# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 2 Time-Frequency Qubit States
### Time-Frequency GKP State  
We define a frequency lattice of period \(\overline{\omega}\). Centered on each of these intervals, we define the ideal time-frequency GKP state as the following frequency comb at the single photon level:  
\[\left|\overline{\mu}_{\omega}\right\rangle_{a}=\sum_{n\in\mathbb{Z}}\left|(2n+ \mu)\overline{\omega}\right\rangle_{a} \tag{7}\]  
where \(\mu=0,1\) index the two logical states. Note that the equal weight superposition of the zero and the one logical time-frequency GKP states in the frequency domain are the zero and one in the temporal domain:  
\[\left|\overline{\tau}_{\omega}\right\rangle_{a}=\frac{1}{\sqrt{2}} \left(\left|\overline{0}_{\omega}\right\rangle_{a}+\left|\overline{1}_{\omega }\right\rangle_{a}\right)=\left|\overline{0}_{t}\right\rangle_{a} \tag{8}\] \[\left|\overline{-}_{\omega}\right\rangle_{a}=\frac{1}{\sqrt{2}} \left(\left|\overline{0}_{\omega}\right\rangle_{a}-\left|\overline{1}_{\omega }\right\rangle_{a}\right)=\left|\overline{1}_{t}\right\rangle_{a}. \tag{9}\]  
The periodicity of the state in the temporal domain: \(\overline{\omega}=2\pi/\overline{\omega}\). Such a state is not physical since the state is an infinite sum of monochromatic states and will require infinite energy to prepare it. The physical time-frequency GKP state can be built upon this ideal state by applying time and frequency noise, which are frequency- and time-displacement operations multiplied by the Gaussian distribution, which is detailed in [1]. The wave function of the two logical states can be written as follows:  
\[\left|\mu_{\omega}\right\rangle_{a}=N_{\mu}\sum_{n\in\mathbb{Z}}\int d\omega G ^{\kappa}(\omega)G^{\sigma}(\omega-(2n+\mu)\overline{\omega})\left|\omega \right\rangle_{a} \tag{10}\]  
where \(G\) are Gaussian functions representing the envelope of the comb of width \(\kappa\) and the peaks of the comb of width \(\sigma\). The frequency probability distribution of the grid state is represented in Figure 1. Alternatively, for large comb \(\overline{\omega}/\sigma\gg 1\)[39], we can write  
\[\left|\mu_{\omega}\right\rangle_{a}=N_{\mu}\sum_{n\in\mathbb{Z}}c_{2n+\mu} \int d\omega G^{\sigma}(\omega-(2n+\mu)\overline{\omega})\left|\omega\right \rangle_{a} \tag{11}\]  
with the envelope coefficients \(c_{n}=\exp(-(n\overline{\omega}/\kappa)^{2}/2)\). \(N_{\mu}\) is the normalization constant found thanks to the relation \(1=\left|{}_{a}\langle\mu_{\omega}|\mu_{\omega}\rangle_{a}|^{2}=N_{\mu}^{2} \sum_{n\in\mathbb{Z}}\left|c_{2n+\mu}\right|^{2}\sqrt{\pi\sigma^{2}}\). In general, the physical GKP state can be in a superposition of the two logical states:  
\[\left|\psi\right\rangle=N_{a\beta}(\alpha|0_{\omega}\rangle_{a}+\beta|1_{ \omega}\rangle_{a}), \tag{12}\]  
Figure 1: Probability density distribution of the two time–frequency qubits. (**a**) Frequency cat state \(\sigma\) is the half-width of the Gaussian peak, and \(\Delta\) is the spectral separation between the two logical states. (**b**) Time–frequency GKP state. The width of each peak is \(\sigma\), the envelope \(\kappa\), and the periodicity of the comb state is \(\overline{\omega}\). In the temporal domain, the periodicity of the state is \(2\pi/\overline{\omega}\), the width of the envelope and the peak are \(\sigma\) and \(\kappa\), respectively. Frequency units are made dimensionless with respect to the spatial separation of the state (**left**) and the periodicity of the state (**right**). where \(N_{a\beta}=(|\alpha|^{2}+|\beta|^{2}+2\mathrm{Re}(\alpha^{*}\beta_{a}\langle 0_{ \omega}|1_{\omega}\rangle_{a})))^{-1/2}\). The two logical states are not orthogonal when a Gaussian wave packet enters the frequency bin of its neighbor. The overlap \({}_{a}\langle 0_{\omega}|1_{\omega}\rangle_{a}\) is different from zero and is equal to  
\[{}_{a}\langle 0_{\omega}|1_{\omega}\rangle_{a}=e^{-\overline{\omega}^{2}/4 \omega^{2}}\frac{\sum_{n}c_{2n}c_{2n+1}^{*}}{(\sqrt{\sum_{n}|c_{2n}|^{2}\sum_{ n}|c_{2n+1}|^{2}})}. \tag{13}\]  
The full state is thus described by five important parameters.