# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## Appendix B Teleportation-Based Error Correction Protocol with Physical Time-Frequency GKP State  
In the following, we employ the GKP coherent picture [51]. The GKP qubit defined by Equation (12) is composed of an envelope of width \(\kappa\) and each peak has a width of \(\sigma\) (resp. \(\kappa_{1},\sigma_{1}\)), while the frequency widths of the EPR state is noted as \(\bar{\kappa}\) and \(\bar{\sigma}\). As noted, when the time-frequency GKP state has a limited bandwidth, the frequency qubit beam splitter is not able to perfectly separate the odd and even components of the comb. This results in some of the state leaking out of each spatial port and not conforming to a GKP state. Furthermore, the state in the right port is also distorted. To address these two challenges, it is necessary to employ frequency filters that enable projection back into the GKP subspace and eliminate undesirable components. Given that the state being teleported and the EPR state have different frequency widths, a frequency filter is placed prior to detection, which establishes the frequency bin and aligns with the reference EPR state. The projector modeling for the frequency-filtering process has the following form:  
\[\hat{\Pi}_{a}=N_{e}^{2}(\sigma,\kappa)\sum_{n\in\mathbb{Z}}\tilde{c}_{2n}\int G _{2n}^{\bar{\sigma}}(\omega)|\omega\rangle\langle\omega| \tag{24}\]  
\[\hat{\Pi}_{a^{\prime}}=N_{o}^{2}(\sigma,\kappa)\sum_{n\in\mathbb{Z}}\tilde{c}_ {2n+1}\int G_{2n+1}^{\bar{\sigma}}(\omega)|\omega\rangle\langle\omega| \tag{25}\]  
The normalization constant is found by using \(\hat{\Pi}^{2}=\mathbb{I}\) and \(\mathrm{Tr}(\hat{\Pi}^{2})=1\), \(N_{e}^{2}(\bar{\sigma},\bar{\kappa})=\sqrt{2\pi}/\bar{\sigma}\sum_{n}|\tilde{ c}_{2n}|^{2}\). In the large comb approximation, we have \(N_{e}(\bar{\sigma},\bar{\kappa})=N_{o}(\bar{\sigma},\bar{\kappa})\). The coincidence destructive measurement is described by the positive operator value measurement:  
\[\hat{\Pi}_{a,a^{\prime}}=\hat{\Pi}_{a}\hat{\Pi}_{a^{\prime}}\otimes|1\rangle \langle 1|. \tag{26}\]  
By assuming that the state is pure, the probability of coincidence in the spatial port \(a,a^{\prime}\), \(P_{aa^{\prime}}(0,1)=\mathrm{Tr}(\hat{\Pi}_{a,a^{\prime}}|\psi\rangle\langle \psi|\hat{\Pi}_{a,a^{\prime}}^{\dagger})\) is  
\[P_{aa^{\prime}}(0,1)= \bigg{|}\frac{1}{2\sqrt{2}}\bigg{|}^{2}\bigg{|}a_{\sigma\bar{ \sigma}}^{01}+b_{\sigma\bar{\sigma}}^{01}\bigg{|}^{2}\times\] \[\bigg{(}\int d\omega(|a|^{2}|\langle\omega|\bar{0}\rangle|^{2}+| \beta|^{2}|\langle\omega||\bar{1}\rangle|^{2}+2\mathrm{Re}(a^{*}\beta\langle \omega||\bar{0}\rangle\langle\omega|\bar{1}\rangle))\bigg{)} \tag{27}\]  
where we used \(|a_{\sigma\bar{\sigma}}^{01}+b_{\sigma\bar{\sigma}}^{01}|=|a_{\sigma\bar{ \sigma}}^{10}+b_{\sigma\bar{\sigma}}^{10}|\), which is shown afterward. For an EPR with a sufficiently narrow distribution, we assume that \(\langle\bar{0}|\bar{1}\rangle=0\), \(\int d\omega|\langle\omega|\bar{0}\rangle|^{2}=\int d\omega|\langle\omega| \bar{1}\rangle|^{2}=1\). These two conditions are important since the probability then does not depend on \(a\) and \(\beta\),  
\[P_{aa^{\prime}}(0,1)=|\frac{1}{2\sqrt{2}}|^{2}|a_{\sigma\bar{\sigma}}^{01}+b_{ \sigma\bar{\sigma}}^{01}|^{2} \tag{28}\]  
since otherwise, information about the quantum state could be extracted during the measurement. The wave function of the state after the detection is \(|\psi\rangle_{c}=\hat{\Pi}_{i,j}|\psi\rangle/\mathrm{Tr}(\hat{\Pi}_{i,j}| \psi\rangle\langle\psi|\hat{\Pi}_{i,j}^{\dagger})\), where \(i,j=a,b;a^{\prime},b^{\prime}\). When coincidences are detected at the spatial port \(a\) and \(a^{\prime}\), the post-selected state is  
\[|\psi\rangle_{c}=\frac{1}{|a_{\sigma\bar{\sigma}}^{01}+b_{\sigma\bar{\sigma}}^ {01}|}(\alpha(a_{\sigma,\sigma}^{1,\sigma}+b_{0,\sigma}^{1,\sigma})|\bar{0} \rangle_{c}+\beta(a_{0,\sigma}^{1,\sigma}+b_{0,\sigma}^{1,\sigma})|\bar{1} \rangle_{c}).\]The expression is similar for the other coincidence events in the other spatial ports, and extra Pauli operations have to be performed.