Quantum Gate Set Tomography: Difference between revisions

Quantum gate set tomography is a robust and powerful method for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device.  This method arose from the observation that Quantum Process tomography [link here] is inaccurate in presence of state preparation and measurement errors. This procedure estimates the complete process matrices of Pauli transfer matrices for the gate set based on experimental data.

'''Output''': Gate set, $G$

* Initialize system to $|\rho\rangle$

* For $i = 1, 2, ..., N$:

** For $j = 1, 2, ..., N$:

*** For $k = 1, 2, ..., K$:

**** For $r = 1, 2, ..., n$

***** Apply gate sequence $F_i \circ G_k \circ F_j$

***** Measure with POVM $E$, get $n_r = 1$ or $n_r = 0$

**** $m_{ijk} = \sum_{r=1}^{n} \frac{n_r}{n}$

**** $(\tilde{G}_k)_{ij} = p_{ijk} = m_{ijk} =  \langle E|F_iG_kF_j|\rho\rangle$

**** if $k==0$ (null gate), $g_{ij} = \langle E|F_iF_j|\rho\rangle$

** $|\tilde{\rho}\rangle_i = \langle E|F_i|\rho\rangle$

** $B_{ij} = \langle i |F_j |\rho\rangle$

** Check that the Gram matrix $g_{ij}$ is non singular, so that it may be inverted

** For the gate set, the estimate is $|\hat{\rho}\rangle = g^{-1}|\tilde{\rho}\rangle, \langle \hat{E}| = \langle \tilde{E} |, \hat{G_k} = g^{-1} \tilde{G}_k$

              \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})\}

          \item Final gate set is: $\hat{G}_k^{*} = \hat{B}^*\hat{G}_k(\hat{B}^*)^{-1}, |\hat{p}^*\rangle = \hat{B}^*|\hat{p}\rangle, \langle \hat{E}^*| = \langle \hat{E}|(\hat{B}^*)^{-1}$

** $\hat{p}_{ijk} = \langle \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t})|\hat{\rho}(\vec{t})\rangle$

**  Find the set of parameters $\vec{t}$ where $l(\hat{G})$ is minimized.

           Minimize: \begin{equation}

           \end{equation}

          \item Minimize the above function with two different commonly used types of parameterisation of gates:

              \item Pauli Process Matrix Optimization problem:

              Minimize: \begin{equation}

              Subject to: \begin{equation}

               \end{equation}

               \end{equation}

               \end{equation}

              \item Pauli Transfer Matrix Representation

              Minimize: \begin{equation}

               \end{equation}

              Subject to: \begin{equation}

               \end{equation}

               \end{equation}

               \end{equation}

               \end{equation}