# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 3 Spatial Separation of the Two Logical Time-Frequency Qubit States
### Spatial Separation of the Two Logical Time-Frequency GKP States  
In this part, we introduce how to separate spatially the odd and the even frequencies with a Mach-Zehnder interferometer. Such a scheme was already proposed for manipulating large quadrature position-momentum continuous variables cluster states [56], and was described in [2; 25]. We start with a time-frequency GKP state with a finite envelop but with infinitely narrow frequency width \(|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{a}+|1\rangle_{a})\), and is introduced into a balanced beam splitter. The spatial output port of the beam splitter is denoted as \(a\) and \(b\). A time-shiftoperation is performed in spatial port \(b\), and then the two spatial ports are recombined into a balanced beam splitter (see Figure 3). The final wave function can be written as  
\[|\psi\rangle=\frac{1}{2}\sum_{n\in\mathbb{Z}}\left[c_{n}(e^{i\pi\omega t}-1)|n \overline{\omega}\rangle_{a}|\Omega\rangle_{b}-(e^{i\pi\omega t}+1)|\Omega \rangle_{a}|n\overline{\omega}\rangle_{b}\right]\!. \tag{20}\]  
If we set \(t=\pi/\overline{\omega}\), after the second beam splitter, the wave function is (see [2])  
\[|\psi\rangle=\frac{1}{\sqrt{2}}(-|1\rangle_{a}|\Omega\rangle_{b}+|\Omega \rangle_{a}|0\rangle_{b}). \tag{21}\]  
The odd and even frequency components are spatially separated, allowing for individual manipulation, such as correcting the phase accumulation of the one logical state. It is possible to achieve the same result in the temporal domain by shifting the frequency instead of the time as shown in Figure 3.  
If each frequency peak is not infinitely narrow, then the output wave function after the Mach-Zehnder interferometer can be written as  
\[|\psi\rangle=\frac{1}{2}\sum_{n\in\mathbb{Z}}c_{n}[(\int d\omega(e^{i\pi\omega/ \overline{\omega}}-1)G^{\sigma}(\omega-n\overline{\omega})|\omega\rangle_{a} |\Omega\rangle_{b}-(\int d\omega(e^{i\pi\omega/\overline{\omega}}+1)G^{\sigma} (\omega-n\overline{\omega})|\Omega\rangle_{a}|\omega\rangle_{b}. \tag{22}\]  
The corresponding probability frequency distributions at spatial port \(a\) and \(b\) are  
\[P_{a}(\omega) =\frac{1}{4}|\sum_{n\in\mathbb{Z}}c_{n}(e^{i\pi\omega/\overline{ \omega}}-1)G^{\sigma}(\omega-n\overline{\omega})|^{2}, \tag{23}\] \[P_{b}(\omega) =\frac{1}{4}|\sum_{n\in\mathbb{Z}}c_{n}(e^{i\pi\omega/\overline{ \omega}}+1)G^{\sigma}(\omega-n\overline{\omega})|^{2}. \tag{24}\]  
We represent in Figures 4a and 5a the spectral probability density distribution of two \(|+_{\omega}\rangle\) states for \(\sigma=0.1\overline{\omega}\) and \(\sigma=0.2\overline{\omega}\). The output probability density distribution after the Mach-Zehnder interferometer if we start with the zero or one logical state is represented in Figure 4b,c. We observe that when \(\sigma=0.1\overline{\omega}\), it is valid to consider the zero and one logical states independently since they do not interfere due to their central frequencies being too far apart to overlap. However, the spatial separation is imperfect; the one logical state does not emerge from the correct spatial port. When \(\sigma=0.2\overline{\omega}\) (see Figure 5b,c), there is an interference term between the zero and one logical state because they now overlap significantly. The resulting state is outside the GKP subspace; it can be seen with the distortion because the probability at the center of the odd and even frequency bins is zero. The addition of a periodic frequency filter can enhance the projection into the  
Figure 3: Spatial separation of the odd and even peaks of the time–frequency GKP state with a Mach–Zehnder interferometer in the frequency (**left**) and in the temporal (**right**) domain. BS stands for balanced beam splitter. The spatial separation is imperfect when the states are no longer perfectly monochromatic. GKP subspace and the choice of a frequency width would be crucial for rejecting the logical states emerging from the incorrect spatial port.