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

==Notation==
* <math>N_M</math>: Finite  number  of  projective  measurements  performed per  random  unitary
* <math>N_U</math>: Finite  number of random unitaries used to infer overlap
* <math>\epsilon</math>: Fixed value of statistical error
* <math>S_1, S_2</math>: Two devices realised on different physical platforms
* <math>N_1, N_2</math>: Spins consisted in <math>S_1</math> and <math>S_2</math> respectively
* <math>U_1, U_2</math>: Quantum operation prepared in <math>S_1</math> and <math>S_2</math> respectively
* <math>\rho_1, \rho_2</math>: Density matrices of <math>U_1</math> and <math>U_2</math> respectively
* <math>\rho_{i,A_i}</math>: Reduced density matrices
* <math>A_i</math>: Sub system of identical size <math>N_{A}</math> where <math>A_i \subseteq S_i</math>
* <math>N_{A}</math>: Size of subset <math>A_i</math>. Subsets <math>A_1</math> and <math>A_2</math> have the equal size <math>N_{A}</math>
* <math>D_A</math>: Associated Hilbert space dimension, <math>D_A = d^{N_A}</math>
* <math>U_A</math>: Random unitary
* <math>U_k</math>: Local random unitaries acting on spins <math>k=1, .., N_A</math>. Here, the <math>U_k</math> are sampled independently from a [[unitary 2-design]] defined on the local Hilbert space <math>C^d</math>
* <math>k</math>: Spin
* <math>|s_A \rangle</math>: This denotes a string of possible measurement outcomes for spins. <math>|s_A \rangle =|s_1, ..., S_{N_A} \rangle </math>
* <math>P_U^{(i)}</math>: Estimate of probabilities of measurement outcome for different spins.
* <math>\overline{P_U^{(i)}(s_A)}</math>: The ensemble average over random unitaries of the form <math>U_A</math>
* <math>D[s_A, s_A']</math>:  Hamming distance between two strings <math>s_A, s_A'</math> is defined as the number of spins where <math>s_k \neq s_k'</math> i.e. <math>D[s_A,s_A'] = |\{k \in \{ 1, .., N_A\}| s_k \neq s_k'\}|</math>
* <math>F_{max}(\rho_1, \rho_2)</math>: Cross platform fidelity between two quantum states