Efficient Posterior Sampling for ${\mathbb{Z}}_2$ Synchronization

Consider the ${\mathbb{Z}}_2$ synchronization problem $$\mathit{Y}=\frac{\lambda }{\sqrt{n}}\theta {\theta }^{\mathrm{\top }}+\mathit{Z},$$ where $\theta$ is uniform on $\{-1,1{\}}^n$ and $\mathit{Z}$ is an independent Gaussian Wigner matrix with off-diagonal variance one. We give a polynomial-time posterior sampling algorithm for every fixed $\lambda >1$, for which the conditional output law converges to the posterior in total variation, in expectation over the observation. The construction combines sequential TAP proposals with an independence Metropolis correction. The key is to control signed overlaps after logarithmic pinning and conditional TAP approximations along a random revealing path, which give an efficiently evaluable proposal with a polynomial density-ratio bound outside a set of vanishing posterior mass. To the best of our knowledge, this is the first polynomial-time posterior sampler for ${\mathbb{Z}}_2$ synchronization with a total-variation guarantee throughout the supercritical regime. For comparison, the diffusion-based sampler of \cite{montanari2023posterior} gives normalized Wasserstein guarantees at sufficiently large fixed signal-to-noise ratio.