Remove the system-size factor in the global metastability-to-ADB bound

Determine whether the factor n in the conversion from global metastability to approximate detailed balance for detailed-balanced Lindbladians is necessary, thereby determining whether the system-size-dependent factor in the Hamiltonian-learning precision floor can be removed.

Background

The paper proves that, for a state σ with global metastability error ε_msglo, the approximate detailed-balance error is bounded by a quantity proportional to ε_msglo[n + log(1/ε_msglo)]. This factor n is the source of the system-size dependence in the resulting Hamiltonian-learning precision floor.

The authors give examples showing that certain other linear system-size factors are necessary, but they do not establish whether the factor n in the global-metastability-to-ADB conversion is intrinsic. Removing it would yield a system-size-independent precision guarantee under global metastability.

References

Whether the system-size-dependent factor between the learning precision η and metastability ε in \Cref{thm:intro-global-metastable-learning} is necessary remains open.

— Efficient learning of quantum interactions from thermal metastable states  (2610.01538 - Wang et al., 1 Oct 2026) in Section 1, subsection “Main results”