Editing Compressed Sensing Tomography (section)

==Procedure Description==

'''Input''': copies of unknown quantum state

'''Output''': Density matrix of the quantum state, <math>\rho</math>

* Consider system of <math>n</math> qubits and dimension <math>d = 2^n</math>
* Define <math>\mathcal{P}</math> and select <math>m</math> operators <math>(P_1, ... , P_m)</math> from <math>\mathcal{P}</math>
* Make <math>t</math> copies of the unknown quantum state <math>\rho</math>
* For <math>i = 1, 2, ..., m</math>:
** For <math>j = 1, 2, ..., t/m</math>:
*** Measure <math>P_i</math> on <math>\rho</math>
** Average measurement results to get <math>Tr(P_i\rho)</math>
** Define sampling operator <math>\mathcal{A}, \mathcal{A}(\rho)_i = \sqrt{\frac{d}{m}} Tr(P_i\rho)</math>
* Output of measurements is defined as the vector <math>y = \mathcal{A}(\rho) + z</math>
* To estimate <math>\rho</math> there are two methods:
** Using trace minimization:
*** Choose <math>\lambda</math> such that <math> ||A^*(z)|| \leq \lambda</math>, then <math>||\hat{\rho}_{DS} -\rho||_{tr} \leq C_0r\lambda + C_1||\rho_c||_{tr}</math>
*** <math>\hat{\rho}_{DS} =</math> argmin<math>_X ||X||_{tr}</math> such that <math>||\mathcal{A}^*(\mathcal{A}(X) - y)|| \leq \lambda</math>
** Using least-squares linear regression with trace-norm regularization:
*** Choose <math>\mu</math> such that <math>||A^*(z)|| \leq \mu</math>, then <math>||\hat{\rho}_{Lasso} -\rho||_{tr} \leq C^{'}_0r\lambda + C^{'}_1||\rho_c||_{tr}</math>
*** <math>\hat{\rho}_{Lasso} =</math> argmin<math>_X \frac{1}{2} ||\mathcal{A}(X)-y||_2^2 + \mu||X||_{tr}</math>