# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 4 Teleportation-Based Error Correction of Time-Frequency Qubit States
Explicitly, after the filtering operation, the state becomes \(|\tilde{\omega}_{1}\tilde{\omega}_{1}\rangle_{a}-|\omega_{1}\tilde{\omega}_ {1}\rangle_{b}+|\tilde{\omega}_{2}\tilde{\omega}_{2}\rangle_{a}-|\omega_{2} \tilde{\omega}_{2}\rangle_{b}\), which leads only to two bunching events, which are ignored when single photon detectors are used. The same analysis can be employed for the second Bell state. For the third and fourth Bell states, they are transformed as  
\[\frac{N_{\text{EPR}}}{2}[ \pm|\omega_{1}\hat{\omega}_{2}\rangle_{aa^{\prime}}\mp|\omega_{1} \hat{\omega}_{2}\rangle_{ab^{\prime}}\pm|\omega_{1}\hat{\omega}_{2}\rangle_{ba ^{\prime}}\mp|\omega_{1}\hat{\omega}_{2}\rangle_{bb^{\prime}}\] \[+|\omega_{2}\hat{\omega}_{1}\rangle_{a^{\prime}a}-|\omega_{2} \hat{\omega}_{1}\rangle_{a^{\prime}b}+|\omega_{2}\hat{\omega}_{1}\rangle_{b^{ \prime}a}-|\omega_{2}\hat{\omega}_{1}\rangle_{b^{\prime}b}]. \tag{28}\]  
The use of frequency filters is again imperative since the measurement of coincidence of \(a,a^{\prime}\) (or \(b,b^{\prime}\)) permits to ensure that the quantum state \(N_{\alpha\beta}(\beta|\hat{\omega}_{1}\rangle+\alpha|\hat{\omega}_{2}\rangle)\) is teleported. In the same way, the measurement of the coincidence of \(a,b^{\prime}\) (or \(b,a^{\prime}\)) allows teleporting the state \(N_{\alpha\beta}(\beta|\hat{\omega}_{1}\rangle-\alpha|\hat{\omega}_{2}\rangle)\). The receiver sends to spatial port \(c\) which detectors have measured coincidences, and then a product of Pauli matrix gates must be applied to recover the initial state of interest. The Pauli matrices for the time-frequency cat states are frequency and temporal shifts operations [1], which can be implemented by either an electro-optical modulator [48,49] and a delay line, respectively. The full optical scheme of the teleportation-based error correction protocol is represented in Figure 6, along with an illustration of the effect of the error correction for qubit cat states. The corresponding probability of each event is \(P=\frac{1}{8(1+e^{-\Delta^{2}/2\sigma^{2}})}\), and the overall probability success of the teleportation is  
\[P=\frac{1}{2(1+e^{-\Delta^{2}/2\sigma^{2}})}, \tag{29}\]  
which is then lower than 50% since only linear optics is used [57] and because the non-orthogonality of the encoding further decreases the probability of success. Note that the use of photon number-resolving (PNR) detectors will be able to not discard the bunching events coming from the first and second Bell states. The single nanowire superconductivity detector can also differentiate two and one single photons since the amplitude of the resistance of the hotspot over time is higher for two photons. With the use of such a PNR detector, the overall probability of success of the teleportation protocol is  
\[|\psi\rangle= \frac{N_{a\beta}N_{\text{EPR}}}{2}(|\omega_{1}\tilde{\omega}_{1} \rangle+|\omega_{2}\tilde{\omega}_{2}\rangle)(\alpha|\tilde{\omega}_{1}\rangle _{c}+\beta|\tilde{\omega}_{2}\rangle_{c})\]0 Schematic of the teleportation-based error correction of frequency qubit states. The state to be teleported and corrected is located at spatial port a. An EPR frequency qubit state which is less noisy than the state of interest, is at spatial ports b and c. FQBS stands for the frequency qubit beam splitter. \(F_{0,1}\) are frequency filters with a central frequency matching either the zero or one logical state, and with a frequency width equal to that of the EPR state. Depending on which detectors have measured coincidences, Pauli operations \(\hat{X},\hat{Z}\) must be performed to recover the state of interest. The frequency qubit states before and after the teleportation are represented. The zero logical (resp. one) state has a blue (resp. red) color.