When is better ground state preparation worthwhile for energy estimation?

Many quantum simulation tasks require preparing a state with overlap $\gamma$ relative to the ground state of a Hamiltonian of interest, such that the probability of computing the associated energy eigenvalue is upper bounded by $\gamma^2$. Amplitude amplification can increase $\gamma$, but the conditions under which this is more efficient than simply repeating the computation remain unclear. Analyzing Lin and Tong's near-optimal state preparation algorithm we show that it can reduce a proxy for the runtime of ground state energy estimation near quadratically. Resource estimates are provided for a variety of problems, suggesting that the added cost of amplitude amplification is worthwhile for realistic materials science problems under certain assumptions.