# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 2 Time-Frequency Qubit States
### Time-Frequency Entangled GKP State and Error Correction Protocols  
The error correction of continuous variables states can be performed with a Steane error correction protocol for the quadrature degree of freedom [51] and for the time-frequency one [1]. For the two encodings, the protocol consists of entangling the state of interest with one less noisy ancilla (a \(|+\rangle\) logical state in one variable, time or frequency) with a beam splitter (resp. with frequency beam splitter that will be explicated below), and performing one homodyne (resp. single-photon frequency measurement) detection at the spatial output of the ancilla for correcting the error along one variable. The protocol is repeated to correct errors in the orthogonally (or canonically) conjugated variable. To achieve this, one must entangle the state of interest with a less noisy state using a \(|+\rangle\) state in the canonically conjugated variable compared to the first step. Then, perform the entangling operation, project a measurement in the canonically conjugated variable (compared to step one), and finally conduct a conditional displacement operation. An important tool for quantifying the threshold of noise from which it is still possible to correct the GKP states, a Steane error correction protocol was used in [50]. Figures of merit for quantifying the probability of measuring the one logical state while this is zero were obtained in [26; 50]. We will refer to these figures of merit when we note one logical state is noisier (or less noisy) than the other. From now on, we develop two entanglement structures of time-frequency GKP state that can be generated in the laboratory, and the method for performing the error correction in each case. The first entanglement structure of the time-frequency GKP state that we can study is the one obtained by using a spontaneous parametric downconversion process (SPDC) from a non-linear crystal placed into an optical cavity [1; 52]. The corresponding wave function can be cast as  
\[|\psi\rangle=\iint d\omega_{s}d\omega_{i}f_{+}(\omega_{+})f_{-}(\omega_{-})f( \omega_{s})f(\omega_{i})|\omega_{s},\omega_{i}\rangle. \tag{14}\]  
The functions \(f_{\pm}\) model, respectively, the energy conservation and the phase matching of the SPDC process, and \(f\) models the cavity function. The joint spectrum intensity is represented in Figure 2c. Instead of the traditional view, where the wave function of two photons is written by applying a quadratic Hamiltonian on the vacuum state, it can also be expressed with a quantum circuit representation as shown in Figure 2a. This starts with two ideal separable GKP states (fictitious), which undergo initial frequency broadening, which can be interpreted as a frequency noise [1]. Then, the state is entangled through the gate performed by the non-linear crystal:  
\[\hat{U}|\omega_{s},\omega_{i}\rangle=\left|\frac{\omega_{s}+\omega_{i}}{\sqrt{2} },\frac{\omega_{s}-\omega_{i}}{\sqrt{2}}\right\rangle \tag{15}\]  
that we called a frequency beam splitter by analogy with the beam splitter, which acts mathematically similarly in the quadrature position-momentum degree of freedom [4; 53]. The state then undergoes a temporal broadening, which can be interpreted as a temporal noise and a final frequency beam-splitter operation is performed. The mathematical reason behind these four successive operations is that the envelope of the grid state is a function of the collective variables \(\omega_{\pm}\), while the cavity of the local variable is \(\omega_{s,i}\). The form of the frequency entanglement between the two single photon grid states generated by the non-linear process (see Equation (15)) allows for the reduction in the temporal broadening of one of the single photons by performing a temporally resolved measurement of the other. The form of the pure wave function after the conditioned operation is written in [1]. After correcting the single-photon state in the temporal domain, the next step is to correct the state in the frequency domain.