A new result proves a complete modified logarithmic Sobolev inequality (CMLSI) with a system-size-independent constant for the Chen-Kastoryano-Gilyén Gibbs samplers on one-dimensional spin chains, at any fixed positive temperature. The theorem covers Hamiltonians with bounded finite-range interactions, including noncommuting and spatially nonuniform cases, and yields trace and diamond-norm mixing times of order O(log N + log(1/ε)).

The underlying question is how long a quantum simulator takes to prepare the Gibbs state of a spin chain at finite temperature. The Gibbs samplers proposed by Chen, Kastoryano and Gilyén are quantum algorithms for simulating thermal evolution, but previous mixing bounds depended on the number of sites N, limiting their scalability. A CMLSI constant independent of N implies that the cost of preparing the thermal state grows only logarithmically with the chain size.

The proof establishes complete spatial entropy factorization and absorbs quasi-local errors as conditional relative entropy at the original joint state and reference. This static estimate also yields a system-size-independent CMLSI for the single-site Petz heat-bath dynamics and for the regularised heat-bath dynamics.

The implications are direct for quantum simulation of condensed-matter systems and for thermal state preparation on quantum hardware: they guarantee that, in one dimension and at finite temperature, the thermalisation of these algorithms is efficient even for long chains. Extending the result to higher dimensions remains open, where phase transitions and slower correlation decay could change the scaling of mixing times.