Editing Direct Fidelity Estimation (section)

==Notation==
* <math>d</math>: Dimension of Hilbert space
* <math>\rho</math>: Original pure quantum state
* <math>\sigma</math>: Final quantum state which has to be compared to the original quantum state
* <math>F(\rho, \sigma) = </math>tr(<math>\rho \sigma</math>): Fidelity of the two quantum states <math>\rho</math> and <math>\sigma</math>
* <math>W_k (k = 1, 2, ..., d^2)</math>: Denotes all possible Pauli operators, (<math>n</math>-fold tensor products of <math>I</math>, <math>\sigma_x</math>, <math>\sigma_y</math> and <math>\sigma_z</math>)
* <math>\chi_p(k)</math>: Characteristic function, <math>\chi_p(k) =</math> tr(<math>\frac{\rho W_k}{\sqrt{d}}</math>)
* Pr<math>(k)</math>: Probability of selecting <math>k</math> from <math>\{1, .. ,d^2\}</math>. This is the importance weighting rule, which is a natural probability distribution. Pr<math>(k) = (\chi_p(k))^2</math>
* <math>\epsilon</math>: Additive Error
* <math>\delta</math>: Failure probability
* <math>l</math>: Number of times the process is repeated to obtain the average, <math>l = \frac{1}{\epsilon^2 \delta}</math>
* <math>m_i</math>: Number of copies of <math>\sigma</math> in a single process.
* <math>A_{ij}</math>: Measurement outcome, <math>A_{ij} \in \{1, -1\}</math>
* <math>\tilde{X}</math>: Expectation over all measurement outcomes
* <math>\tilde{Y}</math>: Average of all expectation values of all measurement outcomes
* <math>E(m)</math>: Expected number of copies of final quantum state