  We propose and analyze a sample-efficient protocol to estimate the fidelity
between an experimentally prepared state and an ideal target state, applicable
to a wide class of analog quantum simulators without advanced sophisticated
spatiotemporal control. Our approach utilizes newly discovered universal
fluctuations emerging from generic Hamiltonian dynamics, and it does not
require any fine-tuned control over state preparation, quantum evolution, or
readout capability. It only needs a small number of experimental measurements,
achieving near optimal sample complexity: in ideal cases, a percent-level
precision is obtained with $\sim 10^3$ measurements independent of system size.
Furthermore, the accuracy of our fidelity estimation improves with increasing
system size. We numerically demonstrate our protocol for a variety of quantum
simulator platforms such as itinerant particles on optical lattices, trapped
ions, and Rydberg atoms. We discuss further applications of our method for
advanced tasks such as multi-parameter estimation of quantum states and
processes.
