  Estimating the fidelity with a target state is important in quantum
information tasks. Many fidelity estimation techniques present a suitable
measurement scheme to perform the estimation. In contrast, we present
techniques that allow the experimentalist to choose a convenient measurement
setting. Our primary focus lies on a method that constructs an estimator with
nearly minimax optimal confidence intervals for any specified measurement
setting. We demonstrate, through a combination of theoretical and numerical
results, various desirable properties of the method: robustness against
experimental imperfections, competitive sample complexity, and accurate
estimates in practice. We compare this method with Maximum Likelihood
Estimation and the associated Profile Likelihood method, a Semi-Definite
Programming based approach, direct fidelity estimation, quantum state
verification, and classical shadows. Our method can also be used for estimating
the expectation value of any observable with the same guarantees.
