Optimal sample complexity for certifying local Gibbs states

Determine the optimal (minimal) sample complexity, as a function of n, k, β, and ε, for certifying whether two Gibbs states at inverse temperature β of k‑local n‑qubit Hamiltonians are equal or differ by at least ε in trace distance, under copy access to the states.

Background

The paper presents a sample‑ and time‑efficient algorithm for certifying Gibbs states of local Hamiltonians, resolving a question posed by Anshu, but notes the absence of matching lower bounds.

They highlight that even in classical settings the exact complexity of local Gibbs state certification is not settled, motivating a precise characterization of optimal sample complexity in the quantum case.