§3.2, p. 17 — along a B-Chebyshev cooling schedule, |⟨π_i|π_{i+1}⟩|² ≥ 1/B for all i

Let $\Omega$ be a finite set of configurations and $H:\Omega\to\{0,1,\dots,n\}$ a Hamiltonian with at least one configuration of energy $0$. Let $Z(\beta)=\sum_{x}e^{-\beta H(x)}$ be its partition function and let $0=\beta_0<\beta_1<\dots<\beta_\ell=\infty$ be a $B$-Chebyshev cooling schedule for $Z$ (Definition 1). For each $i$ let $\pi_i(x)=e^{-\beta_iH(x)}/Z(\beta_i)$ be the Gibbs distribution and $|\pi_i\rangle=\sum_x\sqrt{\pi_i(x)}|x\rangle$ its quantum sample state. Then for every $i\in\{0,\dots,\ell-1\}$,

This is the overlap condition required by the Wocjan–Abeyesinghe quantum sampling theorem (Theorem 9 of the paper, with $p=1/B$): it says that each Gibbs state in a Chebyshev cooling schedule is close enough to its predecessor for quantum walks to prepare it from the previous one, and it is what the paper's partition-function algorithm (Theorem 11) uses.

Formalization Note Inverse temperatures are in $[0,\infty]$ with $e^{-\infty\cdot h}=[h=0]$. The hypothesis that some configuration has $H(x)=0$ is added and disclosed: it makes $Z(\infty)>0$, without which $\pi_\ell$ and the last ratio of Definition 1 would be divisions by zero. No hypothesis on $B$ is made; Definition 1 forces $B\ge1$. The hypothesis $H\le n$ is the paper's model and is kept although the claim does not use it.