Uniform modified log-Sobolev constant for one-dimensional Gibbs samplers
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Problem
Do normalized one-dimensional Chen–Kastoryano–Gilyén Gibbs samplers have a positive modified log-Sobolev constant uniformly in system size at every fixed finite temperature?
Fix \(\beta>0\), a finite on-site dimension, and a class of one-dimensional finite-range Hamiltonians with bounded local strength. Let \(\rho_\beta(H)=e^{-\beta H}/\operatorname{Tr}(e^{-\beta H})\), and let \(\mathcal L_{H,\beta}\) be the Chen–Kastoryano–Gilyén generator using all single-site Pauli jumps, Gaussian-filter width \(\beta^{-1}\), and the unrescaled sum of local terms.
Does a constant \(\alpha_\beta>0\) exist, independent of chain length and of \(H\) in this class, such that for every state \(\sigma\) and every \(t\geq0\)
Relative entropy in Eq. (1) uses natural logarithms; constants may depend on \(\beta\) but not on system size.
Source
Bergamaschi and Chen explicitly leave the uniform modified log-Sobolev question open for this one-dimensional sampler family [Bergamaschi25]. The statement fixes a normalization and rewrites the question self-containedly.
Progress
Published provenance and progress. Bergamaschi and Chen prove a system-size-independent spectral gap for the relevant one-dimensional noncommuting family, and obtain polylogarithmic-depth Gibbs preparation by a different, quasi-adiabatic route [Bergamaschi25], revised January 23, 2026. Their discussion explicitly leaves the corresponding log-Sobolev implication open. High-temperature rapid-mixing results such as [Rouz26] do not cover arbitrary fixed finite temperature.
Critical update: Gao and Guo’s September 14, 2026 preprint establishes positive complete MLSI for finite-dimensional CKG samplers through a coercivity criterion [Gao26]. Therefore, mere positivity for each fixed finite system is no longer the question retained here.
Why the September result does not close this formulation. The comparison in [Gao26] includes an index of the asymptotic conditional expectation. For a primitive semigroup with stationary state \(\rho\), the order constant
\begin{equation} C(E)=\inf\{C:\sigma\leq C\rho\ \text{for every density matrix }\sigma\} \tag{2} \end{equation}satisfies, directly,
\begin{equation} C(E)=\lambda_{\min}(\rho)^{-1}\geq d^n. \tag{3} \end{equation}The displayed definitions, constraints, and target bounds are recorded in Eqs. (2), (3).
Thus a lower bound divided by this quantity does not establish a uniform many-body constant. Moreover, an ancillary-dimension-independent “complete” inequality is not automatically independent of the physical system size.
Comment
Status: Retained as open only in the uniform-in-\(n\) form. A global reset channel is not an admissible substitute for the specified quasi-local dynamics. The distinction between spectral gap, entropy contraction, and circuit preparation is essential.
References
- [Bergamaschi25]
- Thiago Bergamaschi and Chi-Fang Chen, Fast Mixing of Quantum Spin Chains at All Temperatures. first submitted October 9, 2025; checked version v2, January 23, 2026. Locate: Theorem I.1, preparation corollary, and the discussion “Mixing times beyond spectral gaps.”arXiv
- [Rouz26]
- Cambyse Rouzé, Daniel Stilck França, and Álvaro M. Alhambra, Optimal quantum algorithm for Gibbs state preparation. checked version v2, February 10, 2026; Physical Review Letters 136, 060601 (2026). Use: high-temperature algorithmic progress, not an all-temperature uniform-MLSI theorem.arXivDOI