  We describe a protocol for cross-platform verification of quantum simulators
and quantum computers. We show how to measure directly the overlap
$\textrm{Tr}\left[\rho_1 \rho_2\right]$ and the purities
$\textrm{Tr}\left[\rho^2_{1,2}\right]$, and thus a fidelity of two possibly
mixed quantum states $\rho_1$ and $\rho_2$ prepared in separate experimental
platforms. We require only local measurements in randomized product bases,
which are communicated classically. As a proof-of-principle, we present the
measurement of experiment-theory fidelities for entangled $10$-qubit quantum
states in a trapped ion quantum simulator.
