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A promising use of quantum computers is to prepare quantum states that model complex domains, such as correlated electron wavefunctions or the underlying distribution of a complex dataset. Such states need to be verified in view of algorithmic approximations and device imperfections. As quantum computers grow in size, however, verifying the states they produce becomes increasingly problematic. Relatively efficient methods have been devised for verifying sparse quantum states, but dense quantum states have remained costly to verify. Here I present a novel method for estimating the fidelity $F(μ,τ)$ between a preparable quantum state $μ$ and a classically specified target state $τ$, using simple quantum circuits and on-the-fly classical calculation (or lookup) of selected amplitudes of $τ$. Notably, in the targeted regime the method demonstrates an exponential quantum advantage in sample efficiency over any classical method. The simplest version of the method is efficient for anticoncentrated quantum states (including many states that are hard to simulate classically), with a sample cost of approximately $4ε^{-2}(1-F)dp_{\text{coll}}$ where $ε$ is the desired precision of the estimate, $d$ is the dimension of the Hilbert space in which $μ$ and $τ$ reside, and $p_{\text{coll}}$ is the collision probability of the target distribution. I also present a more sophisticated version of the method, which uses any efficiently preparable and well-characterized quantum state as an importance sampler to further reduce the number of copies of $μ$ needed. Though some challenges remain, this work takes a significant step toward scalable verification of complex states produced by quantum processors.