Quantifying $T$-gate-count improvements for ground-state-energy estimation with near-optimal state preparation

We study the question of when investing additional quantum resources in preparing a ground state will improve the aggregate runtime associated with estimating its energy. We analyze Lin and Tong's near-optimal state preparation algorithm and show that it can reduce a proxy for the runtime, the $T$-gate count, of ground state energy estimation near quadratically. Resource estimates are provided that specify the conditions under which the added cost of state preparation is worthwhile.