  As a measure of the 'closeness' of two quantum states, fidelity plays a
fundamental role in quantum information theory. Fidelity estimation protocols
try to strike a balance between information gleaned from an experiment, and the
efficiency of its implementation, in terms of the number of states consumed by
the protocol. Here we adapt a previously reported optimal state verification
protocol (Phys. Rev. Lett. 120, 170502, 2018) for fidelity estimation of
two-qubit states. We demonstrate the protocol experimentally using a
fully-programmable silicon photonic two-qubit chip. Our protocol outputs
significantly smaller error bars of its point estimate in comparison with
another widely-used estimation protocol, showing a clear step forward in the
ability to estimate the fidelity of quantum states produced by a practical
device.
