Efficient Gibbs preparation beyond high temperature

Preparing $\rho_\beta=e^{-\beta H}/\operatorname{tr}(e^{-\beta H})$ is different from proving a master equation valid (O2) or learning its generator (O9). A sampler must have the desired stationary state, mix rapidly from its allowed initial states, and be implementable at controlled circuit or system–bath cost. Fix the generator normalization so that increasing all rates is not counted as an algorithmic speedup.

Published baseline (April 2026): Hahn, Sweke, Deshpande and Shtanko show that spatial truncation and local-circuit implementation preserve rapid Gibbs mixing with bounded error at high temperature. A January 2026 preprint by Slezak et al. proves polynomial convergence for specified system–bath/repeated-interaction generators in high-temperature lattices, weakly interacting fermions and 1D chains. Those regimes are not being proposed as unsolved.

The selected frontier concerns genuinely interacting 2D spin families outside proven uniform high-temperature guarantees. State the generator, inverse-temperature range and any clustering or phase assumptions; derive quantitative spectral-gap or log-Sobolev bounds and implementation errors, or identify bottlenecks forcing slow mixing. No efficient algorithm for every Hamiltonian at every temperature is conjectured here. Noncommuting conserved charges introduce a related but distinct thermalization target under B4/B5.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

published: Local-circuit approximations preserve rapid Gibbs mixing at high temperature with bounded preparation error. Evidence

preprint: Polynomial-time thermalization is proved for particular generator families and high-temperature lattices, weakly interacting fermions and 1D chains, not all 2D low-temperature models. Evidence