  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.
