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Quantum Gate Set Tomography: Difference between revisions

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Quantum Gate Set Tomography (edit)

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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.
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]] is inaccurate in the 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.


'''Tags:''' [[Certification protocol]], [[Tomography]], [[Quantum process density matrix reconstruction]], [[Maximum Likelihood estimation]], [[Linear Inversion]], [[Gate set]]
'''Tags:''' [[Certification protocol]], [[Tomography]], [[Quantum process density matrix reconstruction]], [[Maximum Likelihood estimation]], [[Linear Inversion]], [[Gate set]]
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==Protocol Description==
==Protocol Description==


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


* Initialize system to $|\rho\rangle$
* Initialize system to <math>|\rho\rangle</math>
* For $i = 1, 2, ..., N$:
* For <math>i = 1, 2, ..., N</math>:
** For $j = 1, 2, ..., N$:
** For <math>j = 1, 2, ..., N</math>:
*** For $k = 1, 2, ..., K$:
*** For <math>k = 1, 2, ..., K</math>:
**** For $r = 1, 2, ..., n$
**** For <math>r = 1, 2, ..., n</math>
***** Apply gate sequence $F_i \circ G_k \circ F_j$
***** Apply gate sequence <math>F_i \circ G_k \circ F_j</math>
***** Measure with POVM $E$, get $n_r = 1$ or $n_r = 0$
***** Measure with POVM <math>E</math>, get <math>n_r = 1</math> or <math>n_r = 0</math>
**** $m_{ijk} = \sum_{r=1}^{n} \frac{n_r}{n}$
**** <math>m_{ijk} = \sum_{r=1}^{n} \frac{n_r}{n}</math>
**** $(\tilde{G}_k)_{ij} = p_{ijk} = m_{ijk} =  \langle E|F_iG_kF_j|\rho\rangle$
**** <math>(\tilde{G}_k)_{ij} = p_{ijk} = m_{ijk} =  \langle E|F_iG_kF_j|\rho\rangle</math>
**** if $k==0$ (null gate), $g_{ij} = \langle E|F_iF_j|\rho\rangle$
**** if <math>k==0</math> (null gate), <math>g_{ij} = \langle E|F_iF_j|\rho\rangle</math>
** $|\tilde{\rho}\rangle_i = \langle E|F_i|\rho\rangle$
** <math>|\tilde{\rho}\rangle_i = \langle E|F_i|\rho\rangle</math>
** $B_{ij} = \langle i |F_j |\rho\rangle$
** <math>B_{ij} = \langle i |F_j |\rho\rangle</math>
* For Linear inversion:
* For Linear inversion:
** Check that the Gram matrix $g_{ij}$ is non singular, so that it may be inverted
** 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 $|\hat{\rho}\rangle = g^{-1}|\tilde{\rho}\rangle, \langle \hat{E}| = \langle \tilde{E} |, \hat{G_k} = g^{-1} \tilde{G}_k$
** 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:
** Apply gauge optimization, by minimizing:
          \begin{equation}
<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})\}
              \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>
          \end{equation}
** 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>
          \item Final gate set is: $\hat{G}_k^{*} = \hat{B}^*\hat{G}_k(\hat{B}^*)^{-1}, |\hat{p}^*\rangle = \hat{B}^*|\hat{p}\rangle, \langle \hat{E}^*| = \langle \hat{E}|(\hat{B}^*)^{-1}$
* For Maximum Likelihood Estimation:
* For Maximum Likelihood Estimation:
** $\hat{p}_{ijk} = \langle \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t})|\hat{\rho}(\vec{t})\rangle$
** <math>\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 $\vec{t}$ where $l(\hat{G})$ is minimized.
**  Find the set of parameters <math>\vec{t}</math> where <math>l(\hat{G})</math> is minimized.
           Minimize: \begin{equation}
           <div style="text-align: center;">Minimize: <math>
               l(\hat{G}) = \sum_{ijk} (m_{ijk} - \hat{p}_{ijk}(\vec{t}))^2 / \sigma^2_{ijk}
               l(\hat{G}) = \sum_{ijk} (m_{ijk} - \hat{p}_{ijk}(\vec{t}))^2 / \sigma^2_{ijk}
           \end{equation}
           </math></div>
          \item Minimize the above function with two different commonly used types of parameterisation of gates:
** Minimize the above function with two different commonly used types of parameterisation of gates:
          \begin{enumerate}
*** Pauli Process Matrix Optimization problem:
              \item Pauli Process Matrix Optimization problem:
      <div style="text-align: center;">Minimize: <math>
              Minimize: \begin{equation}
                 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}
                 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}
              \end{equation}
      </math></div>
              Subject to: \begin{equation}
      <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
                   \sum_{mn}(\chi_G)_{mn}Tr\{P_mP_rP_n\} - \delta_{0r} = 0, (r = 1, .. , d^2), \forall G \in Gateset
               \end{equation}
               </math></div>
              \begin{equation}
      <div style="text-align: center;"><math>
                   Tr\{\rho\} = 1
                   Tr\{\rho\} = 1
               \end{equation}
               </math></div>
              \begin{equation}
      <div style="text-align: center;"><math>
                   1 - E \geqslant 0
                   1 - E \geqslant 0
               \end{equation}
               </math></div>
              \item Pauli Transfer Matrix Representation
*** Pauli Transfer Matrix Representation
              Minimize: \begin{equation}
<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}
                 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}
               \end{equation}
               </math></div>
              Subject to: \begin{equation}
<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
                   \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
               \end{equation}
               </math></div>
              \begin{equation}
<div style="text-align: center;"><math>
                   (R_G)_{0i} = \delta_{0i}, \forall G \in Gateset
                   (R_G)_{0i} = \delta_{0i}, \forall G \in Gateset
               \end{equation}
               </math></div>
              \begin{equation}
<div style="text-align: center;"><math>
                   (R_G)_{ij} \in [-1, 1], \forall G, i, j
                   (R_G)_{ij} \in [-1, 1], \forall G, i, j
               \end{equation}
               </math></div>
              \begin{equation}
<div style="text-align: center;"><math>
                   Tr\{\rho\} = 1
                   Tr\{\rho\} = 1
              \end{equation}
              </math></div>
              \begin{equation}
<div style="text-align: center;"><math>
                   1 - E \geqslant 0
                   1 - E \geqslant 0
               \end{equation}
               </math></div>
          \end{enumerate}
==Further Information==
      \end{enumerate}
 
==Related Papers==
* Daniel Greenbaum (2015) arXiv:1509.02921v1: Introduction to Quantum Gate Set Tomography
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