Editing Quantum Gate Set Tomography (section)

==Procedure Description==

'''Output''': Gate set, <math>G</math>

* Initialize system to <math>|\rho\rangle</math>
* For <math>i = 1, 2, ..., N</math>:
** For <math>j = 1, 2, ..., N</math>:
*** For <math>k = 1, 2, ..., K</math>:
**** For <math>r = 1, 2, ..., n</math>
***** Apply gate sequence <math>F_i \circ G_k \circ F_j</math>
***** Measure with POVM <math>E</math>, get <math>n_r = 1</math> or <math>n_r = 0</math>
**** <math>m_{ijk} = \sum_{r=1}^{n} \frac{n_r}{n}</math>
**** <math>(\tilde{G}_k)_{ij} = p_{ijk} = m_{ijk} =  \langle E|F_iG_kF_j|\rho\rangle</math>
**** if <math>k==0</math> (null gate), <math>g_{ij} = \langle E|F_iF_j|\rho\rangle</math>
** <math>|\tilde{\rho}\rangle_i = \langle E|F_i|\rho\rangle</math>
** <math>B_{ij} = \langle i |F_j |\rho\rangle</math>
* For Linear inversion:
** Check that the Gram matrix <math>g_{ij}</math> is non singular, so that it may be inverted
** For the gate set, the estimate is <math>|\hat{\rho}\rangle = g^{-1}|\tilde{\rho}\rangle, \langle \hat{E}| = \langle \tilde{E} |, \hat{G_k} = g^{-1} \tilde{G}_k</math>
** Apply gauge optimization, by minimizing:
<div style="text-align: center;"><math>\hat{B}^* = argmin_{\hat{B}}\sum^{K+1}_{k=1}Tr\{(\hat{G}_k - \hat{B}^{-1}T_k\hat{B})^T(\hat{G}_k - \hat{B}^{-1}T_k\hat{B})\}
</math></div>
** Final gate set is: <math>\hat{G}_k^{*} = \hat{B}^*\hat{G}_k(\hat{B}^*)^{-1}, |\hat{p}^*\rangle = \hat{B}^*|\hat{p}\rangle, \langle \hat{E}^*| = \langle \hat{E}|(\hat{B}^*)^{-1}</math>
* For Maximum Likelihood Estimation:
** <math>\hat{p}_{ijk} = \langle \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t})|\hat{\rho}(\vec{t})\rangle</math> 
**  Find the set of parameters <math>\vec{t}</math> where <math>l(\hat{G})</math> is minimized.
           <div style="text-align: center;">Minimize: <math>
               l(\hat{G}) = \sum_{ijk} (m_{ijk} - \hat{p}_{ijk}(\vec{t}))^2 / \sigma^2_{ijk}
           </math></div>
** Minimize the above function with two different commonly used types of parameterisation of gates:
*** Pauli Process Matrix Optimization problem:
       <div style="text-align: center;">Minimize: <math>
                 l(\hat{G}) = \sum_{ijk} (m_{ijk} - \sum_{mnrstu}(\chi_{F_i})_{tu}  (\chi_{G_k})_{rs} (\chi_{F_j})_{mn} Tr\{EP_t P_r P_m \rho P_n P_s P_u\} )^2 / \sigma^2_{ijk}
      </math></div>
      <div style="text-align: center;">Subject to: <math>
                   \sum_{mn}(\chi_G)_{mn}Tr\{P_mP_rP_n\} - \delta_{0r} = 0, (r = 1, .. , d^2), \forall G \in Gateset
               </math></div>
      <div style="text-align: center;"><math>
                   Tr\{\rho\} = 1
               </math></div>
      <div style="text-align: center;"><math>
                   1 - E \geqslant 0
               </math></div>
*** Pauli Transfer Matrix Representation
<div style="text-align: center;">   Minimize: <math>
                 l(\hat{G}) = \sum_{ijk} (m_{ijk} - \langle \hat{E}(\vec{t})|\hat{R}_{F_i}(\vec{t})\hat{R}_{G_k}(\vec{t})\hat{R}_{F_j}(\vec{t})|\hat{\rho}(\vec{t})\rangle )^2 / \sigma^2_{ijk}
               </math></div>
<div style="text-align: center;">Subject to: <math>
                   \rho_G = \frac{1}{d^2} \sum_{i,j=1}^{d^2}(R_G)_{ij}P^T_G \otimes P_i \geqslant 0, \forall G \in Gateset
               </math></div>
<div style="text-align: center;"><math>
                   (R_G)_{0i} = \delta_{0i}, \forall G \in Gateset
               </math></div>
<div style="text-align: center;"><math>
                   (R_G)_{ij} \in [-1, 1], \forall G, i, j
               </math></div>
<div style="text-align: center;"><math>
                   Tr\{\rho\} = 1
              </math></div>
<div style="text-align: center;"><math>
                   1 - E \geqslant 0
               </math></div>