Rapid mixing for high-temperature Gibbs states with arbitrary external fields

Gibbs states are a natural model of quantum matter at thermal equilibrium. We investigate the role of external fields in shaping the entanglement structure and computational complexity of high-temperature Gibbs states. External fields can induce entanglement in states that are otherwise provably separable, and the crossover scale is $h\asymp \beta^{-1} \log(1/\beta)$, where $h$ is an upper bound on any on-site potential and $\beta$ is the inverse temperature. We introduce a quasi-local Lindbladian that satisfies detailed balance and rapidly mixes to the Gibbs state in $\mathcal{O}(\log(n/\epsilon))$ time, even in the presence of an arbitrary on-site external field. Additionally, we prove that for any $\beta<1$, there exist local Hamiltonians for which sampling from the computational-basis distribution of the corresponding Gibbs state with a sufficiently large external field is classically hard, under standard complexity-theoretic assumptions. Therefore, high-temperature Gibbs states with external fields are natural physical models that can exhibit entanglement and classical hardness while also admitting efficient quantum Gibbs samplers, making them suitable candidates for quantum advantage via state preparation.