Rapid mixing for one-dimensional non-commuting quantum systems

Prove rapid mixing for one-dimensional non-commuting quantum systems at all temperatures, including the modified log-Sobolev inequalities needed to establish such mixing behavior.

Background

The paper studies a quantum Gibbs-sampling Lindbladian whose stationarity properties support Hamiltonian learning. Rapid mixing would provide substantially stronger dynamical information and is closely connected to modified log-Sobolev inequalities, which relate Fisher information to quantum relative entropy.

The authors identify rapid mixing in one-dimensional non-commuting systems at all temperatures as an unresolved problem. They explain that proving the relevant modified log-Sobolev inequalities is particularly difficult and note that their approximate-convexity analysis was motivated in part by this challenge.

References

Finally, it remains a remarkable open problem to prove rapid-mixing in 1D non-commuting quantum systems at all temperatures; see e.g. for related efforts.

— The stationarity test: a framework for learning quantum many-body systems from their thermal states  (2610.01074 - Bergamaschi, 1 Oct 2026) in Section 1, subsection “Discussion and related work,” paragraph “Open questions”

For 1D Hamiltonians with sufficiently rapidly decaying power-law interactions, quasipolynomial-time algorithms for MPO construction and quantum Gibbs-state preparation are known, and polynomial-bond-dimension approximations of purified Gibbs states were recently shown to exist, although efficient constructions remain open; this raises the question of whether our dynamics and algorithm can be adapted to yield polynomial-depth state-preparation circuits.

— Rapid mixing of quantum spin chains at any finite temperature  (2610.01190 - Kim, 1 Oct 2026) in Section Discussion and outlook