# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 2 Time-Frequency Qubit States
### Sources of Time-Frequency Noise  
In this section, we discuss the physical processes that lead to temporal-spectral broadening or distortion. Coherent and incoherent errors will lead to either pure or mixed states, respectively.  
Temporal errors for both codes arise from the temporal spreading of each wave packet composing the state due to the linear dispersion effect, described by a coherent model (see, for instance, [41] for an example in the single photon regime). After such a second-order dispersion effect, the temporal width of the Gaussian wave packet becomes \(\tau=\tau_{0}\sqrt{1+\tau_{c}^{4}/\tau_{0}^{4}}\), where \(\tau_{0}\) is the initial width of the pulse and \(\tau_{c}=\sqrt{\beta_{2}L}\), \(L\) being the length of the dispersive medium and \(\beta_{2}\) the dispersion coefficient. Such a dispersion process is described by a unitary operation, and it can, in principle, be undone by a reverse transformation. However, it requires knowledge of the full characterization of the propagation channel. For the time-frequency GKP state, the dispersive effect not only leads to a temporal spreading of each wave packet, but also leads to the formation of replica temporal images, called the temporal Talbot effect (see, for instance, [42]). One has to consider the specific length of the fiber or dispersive coefficient to recover the initial state, which will be also temporally broadened. Polarizing mode dispersion is one of incoherent temporal broadening [43; 44; 45]. Due to a coupling between the polarization and frequency degree of freedom, if the polarization is not measured, the single-photon state becomes mixed in frequency [46]. Thus, the error correction protocol that is presented in Section 2.4 becomes particularly relevant because we cannot cancel the error simply by a unitary operation.  
Frequency noise that causes spectral broadening while the spectral distribution remains Gaussian is not typically dominant at the single-photon level. Spectral broadening induced by the self-phase modulation effect [47] results from the accumulated phase \(\phi_{NL}(t)=\frac{2\pi}{\lambda}n_{2}I(t)L\), where \(n_{2}\) is the non-linear refractive index, and \(L\) is the length of the medium and depends on the intensity \(I(t)\) of the field, leading to a non-Gaussian spectral distribution. At the single-photon level, this non-linear process does not occur naturally. In [1], we also argue that the frequency noise arises from frequency broadening caused by the generation of photon pairs itself, and we describe one method to correct such noise. There are numerous processes that can distort the spectral distribution of single photons, such as distortions caused by frequency shifts induced by electro-optic modulators [48; 49], or the presence of a filter during single-photon heralding.  
An error correction protocol is implemented to mitigate the effects of potential errors from various sources. Its objective is to restore the Gaussian distribution of each frequency peak. This is because Gaussian probability distributions are better understood for setting confidence intervals and error thresholds as discussed in [50]. Both qubits are sensitive to temporal and frequency broadening. While the time-frequency cat state is designed to be robust against errors over one variable, the time-frequency GKP state can correct errors along both orthogonal variables. It is not necessary to use the time-frequency GKP state if the main error is in the temporal domain. We now develop two error correction methods, one based on a direct frequency entanglement between the noisy state of interest and a less noisy ancilla (Section 2.4), and the other method using only linear optics and also less noisy ancilla (Section 4).