# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 4 Teleportation-Based Error Correction of Time-Frequency Qubit States  
In this section, we propose a protocol for correcting and teleporting frequency qubit states without relying on frequency-entangling operations. This protocol is similar to the one used for teleporting polarization qubit states as described in [57], and is inspired by the GKP analog [28; 29]. Since the EPR state has a lower level of noise in both temporal and frequency variables, the protocol includes an additional component for correcting errors in the state being teleported. The single-photon state \(|\psi\rangle\) to be teleported and corrected is described by the wave function \(|\psi\rangle=N_{a\beta}(\alpha|0\rangle_{a}+\beta|1\rangle_{a})\), where the two logical states are either the time-frequency cat or GKP qubits (see Equation (12)). The wave function of the entangled EPR time-frequency state in spatial ports \(b\) and \(c\) that assists with the teleportation and the correction is  
\[\big{|}\phi^{+}\big{\rangle}_{bc}=N_{\text{EPR}}(\big{|}0\bar{1}\big{\rangle}_{ bc}+\big{|}\bar{1}0\big{\rangle}_{bc}) \tag{25}\]  
composed of logical states which are less noisy, indicated with the tilde notation, than the state to be teleported. Upon completion of the protocol, the single-photon state \(|\psi\rangle\) is localized to spatial port \(c\), and the correction is automatically performed since the EPR state is less noisy than the state of interest. We now write the protocol for the error correction and teleportation of the time-frequency cat state, considering that the Bell measurement perfectly separates the two logical states. This protocol allows for correcting frequency errors affecting the frequency qubit states. The protocol is shown in Figure 6, and we will now proceed to write the evolution of the wave function at each step of the protocol. The initial wave function, composed of the state to be corrected and teleported, and the EPR state, written in Bell's basis is  
\[|\psi\rangle= \frac{N_{a\beta}N_{\text{EPR}}}{2}(|\omega_{1}\tilde{\omega}_{1} \rangle+|\omega_{2}\tilde{\omega}_{2}\rangle)(\alpha|\tilde{\omega}_{1}\rangle _{c}+\beta|\tilde{\omega}_{2}\rangle_{c})\] \[+(|\omega_{1}\tilde{\omega}_{1}\rangle-|\omega_{2}\tilde{\omega}_ {2}\rangle)(\alpha|\tilde{\omega}_{1}\rangle_{c}-\beta|\tilde{\omega}_{2} \rangle_{c})\] \[+(|\omega_{1}\tilde{\omega}_{2}\rangle+|\omega_{2}\tilde{\omega}_ {1}\rangle)(\beta|\tilde{\omega}_{1}\rangle_{c}+\alpha|\tilde{\omega}_{2} \rangle_{c})\] \[+(-|\omega_{1}\tilde{\omega}_{2}\rangle+|\omega_{2}\tilde{\omega}_ {1}\rangle)(\beta|\tilde{\omega}_{1}\rangle_{c}-\alpha|\tilde{\omega}_{2} \rangle_{c}). \tag{26}\]  
The single-photon state and one member of the EPR pair are then combined into a beam splitter, followed by two parity-frequency beam splitters placed in the spatial ports \(a^{\prime}\) and \(b^{\prime}\). The first and second Bell states are transformed as  
\[\frac{N_{\text{EPR}}}{2}[|\omega_{1}\tilde{\omega}_{1}\rangle_{a} -|\omega_{1}\rangle_{a}|\tilde{\omega}_{1}\rangle_{b}+|\omega_{1} \rangle_{b}|\tilde{\omega}_{1}\rangle_{a}-|\omega_{1}\tilde{\omega}_{1} \rangle_{b}\] \[+|\omega_{2}\tilde{\omega}_{2}\rangle_{a}-|\omega_{2}\rangle_{a} |\tilde{\omega}_{2}\rangle_{b}+|\omega_{2}\rangle_{b}|\tilde{\omega}_{2} \rangle_{a}-|\omega_{2}\tilde{\omega}_{2}\rangle_{b}]. \tag{27}\]  
The presence of a single photon in each port is a consequence of the distinguishability of the photons. While if the photons are indistinguishable, only bunching events are measured, as in the case of polarization encoding for two orthogonal logical states. In order to suppress these coincidence events that lead to errors in the teleportation protocol, the use of frequency filters with the same frequency width (envelope and peak) as the EPR state is a possible solution.