# Teleportation-Based Error Correction Protocol of Time-Frequency Qubit States
## 1 Introduction  
Quantum information harnesses the foundational principles of quantum mechanics, such as superposition and entanglement, to manipulate and transmit data in fundamentally novel ways. For instance, quantum entanglement establishes correlations between distant particles, a phenomenon exploited in quantum communication for secure key distribution and teleportation protocols. The pertinence of single photons in encoding quantum information emanates from their unique attributes, including minimal susceptibility to environmental decoherence, and the fact that single photons travel at the speed of light. Quantum information can be encoded in various degrees of freedom of single photons, which can be described by either discrete or continuous variables (CV). Frequency (or energy) and time of arrival are natural pairs of conjugate quantum continuous variables in the single photon subspace, along with the transverse position and momentum degrees of freedom [1; 2; 3; 4]. Discretizing the frequency or time of arrival into temporal or frequency bins, or performing the mode decomposition of the continuous variable distribution of the single photon, can be experimentally motivated due to the finite resolution of detection devices or specific requirements of a quantum protocol, such as in quantum metrology for super-resolution [5]. We should stress that in any dimensional single-photon encoding, photon losses do not correspond to a logical error. The second way of encoding information is through particle number sensitive encoding, which can be used to define physical systems with either discrete or continuous variables. In this encoding, CV corresponds to the quadratures of the electromagnetic field, i.e., the amplitude and phase of the quantum field, in a particular mode. With the particle-sensitive encoding, photon loss corresponds to a logical error. Mathematically, the quadrature of an electromagnetic field in a given mode can be treated as the continuous degree of freedom of a single photon [4], as long as an auxiliary discrete mode is occupied by only one single photon. Error correction code for continuous variables encoding is defined by discretizing them. Three bosonic qubit codes have been studied, such as the cat code [6; 7; 8], Gottesman, Kitaev, and Preskill (GKP) code [9; 10; 11; 12; 13; 14; 15] and the binomial code [16; 17]. Cat and GKP codes are candidates for achieving universal quantum computation; see [7; 18; 19; 20]. GKP code could be employed for building quantum repeaters [21], and for sensing applications [22; 23]. The mathematical analogy between time-frequency and quadrature CV allows defining time-frequency qubit states, called the time-frequency cat state [2; 24] and time-frequency GKP state [1; 2; 25]. Both of these types of code are ways to discretize time-frequency continuous variables at the single photon level to define a qubit. CV or time-frequency CV codes possess an equivalent mathematical structure; they are common eigenvectors of non-commuting displacements operators [1] and they are thus designed to be robust against a small shift in one continuous variable (cat state) and the two canonically conjugated ones (GKP state). In this paper, we start by reminding the reader of the mathematical structure of the time-frequency cat and GKP codes and discuss the temporal and frequency errors that they were designed to be robust against. From the standard method for quantum error correction for quadrature GKP states [26], we propose the analogous protocol to perform correcting the time-frequency GKP state. In such a quantum error correction scheme, a temporally and spectrally broadened (corresponding to errors in this encoding) time-frequency GKP state is first entangled with a less noisy ancilla GKP, and finally temporal- or frequency-resolved detection is performed on the ancilla state. The resulting time-frequency GKP state is either corrected in the time (resp. frequency) if the frequency (resp. temporal) resolved measurement has been performed.