# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 5 Conclusions  
In this paper, we started to present two time-frequency qubit states, time-frequency cat and GKP states, and the errors from which they are robust against. Then, we explained two different optical scenarios that lead to different entanglement structures of time-frequency GKP states and gave different periodicities for the joint spectral intensity of the photon pair. This time-frequency GKP state entanglement structure leads naturally to a first error correction method that requires non-linear processes [1] or quantum dot embedded into a wave guide [27] to realize it. We then developed a method to perform the Bell measurement for these two time-frequency qubits using Mach-Zehnder interferometry. Since this error correction method is not scalable and difficult to implement experimentally, we discussed the teleportation-based error correction protocol for the frequency qubit state, and developed the case of the time-frequency GKP state in the appendix. The teleportation-based error correction protocol is well suited for the single-photon encoding, as it requires only linear optics. The single-photon state carrying the quantum information of interest is first combined into a balanced beam splitter with one member of a less noisy pair, and then the Bell measurement is performed. Therefore, if the initial state is broadened in the temporal or frequency domain, because the remaining single-photon state belongs to a less noisy entangled pair, such a state is less broadened in the temporal (or frequency) domain and carries the information of interest (see Figure 6). By contrast with the polarization degree of freedom, since the frequency qubit is defined from the discretization of a continuous variable, the two logical states composing the qubit are intrinsically non-orthogonal such that this fact leads to a decrease in the overall probability of success of the teleportation protocol. We can envision the scenario where the single-photon frequency is the carrier of the quantum information and is moving through different nodes of a quantum network, and the entangled states could be generated locally and allow healing the temporally and frequency-broadened single-photon state carrying the quantum information of interest. This teleportation-based error correction optical scheme has the same objective as a quantum relay, that is, reducing the error rate of a quantum key distribution protocol by reducing the overlap of the two logical states composing the qubit. One perspective is to study the case where the logical qubit states are not infinitely frequency-narrowed. The Bell measurement will lead to wrong detections because the spatial separation of the two logical states into two spatial ports, realized with Mach-Zehnder interferometry, is imperfect. This can be corrected by using frequency filters but at the cost of losing single photon detection events. To tackle this issue, the use of frequency-resolved detection and the fault-tolerance threshold defined in [50] could be valuable for avoiding the use of frequency filters. We showed that our protocol can correct the errors of the qubit composed of two colors by reducing their spectral width. As a direct perspective, a single photon qubit composed of two Gaussian centered on two temporal bins (see Equation (2)) could be corrected from the temporal dispersion with the teleportation-based error correction scheme by using a less temporally broadened entangled state, thereby reducing the error rate of quantum communication protocols [61; 62; 63]. This project has received funding from the European Union's Horizon Europe research and innovation program under the project \"Quantum Security Networks Partnership\" (QSNP, grant agreement No 101114043). Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or Telecom Paris. Neither the European Union nor the granting authority can be held responsible for them. Not applicable.