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When is better ground state preparation worthwhile for energy estimation?

Many quantum simulation tasks require preparing a state with overlap γ relative to the ground state of a Hamiltonian of interest, such that the probability of computing the associated energy eigenvalue is upper bounded by γ 2 . Amplitude ampliﬁcation can increase γ , 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.