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington-Kirkpatrick model with negligible total-variation distance (TVD) error up to inverse temperature $β< 1/2$. Prior work obtained TVD error guarantees only up to $β\approx 0.295$, while results covering the entire replica-symmetric regime $β< 1$ gave guarantees only in Wasserstein distance. Our approach demonstrates that the same potential Hessian ascent previously developed for optimization also functions as a sampling algorithm by implementing algorithmic stochastic localization at high temperature. By estimating the covariance of the tilted Gibbs distribution via Gaussian integration by parts, overlap concentration, and precise cavity estimates, we show that a Hessian-ascent process achieves an $O(1)$ Wasserstein error guarantee for finite-time localization, improving on the previous $o(n)$. A careful comparison of stochastic localization with the Hessian ascent process and a free probability argument controlling the diagonal sub-algebra of the Hessian improves this to $O(1)$ in KL divergence. We then use Jarzynski's equality with rejection sampling, along with entropy contraction on the time-$T$ localized distribution, to refine the error to $o(1)$ in TVD up to a constant time $T$ and to complete the sampling with the polarized walk.

翻译：我们提出了一种多项式时间算法，用于在逆温β < 1/2下以可忽略的全变差距离（TVD）误差从Sherrington–Kirkpatrick模型的吉布斯测度中采样。先前的工作仅在β ≈ 0.295范围内获得TVD误差保证，而覆盖整个复制对称区域β < 1的结果仅给出了Wasserstein距离下的保证。我们的方法表明，先前为优化开发的相同势能海森上升过程可通过在高温下实现算法随机定域化来作为采样算法。通过高斯分部积分、重叠集中性和精确腔估计来估计倾斜吉布斯分布的协方差，我们证明了海森上升过程在有限时间定域化下可达到O(1)的Wasserstein误差保证，优于先前的o(n)结果。通过仔细比较随机定域化与海森上升过程，并利用控制海森对角子代数的自由概率论证，我们将此改进至KL散度下的O(1)。随后，我们使用带有拒绝采样的Jarzynski等式，结合时间-T定域化分布上的熵收缩，将误差优化至恒定时间T下的全变差距离o(1)，并借助极化游走完成采样。