Editing Cross-Platform verification of Intermediate Scale Quantum Devices (section)

==Properties==
* '''Figure of merit''': Cross-platform fidelity of two quantum states
* We can estimate the density matrix overlap of two quantum states here as well as their purities.
* The present protocol scales, although exponentially, much more favorably with the (sub)system size, allowing practical cross-platform verification for (sub)systems involving tens of qubits on state-of-the-art quantum devices.
* This protocol can be used to perform fidelity estimation towards known target theoretical states, as an experiment-theory comparison.
* In practice, from a finite number of projective measurements performed per random unitary (<math>N_M</math>), a statistical error of the estimated fidelity arises. With that, a finite number (<math>N_U</math>) of random unitaries used to infer overlap and purities can also cause a  statistical error while estimating fidelity. Therefore, the scaling of the total number of experimental runs <math>N_MN_U</math>, which are required to reduce this statistical error below a fixed value of <math>\epsilon</math>
* In the regime <math>N_M \leq D_A</math>  and <math>N_U \gg 1</math>, the average statistical error <math>|[F_{max}(\rho_A, \rho_A)]_e - 1| ~ 1/(N_M \sqrt{N_U})</math>. For unit target fidelity, the optimal allocation of the total measurement budget <math>N_U N_M</math> is thus to keep <math>N_U</math> small and fixed.
* The fidelity estimation of PR (entangled) states is thus less prone to statistical errors which we attribute to the fact that fluctuations across random unitaries are reduced due to the mixedness of the subsystems.
* The optimal allocation of <math>N_U</math> vs. <math>N_M</math> for given <math>N_UN_M</math> depends on the quantum states, in particular their fidelity and the allowed statistical error <math>\epsilon</math>,and is thus a priori not known
* In larger quantum systems, it gives access to the fidelities of all possible subsystems up to a given size – determined by the accepted statistical error and the measurement budget – and thus enables a fine-grained comparison of large quantum systems.