Quasi-optimal sampling from Gibbs states via non-commutative optimal transport metrics

Understand

We study the problem of sampling from and preparing quantum Gibbs states of local commuting Hamiltonians on hypercubic lattices of arbitrary dimension.

• We prove that any such Gibbs state which satisfies a clustering condition that we coin decay of matrix-valued quantum conditional mutual information (MCMI) can be quasi-optimally prepared on a quantum computer.
• We do this by controlling the mixing time of the corresponding Davies evolution in a normalized quantum Wasserstein distance of order one.
• To the best of our knowledge, this is the first time that such a non-commutative transport metric has been used in the study of quantum dynamics, and the first time quasi-rapid mixing is implied by solely an explicit clustering condition.