Table 2.

Summary of the classical and quantum run-times for thermodynamic limits needed to obtain the hardware-limited precision with hardware error/noise δ

Problem	Error model	Assumption	Classical run-time	Quantum run-time
Provable noisy quantum advantage over Krylov Subspace/Exact methods
Dynamics	General errors	None	$\exp \left(\tilde{\mathrm{\Omega }}\left({\log }^d\left(\mathrm{\Theta }\left({\delta }^{-1}\right)\right)\right)\right)$	O(poly(δ−1))
Ground state	Coherent	• Stable gap	$\exp \left(\mathrm{\Omega }\left({\log }^d\left(\mathrm{\Theta }\left({\delta }^{-1}\right)\right)\right)\right)$	O(poly(δ−1))
	Hamiltonian	• Logarithmic convergence
	errors	• Adiabatic path with a gap  > Ω(1/poly(n))
Fixed points	General errors	Rapid mixing	$\exp \left(\mathrm{\Omega }\left({\log }^d\left(\mathrm{\Theta }\left({\delta }^{-1}\right)\right)\right)\right)$	O(poly(δ−1))
Conjectured noisy quantum advantage
Ground states	Coherent	• Stable Ω(1/poly(n)) gap	$\exp \left(\mathrm{\Omega }\left(\text{poly}\left({\delta }^{-1}\right)\right)\right)$	O(poly(δ−1))
	Hamiltonian	• Power-law convergence
	errors	• Adiabatic path with a gap  > Ω(1/poly(n))
Fixed points	Coherent and incoherent Markovian errors	• Reaches ε—close to fixed point in O(poly(n, 1/ε)) time • Power-law convergence	$\exp \left(\mathrm{\Omega }\left(\text{poly}\left({\delta }^{-1}\right)\right)\right)$	O(poly(δ−1))

For classical run-times, we only consider Krylov subspace methods or exact diagonalization as the classical algorithm for the provided lower bounds, and not heuristics which do not have rigorous convergence guarantees. Note that $\tilde{\mathrm{\Omega }}$ suppresses $\log \log \left({\delta }^{-1}\right)$ factors.