Table 1.

Summary of the stability results for dynamical and equilibrium many-body problems, together with the required assumptions on the many-body model and errors

Problem	Error	Assumption	Observable	Stability, f(δ)
Dynamics	General errors	GF: None.	GF: k-local observables.	GF: O(δt).
		SS: None.	SS: k − local observables.	SS: O(δtd+1). (Established in ref. 38 for Markovian errors).
Ground state	Coherent Hamiltonian errors	GF: Assumption 1.	GF: Translationally invariant k − local observables.	GF: O(δβ), where β is a model dependent constant. (Established in ref. 37 for gapped models).
		SS: Stable gap.	SS: Local observables	SS: O(δ) (Established in ref. 54).
Gibbs state	Coherent Hamiltonian errors	GF: No assumption.	GF: Translationally invariant k − local observables.	GF: $O\left(\sqrt{\delta }\right)$.
		SS: Stably exponentially clustered correlations.	SS: Local observables.	SS: $O\left(\delta {\log }^d\left(1/\delta \right)\right)$. (Established in ref. 46).
Fixed points	GF: Coherent and Incoherent Markovian errors.	GF: Assumption 2.	GF: Translationally invariant k − local observables	GF: O(δβ), where β is a model dependent constant.
	SS: General errors	SS: Rapid Mixing.	SS: Local observables	SS: O(δ). (Established in ref. 38 for Markovian errors).

Both results for Gaussian fermions and spin systems are summarized—“GF” indicates Gaussian fermions and ‘SS’ indicates spin systems. Note that observables for the Gaussian fermionic problems are all quadratic.