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

==Outline==
The aim here to perform cross-platform verification by measuring the overlap of quantum states produced with two different experimental setups, potentially realized on very different physical platforms, without any  prior assumptions on the quantum states themselves. This can be used to
whether two quantum devices have prepared the same quantum state. Here, the cross-platform fidelity is inferred from the statistical correlations between randomized measurements performed on the first and second device. 

This protocol to measure the cross-platform fidelity of two quantum states requires only classical communication of random unitaries and measurement outcomes between the two platforms, with the experiments possibly taking place at very different points in time and space.

This protocol consists of the following steps:

* We start with two quantum devices which are based on different physical platforms, each consisting of two different spins. Two quantum operations are prepared in these quantum devices, which are each described by a density matrix.
* We find the reduced density matrices for the sub-systems of identical size for each device using partial trace operator over that sub-system. 
* We apply a same random unitary is applied to the two quantum states. This random unitary is defined as the product of local random unitaries acting on all spins of the subsystem. Here, the local random unitaries are sampled independently from a [[unitary 2-design]] defined on the local Hilbert space and sent via classical communication to both devices.
* Now projective measurements in a computational basis are performed for both the systems.
* Repeating these measurements for the fixed random unitary provides us with the estimates of probability of measurement outcomes for the both the states.
* This entire procedure is then repeated for many different random unitaries.
* Finally we estimate the density matrix from the second order cross-correlations between the two platforms using the ensemble average of probabilities over random unitaries from the above procedure. 
* The purities for the two sub systems are obtained as second-order auto-correlations of the probabilities.