# Teleportation-based error correction protocol of time-frequency qubits
states

N<EMAIL_ADDRESS>Telecom Paris, Institut
Polytechnique de Paris, 19 Place Marguerite Perey, 91120 Palaiseau, France

###### Abstract

We present a linear optical protocol for teleporting and correcting both
temporal and frequency errors in two time-frequency qubit states. The first
state is the frequency (or time-of-arrival) cat qubit, which is a single
photon in a superposition of two frequencies (or time-of-arrival), while the
second is the time-frequency Gottesman-Kitaev-Preskill (GKP) state, which is a
single photon with a frequency comb structure. The proposed optical scheme
could be valuable for reducing error rate in quantum communication protocols
involving one of these qubits.

##  I Introduction

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 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 a 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] and [19, 20]. GKP
codes could be employed for building quantum repeaters [21], and for sensing
application [22, 23]. The mathematical analogy between time-frequency and
quadrature CV allows defining time-frequency qubits states, called time-
frequency cat state [2, 24] and time-frequency GKP state [1, 2, 25]. Both of
these codes 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 small shift in one continuous variable (cat state) and the two
canonically conjugated ones (GKP state).

In this paper, we start by reminding the mathematical structure of the time-
frequency cat and GKP codes, and discuss the temporal and frequency errors
from which they are designed to be robust against. We then analyse two
different entanglement structures of time-frequency GKP state which can be
generated experimentally, and can be interpreted as the entanglement of a
noisy state of interest with a less noisy ancilla. However, these entanglement
structures which lead to a natural error correction strategy for single photon
encoding is difficult to realize experimentally with the current technology,
since it requires to perform a frequency entanglement operation between two
single photons. Therefore, the standard method for error correction for
quadrature qubits cannot be applied straightforwardly. Inspired by the
teleportation-based error correction protocol for quadrature variables [26,
27], we develop a teleportation-based error correction protocol of frequency
qubits states, using only linear optical elements, and allows correcting both
time and frequency variables at once. The error correction is naturally
performed because the ancilla EPR state which assists to the teleportation is
less noisy in the temporal and frequency domains compared to the state of
interest. Since the protocol requires the use of Bell’s measurement, we also
describe how to experimentally implement such a measurement for the two type
of frequency qubits. The proposed protocol is intrinsically probabilistic but
consist on the teleportation of non-orthogonal states [28], instead of
orthogonal ones [29, 30]. The non-orthogonality of the state reduces the
efficiency of the teleportation protocol, and we mention that photon number
resolving detectors can help increase the probability of success by reducing
the number of rejected measurement events. The presented teleportation
protocol is a new solution for correcting temporal broadening caused by
dispersion effects affecting time-bin qubit states, thereby reducing the error
rate of quantum communication protocols [31, 32, 33].

The paper is organized as follows. In Sec. II, we provide a reminder of the
definition of the time-frequency cat and GKP states and the reason of the
experimental difficulty behind the natural error correction scheme of the
time-frequency GKP states which requires frequency entanglement gates. In Sec.
III, we explain the optical equivalent of the polarizing beam-splitter, a
Mach-Zehnder interferometer, which allows separating spatially the two logical
states of the time-frequency cat and GKP qubits. Such an interferometer is
crucial for implementing Bell’s measurement in the time-frequency degree of
freedom. In Sec. IV, we present a teleportation-based error correction
protocol for the time-frequency GKP state, which makes use of the Bell’s
measurement. The protocol is probabilistic and can be achieved with current
experimental devices. Finally, in Sec. V, we summarize our results and present
new perspectives.

##  II Time-frequency qubits states

###  II.1 Time-frequency cat state

We will denote $\ket{\Omega}$ the vacuum state. A single photon state at
frequency $\omega$ in the spatial port $a$ is denoted as
$\ket{\omega}_{a}=\hat{a}^{\dagger}(\omega)\ket{\Omega}$. The frequency cat
state as introduced in [24], is defined as the superposition of a single
photon into two different frequency Gaussian distribution:

$\ket{\psi}=N_{\alpha\beta}(\alpha\ket{\omega_{1}}_{a}+\beta\ket{\omega_{2}}_{a})=N_{\alpha\beta}(\alpha\ket{0}_{a}+\beta\ket{1}_{a}),$
(1)

where $\ket{\omega_{1}}_{a}=\frac{1}{\sqrt{2\pi\sigma^{2}}}\int
d\omega\text{exp}(-(\omega-\omega_{1})^{2}/2\sigma^{2})\ket{\omega}_{a}$ and
the normalization of the state is given by
$1=N_{\alpha\beta}^{2}(\absolutevalue{\alpha}^{2}+\absolutevalue{\beta}^{2}+2\text{Re}(\alpha\beta^{*})e^{-(\omega_{1}-\omega_{2})^{2}/2\sigma^{2}})$.
This is a non-orthogonal qubit state as their overlap is
${}_{a}\bra{0}\ket{1}_{a}=\text{exp}(-(\omega_{1}-\omega_{2})^{2}/2\sigma^{2})$.
Experimental proposal for manipulating such a state was proposed in [34, 35],
with pulse shapers and electro-optic modulators were used in cascade. The
results showed an operation fidelity close to 100% but the successive optical
elements decrease drastically the probability of single photon detection. The
wavefunction of the time cat state is defined as:

$\ket{\psi}=N_{\alpha\beta}(\alpha\ket{t_{1}}_{a}+\beta\ket{t_{2}}_{a}).$ (2)

Note that for avoiding to have a normalization constant depending in the
coefficients $\alpha,\beta$, and writing the wavefunction in an orthogonal
basis, we can employ the Gram-Schmidt decomposition procedure. The normalized
orthogonal basis $\ket{a}$ and $\ket{b}$ can be written as:

$\ket{a}=\ket{0},\ \ket{b}=N[\ket{1}-\bra{0}\ket{1}\ket{0}],$ (3)

where $N=1/\sqrt{1-r^{2}}$ where $r=\absolutevalue{\bra{0}\ket{1}}$. The GKP
input state Eq. (12) can be written in the orthogonal basis as:

$\ket{\psi}=(\alpha+\beta\bra{0}\ket{1})\ket{a}+\frac{\beta}{N}\ket{b},$ (4)

where we have now
$\absolutevalue{(\alpha+\beta\bra{0}\ket{1})}^{2}+\absolutevalue{\frac{\beta}{N}}^{2}=1$.

The frequency entangled cat state, an EPR state could be written as:

$\displaystyle\ket{\phi^{\pm}}=N_{\text{EPR}}(\ket{\omega_{1}\omega_{1}}_{ab}\pm\ket{\omega_{2}\omega_{2}}_{ab})$
(5)
$\displaystyle\ket{\psi^{\pm}}=N_{\text{EPR}}(\ket{\omega_{1}\omega_{2}}_{ab}\pm\ket{\omega_{2}\omega_{1}}_{ab})$
(6)

where $N_{\text{EPR}}^{2}(2+2e^{-(\omega_{1}-\omega_{2})^{2}/2\sigma^{2}})=1$.
When $\omega_{1}-\omega_{2}\gg\sigma$ we recover the normalization of an EPR
state composed of orthogonal qubits $N_{\text{EPR}}=1/\sqrt{2}$. The frequency
cat state can be produced by integrated optical waveguide [36], and bulk
system [37]. The wavefunction of a temporal entangled EPR state Eq. (5) has
the same mathematical structure, and such a quantum state can be produced by
quantum dots for instance [38]. This type of quantum state has potential
applications in quantum communications [39].

###  II.2 Time-frequency GKP state

We define a frequency lattice of period $\overline{\omega}$. Centered on each
of this interval, we define the ideal time-frequency GKP state as the
following frequency comb at the single photon level:

$\ket{\overline{\mu}_{\omega}}_{a}=\sum_{n\in\mathds{Z}}\ket{(2n+\mu)\overline{\omega}}_{a}$
(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:

$\displaystyle\ket{\overline{+}_{\omega}}_{a}=\frac{1}{\sqrt{2}}(\ket{\overline{0}_{\omega}}_{a}+\ket{\overline{1}_{\omega}}_{a})=\ket{\overline{0}_{t}}_{a}$
(8)
$\displaystyle\ket{\overline{-}_{\omega}}_{a}=\frac{1}{\sqrt{2}}(\ket{\overline{0}_{\omega}}_{a}-\ket{\overline{1}_{\omega}}_{a})=\ket{\overline{1}_{t}}_{a}.$
(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 state and will require an
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 Gaussian distribution
which is detailed in [1]. The wavefunction of the two logical states can be
written as follows:

$\ket{\mu_{\omega}}_{a}=N_{\mu}\sum_{n\in\mathds{Z}}\int d\omega
G^{\kappa}(\omega)G^{\sigma}(\omega-(2n+\mu)\overline{\omega})\ket{\omega}_{a}$
(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 Fig. 1.
Alternatively for large comb $\overline{\omega}/\sigma\gg 1$ [40], we can
write:

$\ket{\mu_{\omega}}_{a}=N_{\mu}\sum_{n\in\mathds{Z}}c_{2n+\mu}\int d\omega
G^{\sigma}(\omega-(2n+\mu)\overline{\omega})\ket{\omega}_{a}$ (11)

with the envelope coefficients
$c_{n}=\text{exp}(-(n\overline{\omega}/\kappa)^{2}/2)$. $N_{\mu}$ is the
normalization constant found thanks to the relation
$1=\absolutevalue{{}_{a}\bra{\mu_{\omega}}\ket{\mu_{\omega}}_{a}}^{2}=N_{\mu}^{2}\sum_{n\in\mathds{Z}}\absolutevalue{c_{2n+\mu}}^{2}\sqrt{\pi\sigma^{2}}$.

Figure 1: Probability 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 is $\sigma$ 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).

In general, the physical GKP state can be in a superposition of the two
logical states:

$\ket{\psi}=N_{\alpha\beta}(\alpha\ket{0_{\omega}}_{a}+\beta\ket{1_{\omega}}_{a}),$
(12)

where
$N_{\alpha\beta}=(\absolutevalue{\alpha}^{2}+\absolutevalue{\beta}^{2}+2\text{Re}(\alpha^{*}\beta\bra{0}\ket{1})))^{-1/2}$.
The two logical states are not orthogonal when a Gaussian wavepacket enters in
the frequency bin of its neighbour. The overlap
${}_{a}\bra{0_{\omega}}\ket{1_{\omega}}_{a}$ is different than zero and is
equal to:

${}_{a}\bra{0_{\omega}}\ket{1_{\omega}}_{a}=e^{-\overline{\omega}^{2}/4\sigma^{2}}\frac{\sum_{n}c_{2n}c^{*}_{2n+1}}{(\sqrt{\sum_{n}\absolutevalue{c_{2n}}^{2}\sum_{n}\absolutevalue{c_{2n+1}}^{2})}}.$
(13)

The full state is thus described by five importants parameters. The complex
parameters $\alpha$ and $\beta$ where the quantum information is encoded, the
frequency width $\sigma,\kappa$ and the periodicity of the state
$\overline{\omega}$. If the state is not too much noisy, meaning that
${}_{a}\bra{0_{\omega}}\ket{1_{\omega}}_{a}\sim 0$, then the normalisation
condition of Eq. (12) is given by
$\absolutevalue{\alpha}^{2}+\absolutevalue{\beta}^{2}=1$. Finally, in [1, 4],
we define the time-of-arrival and frequency operators which do not commute and
verify an Heisenberg algebra. This is mathematically equivalent to the non-
commutativity of time-frequency displacement operators. This property leads to
consider temporal and frequency bandwidth as quantum noise at the single
photon level.

The GKP states which are defined as the sum of squeezed states in a given mode
[9, 41], are designed to be robust against small shift in position and
momentum, which can be caused by a Gaussian quantum channel, but they are also
robust against photon losses [41]. On the other hand, time-frequency GKP
states are designed to be robust against small shift in time and frequency.
The major difference between GKP states and time-frequency GKP states is that
photon losses do not result in errors for the latter. In general, temporal
errors are the dominant source of errors, while frequency is considered a
robust degree of freedom, as it is barely affected by linear physical
processes. Additionally, GKP states can be used for fault-tolerant universal
quantum computation [20]. Due to their mathematical similarity with time-
frequency GKP states, it is expected that they would lead to the same
mathematical result. However, the generation of non-Gaussian states using the
degree of freedom of a single photon is relatively simple to implement
experimentally. As a result, the experimental implementation of a time-
frequency GKP state is considered straightforward. In contrast, creating
entanglement gates between two single photons is a more challenging task. For
particle-number sensitive encoding, non-Gaussian operations involving the
quadrature degree of freedom can be difficult to implement, while two-mode
Gaussian operations, such as with a beam splitter, are easier to perform.

###  II.3 Sources of time-frequency noise

In this section, we discuss the physical processes that lead to temporal-
spectral broadening or distorsion. Coherent and incoherent errors will lead to
either pure and mixed state respectively.

Temporal errors for both codes arise from temporal spreading of each
wavepacket composing the state due to linear dispersion effect, described by a
coherent model (see for instance [42] for an example in the single photon
regime). After such second order dispersion effect, the temporal width of the
Gaussian wavepacket 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 an unitary operation, and it can in principle be undone by a
reverse transformation. However, it requires the knowledge of the full
characterization of the propagation channel. For the time-frequency GKP state,
the dispersive effect not only lead to a temporal spreading of each
wavepacket, but also lead to the formation of replica temporal images, called
the temporal Talbot effect (see for instance [43]). One has to consider
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 [44, 45, 46]. 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 [47]. Thus, the
error correction protocol that will be presented in Sec. II.4 becomes
particularly relevant because we can not 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 [48]
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, $L$ 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 frequency noise arises from
frequency broadening caused by the generation of photon pairs itself, and we
describe one method to correct such a 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 [49, 50], 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 [52]. 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 Sec. II.4, the other method by using only linear optics and also
less noisy ancilla Sec. IV.

###  II.4 Time-frequency entangled GKP state and error correction protocols

Error correction of continuous variables states can be done with a Steane
error correction protocol, for the quadrature degree of freedom [53] 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 $\ket{+}$
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 $\ket{+}$ 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. Important
tools for quantifying the threshold of noise from which it is still possible
to correct the GKP states, using such a Steane error correction protocol was
done in [52]. Figures of merit for quantifying the probability of measuring
the one logical state while this is the zero that should be obtained was done
in [54, 52]. We will refer to these figures of merit, when we will note one
logical state is more (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 how to perform the error correction in each case.

The first entanglement structure of time-frequency GKP state that we can
studied is the one obtained by using a spontaneous parametric down conversion
process (SPDC) from a non-linear crystal placed into an optical cavity [1,
55]. The corresponding wavefunction can be cast as:

$\ket{\psi}=\iint
d\omega_{s}d\omega_{i}f_{+}(\omega_{+})f_{-}(\omega_{-})f(\omega_{s})f(\omega_{i})\ket{\omega_{s},\omega_{i}}.$
(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 Fig. 2(c). Instead of the
traditional view where the wavefunction 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 Fig. 2(a). 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:

$\hat{U}\ket{\omega_{s},\omega_{i}}=\ket{\frac{\omega_{s}+\omega_{i}}{\sqrt{2}},\frac{\omega_{s}-\omega_{i}}{\sqrt{2}}}$
(15)

that we called a frequency beam-splitter by analogy with the beam-splitter
which acts mathematically similarly into the quadrature position-momentum
degree of freedom [56, 4]. 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 envelop of the grid state are function of
the collective variables $\omega_{\pm}$, while the cavity of the local
variable $\omega_{s,i}$. The form of the frequency entanglement between the
two single photon grid state generated by the non-linear process (see Eq.
(15)) allows for the reduction of the temporal broadening of one of the single
photons by performing a temporally-resolved measurement of the other. The form
of the pure wavefunction after the conditioned operation has been written in
[1].

Figure 2: Different joint spectral intensity
$\absolutevalue{\bra{\omega_{s},\omega_{i}}\ket{\psi}}^{2}$ of time-frequency
entangled GKP states (c),(d) that can be generated experimentally and their
quantum circuit representation (a), (b). The situation (a) corresponds to a
photon pair produced by a SPDC process, while (b) corresponds to two single
photons with a frequency comb structure produced by two independent processes
which are then frequency entangled. The position of frequency beam-splitter
operations modifies the periodicity of the grid state from a factor
$\sqrt{2}$. $\hat{D}(\omega)$ and ${\cal{\hat{D}}}(t)$ are frequency and
temporal displacement operations defined in [1].

After correcting the single photon state in the temporal domain, the next step
is to correct the state in the frequency domain. For that, we must first
entangle the single photon to correct with a less noisy ancilla single photon
state using the entanglement gate Eq. (15). Since now we have two separable
single photon states, we can not directly entangle them by using a non-linear
crystal, which is non-efficient. Nevertheless, such a frequency entangled gate
could be implemented with a quantum emitter embedded into a waveguide, which
assists to the interaction between the two single photons [57, 58].

The second entanglement structure of time-frequency GKP state that can be
considered, is to start with two initially separable GKP states, with one
being less noisy than the other. These states are then entangled using Eq.
(15). The resulting wavefunction is:

$\ket{\psi^{\prime}}=\iint
d\omega_{s}d\omega_{i}f_{+}(\omega_{+})f_{-}(\omega_{-})f(\omega_{+})f(\omega_{-})\ket{\omega_{s},\omega_{i}}.$
(16)

In this new spectral function (see Fig. 2(d)), the cavity function is now
dependent on the collective variables, thus the periodicity of the grid state
is not the same as in Eq. (14). The corresponding quantum circuit
representation is pictured in Fig. 2(b). The error correction protocol is
followed by a temporally-resolved measurement followed by a conditional
displacement operation to correct only the temporal noise in the state of
interest. A second entanglement operation is performed using a less noisy
ancilla state, followed by a resolved-frequency measurement followed by a
conditional displacement operation to correct the frequency noise. The
difference in the joint spectral amplitude of the photon pairs results - Eq.
(14) and Eq. (16) - in a different wave function when one of the photons
undergoes a temporally-resolved (and frequency) measurement, and a conditional
displacement operation.

##  III Spatial separation of the two logical time-frequency qubit states

In this section, we develop the equivalent of the polarizing beam-slitter
operation for the time-frequency cat and GKP state, which is the crucial
optical component for the teleportation-based error correction described in
the next section. Such an optical element will be called the frequency qubit
beam-splitter (FQBS) in what follows.

###  III.1 Spatial separation of the two logical time-frequency cat states

In this section, we explicit how to separate spatially the two Gaussian
wavepackets with linear optics, using a Mach-Zehnder interferometer. The
frequency cat state as described by Eq. (1) is introduced into a balanced
beam-splitter and the associated wavefunction is:

$\ket{\psi}=\frac{1}{2}(\ket{\omega_{1}}_{a}+\ket{\omega_{2}}_{a}+\ket{\omega_{1}}_{b}+\ket{\omega_{2}}_{b}).$
(17)

We assumed for simplicity that the two frequency state are well separated
$\omega_{1}-\omega_{2}\gg\sigma$. Then, a pulse shaper is placed at the
spatial port $b$, described by the following unitary operation:

$\hat{U}\ket{\omega_{1}}_{b}=\ket{\omega_{1}}_{b},\
\hat{U}\ket{\omega_{2}}_{b}=e^{i\phi}\ket{\omega_{2}}_{b}.$ (18)

Such an operation can be implemented for instance by mapping the spectral to
the spatial degree of freedom with a grating, a spatial light modulator at the
focal length of two lenses, and then performing back the mapping from the
spatial to spectral degree of freedom [2, 59]. We assume that the frequency
peaks are spaced enough so that the pulse shaper acts on each logical state
independently. The two spatial paths are then recombined to another balanced
beam-splitter. The output state has the final form:

$\ket{\psi}=\frac{1}{\sqrt{2}}\ket{\omega_{1}}_{a}\ket{\Omega}_{b}+\frac{1}{2\sqrt{2}}((1+e^{i\phi})\ket{\omega_{2}}_{a}\ket{\Omega}_{b}+(1-e^{i\phi})\ket{\Omega}_{a}\ket{\omega_{2}}_{b}).$
(19)

If $\phi=\pi$, the two logical states are spatially separated
$\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{\omega_{1}}_{a}\ket{\Omega}_{b}+\ket{\Omega}_{a}\ket{\omega_{2}}_{b})$.
This optical interferometer is the equivalent of the polarizing beam-splitter
that separates the vertical and horizontal polarization of optical fields into
two distinct spatial paths.

###  III.2 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 [60], and was described in [2, 25].

We start with a time-frequency GKP state with a finite envelop but with
infinitely narrow frequency width
$\ket{\tilde{+}}=\frac{1}{\sqrt{2}}(\ket{\tilde{0}}_{a}+\ket{\tilde{1}}_{a}$),
and is introduced into a balanced beam-splitter. The spatial output port of
the beam-splitter is noted $a$ and $b$. A time-shift operation is performed in
spatial port $b$, and then the two spatial ports are recombined into a
balanced beam-splitter (see Fig. 3). The final wave function can be written
as:

$\ket{\psi}=\frac{1}{2}\sum_{n\in\mathds{Z}}[c_{n}(e^{in\overline{\omega}t}-1)\ket{n\overline{\omega}}_{a}\ket{\Omega}_{b}-(e^{in\overline{\omega}t}+1)\ket{\Omega}_{a}\ket{n\overline{\omega}}_{b}].$
(20)

If we set $t=\pi/\overline{\omega}$, after the second beam-splitter the
wavefunction is (see [2]):

$\ket{\psi}=\frac{1}{\sqrt{2}}(-\ket{1}_{a}\ket{\Omega}_{b}+\ket{\Omega}_{a}\ket{0}_{b}).$
(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 Fig. 3.

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.

If each frequency peak is not infinitely narrow, then the output wavefunction
after the Mach-Zehnder interferometer can be written as:

$\ket{\psi}=\frac{1}{2}\sum_{n\in\mathds{Z}}c_{n}[(\int
d\omega(e^{i\pi\omega/\overline{\omega}}-1)G^{\sigma}(\omega-n\overline{\omega})\ket{\omega}_{a}\ket{\Omega}_{b}\\\
-(\int
d\omega(e^{i\pi\omega/\overline{\omega}}+1)G^{\sigma}(\omega-n\overline{\omega})\ket{\Omega}_{a}\ket{\omega}_{b}.$
(22)

The corresponding probability frequency distribution at spatial port $a$ and
$b$ are

$\displaystyle
P_{a}(\omega)=\frac{1}{4}\absolutevalue{\sum_{n\in\mathds{Z}}c_{n}(e^{i\pi\omega/\overline{\omega}}-1)G^{\sigma}(\omega-n\overline{\omega})}^{2},$
(23) $\displaystyle
P_{b}(\omega)=\frac{1}{4}\absolutevalue{\sum_{n\in\mathds{Z}}c_{n}(e^{i\pi\omega/\overline{\omega}}+1)G^{\sigma}(\omega-n\overline{\omega})}^{2}.$
(24)

We represent in Fig. 4(a) (b) the spectral distribution of two $\ket{+}$
states for $\sigma=0.1\overline{\omega}$ and $\sigma=0.2\overline{\omega}$
along with the output probability distribution after the Mach-Zehnder
interferometer in Fig. 4(c),(d). 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 for 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 Fig. 4), 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 distorsion and 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 GKP subspace and the choice of a
frequency width would be crucial for rejecting the logical states emerging
from the incorrect spatial port.

Figure 4: (a) Input state of the Mach-Zehnder interferometer for
$0.1\overline{\omega}=\sigma$ and $\kappa=0.1$. (b), (c) Probability
distribution in the spatial port $b$ (resp. (a)) if we start with either the
zero (blue) or one (one) logical state or with the equal superposition of the
zero and one logical state. Their overlap is not important enough to observe
an interference effect. (d) Input state of the Mach-Zehnder interferometer for
$0.2\overline{\omega}=\sigma$ and $\kappa=0.1$. (e), (f) Probability
distribution in the spatial port $b$ (resp. (a)) if we start with either the
zero (blue) or one (one) logical state or with the equal superposition of the
zero and one logical state. Their overlap is important enough to observe an
interference effect to see a distorsion of the state which leaks to the
incorrect spatial port.

In Appendix A, we also investigate the separation of the odd and even
components of the comb when the two-photon state is an input of the frequency
qubit beam-splitter. This is relevant to the teleportation-based error
correction protocol because the state to be teleported is combined with a
single photon from an EPR pair during the Bell measurement. The spatial
separation is, in that case, also completely effective when each Gaussian
distribution approaches a Dirac distribution, and not otherwise. Explicitly,
for the two photons states we obtain:
$\ket{0\tilde{1}}_{aa}\rightarrow\ket{0_{G}}_{a}\ket{\tilde{1}_{G}}_{a^{\prime}}+\ket{0_{E}}_{a^{\prime}}\ket{\tilde{1}_{E}}_{a}$,
whose the corresponding expressions in given in Appendix A. The case to
combine two different single photon states also plays a role in quantum
communication scenarios, where the second qubit is from an attacker which
tries to collect the information about the qubit carrying the information of
interest.

##  IV Teleportation-based error correction of time-frequency qubits states

In this section, we propose a protocol for correcting and teleporting
frequency qubit states without relying on frequency entangling operations.
This protocol is similar to the one used for teleporting polarization qubit
states as described in [61], and is inspired from the GKP analog [26, 27].
Since the EPR state has a lower level of noise in both temporal and frequency
variables, the protocol includes an additional component for correcting errors
in the state being teleported.

The single photon state $\ket{\psi}$ to be teleported and corrected is
described by the wavefunction
$\ket{\psi}=N_{\alpha\beta}(\alpha\ket{0}_{a}+\beta\ket{1}_{a})$, where the
two logical state are either the time-frequency cat or GKP qubits (see Eq.
(12)). The wavefunction of the entangled EPR time-frequency state in spatial
port $b$ and $c$ which assists for the teleportation and the correction is:

$\ket{\phi^{+}}_{bc}=N_{\text{EPR}}(\ket{\tilde{0}\tilde{1}}_{bc}+\ket{\tilde{1}\tilde{0}}_{bc})$
(25)

composed of logical states which are less noisy, indicated with the tilde
notation, than the state to be teleported. Upon completion of the protocol,
the single-photon state $\ket{\psi}$ will be localized to the spatial port
$c$, and the correction is automatically performed since the EPR state is less
noisy than the state of interest.

We now write the protocol for the error correction and teleportation of the
time-frequency cat state, considering that the Bell’s measurement perfectly
separate the two logical states. This protocol allows correcting frequency
errors affecting a frequency qubit states. The protocol is represented in Fig.
5, and we will now proceed to write the evolution of the wavefunction at each
step of the protocol. The initial wavefunction, composed of the state to be
corrected and teleported, and the EPR state, written in the Bell’s basis is:

$\ket{\psi}=\frac{N_{\alpha\beta}N_{\text{EPR}}}{2}(\ket{\omega_{1}\tilde{\omega}_{1}}+\ket{\omega_{2}\tilde{\omega}_{2}})(\alpha\ket{\tilde{\omega_{1}}}_{c}+\beta\ket{\tilde{\omega_{2}}}_{c})\\\
+(\ket{\omega_{1}\tilde{\omega}_{1}}-\ket{\omega_{2}\tilde{\omega}_{2}})(\alpha\ket{\tilde{\omega_{1}}}_{c}-\beta\ket{\tilde{\omega_{2}}}_{c})\\\
+(\ket{\omega_{1}\tilde{\omega}_{2}}+\ket{\omega_{2}\tilde{\omega}_{1}})(\beta\ket{\tilde{\omega_{1}}}_{c}+\alpha\ket{\tilde{\omega_{2}}}_{c})\\\
+(-\ket{\omega_{1}\tilde{\omega}_{2}}+\ket{\omega_{2}\tilde{\omega}_{1}})(\beta\ket{\tilde{\omega_{1}}}_{c}-\alpha\ket{\tilde{\omega_{2}}}_{c}).$
(26)

The single photon state and one member of the EPR pair are then combined into
a beam-splitter, followed by two parity-frequency beam-splitter are placed in
spatial port $a^{\prime}$ and $b^{\prime}$. The first and second Bell’s state
are transformed as:

$\frac{N_{\text{EPR}}}{2}[\ket{\omega_{1}\tilde{\omega}_{1}}_{a}-\ket{\omega_{1}}_{a}\ket{\tilde{\omega}_{1}}_{b}+\ket{\omega_{1}}_{b}\ket{\tilde{\omega}_{1}}_{a}-\ket{\omega_{1}\tilde{\omega}_{1}}_{b}\\\
+\ket{\omega_{2}\tilde{\omega}_{2}}_{a}-\ket{\omega_{2}}_{a}\ket{\tilde{\omega}_{2}}_{b}+\ket{\omega_{2}}_{b}\ket{\tilde{\omega}_{2}}_{a}-\ket{\omega_{2}\tilde{\omega}_{2}}_{b}].$
(27)

The presence of a single photon in each port is a consequence of the
distinguishability of the photons. While if the photons were
indistinguishable, when the two logical states are orthogonal as it is the
case for the polarization encoding, only bunching event will be measured. In
order to suppress this coincidence events that leads to errors in the
teleportation protocol, the utilisation of frequency filters with the same
frequency width (envelope and peak) as the EPR state is a potential solution.
Explicitly, after the filtering operation, the state becomes
$\ket{\tilde{\omega}_{1}\tilde{\omega}_{1}}_{a}-\ket{\omega_{1}\tilde{\omega}_{1}}_{b}+\ket{\tilde{\omega}_{2}\tilde{\omega}_{2}}_{a}-\ket{\omega_{2}\tilde{\omega}_{2}}_{b}$,
which leads only to two bunching events, that are ignored if single photon
detectors are used. The same analysis can be employed for the second Bell’s
state.

For the third and fourth Bell’s state, they are transformed as:

$\frac{N_{\text{EPR}}}{2}[\pm\ket{\omega_{1}\tilde{\omega}_{2}}_{aa^{\prime}}\mp\ket{\omega_{1}\tilde{\omega}_{2}}_{ab^{\prime}}\pm\ket{\omega_{1}\tilde{\omega}_{2}}_{ba^{\prime}}\mp\ket{\omega_{1}\tilde{\omega}_{2}}_{bb^{\prime}}\\\
+\ket{\omega_{2}\tilde{\omega}_{1}}_{a^{\prime}a}-\ket{\omega_{2}\tilde{\omega}_{1}}_{a^{\prime}b}+\ket{\omega_{2}\tilde{\omega}_{1}}_{b^{\prime}a}-\ket{\omega_{2}\tilde{\omega}_{1}}_{b^{\prime}b}].$
(28)

The use of frequency filter is again imperative, since the measurement of
coincidence once filtered of $a,a^{\prime}$ (or $b,b^{\prime}$) permits to
ensure that the quantum state
$N_{\alpha\beta}(\beta\ket{\tilde{\omega_{1}}}+\alpha\ket{\tilde{\omega_{2}}})$
has been teleported. In the same way, the measurement of the coincidence of
$a,b^{\prime}$ (or $b,a^{\prime}$) allows to teleport the state
$N_{\alpha\beta}(\beta\ket{\tilde{\omega_{1}}}-\alpha\ket{\tilde{\omega_{2}}})$.
The receiver sends to spatial port $c$ which detectors have measured
coincidences, and then a product of Pauli matrix gates must be applied to
recover the initial state of interest. Pauli matrices for the time-frequency
cat states are frequency and temporal shifts operations [1], which can be
implemented by either a electro-optical modulator [49, 50] and a delay line
respectively. The full optical scheme of the teleportation-based error
correction protocol is represented in Fig. 5, along with a illustration of the
effect of the error correction for qubit cat states.

The corresponding probability of each event is
$P=\frac{1}{8(1+e^{-\Delta^{2}/2\sigma^{2}})}$, and the overall probability
success of the teleportation is

$P=\frac{1}{2(1+e^{-\Delta^{2}/2\sigma^{2}})},$ (29)

which is then lower than 50 %, since only linear optics is used [61], and
because the non-orthogonality of the encoding further decrease the probability
of success. Note that the use of photon-number-resolving (PNR) detectors will
able to not discard the bunching events coming from the first and second
Bell’s state. With the use of such a PNR detector, the overall probability of
success of the teleportation protocol is:

$P_{\text{PNR}}=\frac{3}{4(1+e^{-\Delta^{2}/2\sigma^{2}})}$ (30)

and thus allows to increase the probability of success of the protocol despite
the non-orthogonality of the state. Experimentally, the choice of the PNR
detector could be the one described in [62]. Reaching a $3/4$ probability of
success has also been found by using non-linear process or ancilla entangled
states [63, 64].

We can formulate the previous protocol for time-frequency GKP states. If we
employ a EPR state which is less noisy in the temporal and frequency domain,
the teleported state is corrected in both temporal and frequency variables at
once. It is in contrast with the error correction protocol based on frequency
entanglement described in Sec. II.4, where we have to repeat twice the same
protocol to correct both variables. In the Appendix B, we tackle the case of
the teleportation error correction protocol when the spatial separation of the
non-orthogonal qubit state is imperfect, discussing the special case of time-
frequency GKP state and the Bell’s measurement relying on the spatial
separation of the two logical states described in Sec. III.2. The imperfect
spatial separation affects both the efficiency and the fidelity of the
teleportation protocol. The fidelity is always equal to one when the logical
states are orthogonal, but this is not the case for non-orthogonal time-
frequency qubit states. The use of frequency filters can eliminate detection
events caused by imperfect spatial separation, but it comes at the cost of
decreased brightness.

Figure 5:  Schematic of the teleportation-based error correction of frequency
qubit states. The state to be teleported and corrected is located in spatial
port A. An EPR frequency qubit states which is less noisy than the state of
interest, is at spatial ports B and C. FQBS stands for frequency qubit beam-
splitter. $F_{0,1}$ are frequency filters with central frequency matching
either the zero and one logical state, and with a frequency width equal to
that of the EPR state. Depending on which detectors have measured
coincidences, Pauli operations $\hat{X},\hat{Z}$ must be performed to recover
the state of interest. The frequency qubit states before and after the
teleportation are represented. The zero logical (resp. one) state has a blue
(resp. red) color.

##  V Conclusion

In this paper, we have analyzed the teleportation-based error correction
protocol for two types of frequency qubit states: time-frequency cat states
and time-frequency GKP states. This optical scheme has the same goal as a
quantum relay, reducing the error rate of wrong detection by decreasing the
overlap of the two logical states composing the qubit. We have discussed the
experimental realization of Bell’s measurement for these two types of qubits.
The advantage of discretizing a grid state into a qubit state by combining the
even and odd peaks, rather than considering the state as a time-frequency
qudit, is convenient because it simplifies the optical implementation of grid
state manipulations. When the states are not infinitely frequency narrowed,
the Bell’s measurement leads to wrong detection, as the spatial separation of
the two logical states into two spatial ports is imperfect. This can be
corrected by using frequency filters, at the cost of losing single photon
detection events. To tackle this issue, the use of frequency resolved
detection and the fault-tolerance threshold defined in [52] could be valuable
for avoiding the use of frequency filters. We have illustrate that our
protocol can correct the errors of the qubit composed of two colors, but it
could be also done for correcting the temporal error coming from broadening
and dispersion, of a qubit composed of two Gaussian centered at two temporal
bins (see Eq. (2)). In this context, it is important to study and evaluate the
overall effectiveness of the teleportation protocol, as well as the final
quality of the state being corrected, given the level of accuracy in
separating the two logical states.

## ACKNOWLEDGMENT

N. Fabre acknowledges useful discussions with Filip Rozpędek, Arne Keller and
Pérola Milman for the completion of this manuscript.

##  Appendix A Spatial separation of a two-photon state

We show in this section that a two-photon state as input can also be separated
into the even and the odd components in two distinct spatial ports. We
consider an initial separable two photon idea time-frequency GKP state as:

$\ket{0\tilde{0}}_{aa}=\sum_{n,m\in\mathds{Z}^{2}}c_{2n}\tilde{c}_{2m}\hat{a}^{\dagger}(2n\overline{\omega})\hat{a}^{\dagger}(2m\overline{\omega})\ket{\Omega}.$
(31)

The tilde notation is here to indicate that the two states are not identical,
one of them can be more noisy compared to the other. After the first beam-
splitter and the time-displacement operator, the wave function of the two-
photon state is:

$\frac{1}{2}\sum_{n,m\in\mathds{Z}^{2}}c_{2n}\tilde{c}_{2m}(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{b}^{\dagger}(2n\overline{\omega})\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})\\\
+\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})\hat{b}^{\dagger}(2m\overline{\omega})+\hat{b}^{\dagger}(2m\overline{\omega})\hat{b}^{\dagger}(2n\overline{\omega}))\ket{\Omega}.$
(32)

where
$\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})=e^{i2n\overline{\omega}\tau}\hat{a}^{\dagger}(2n\overline{\omega})$.
The output wave function after the second beam-splitter is:

$\frac{1}{4}\sum_{n,m\in\mathds{Z}^{2}}c_{2n}\tilde{c}_{2m}(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{b}^{\dagger}_{\tau}(2n\overline{\omega}))(\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{b}^{\dagger}_{\tau}(2m\overline{\omega}))\\\
+(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})-\hat{b}^{\dagger}_{\tau}(2n\overline{\omega}))(\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{b}^{\dagger}_{\tau}(2m\overline{\omega}))\\\
+(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{b}^{\dagger}_{\tau}(2n\overline{\omega}))(\hat{a}^{\dagger}(2m\overline{\omega})-\hat{b}^{\dagger}(2m\overline{\omega}))\\\
+(\hat{a}^{\dagger}(2n\overline{\omega})-\hat{b}^{\dagger}(2n\overline{\omega}))(\hat{a}^{\dagger}(2m\overline{\omega})-\hat{b}^{\dagger}(2m\overline{\omega}))\ket{\Omega}.$
(33)

We rearrange and post-select only the coincidence terms:

$\frac{1}{4}\sum_{n,m\in\mathds{Z}^{2}}c_{2n}\tilde{c}_{2m}(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega}))\hat{b}_{\tau}^{\dagger}(2m\overline{\omega})\\\
-(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega}))\hat{b}^{\dagger}(2m\overline{\omega})\\\
+(\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{a}^{\dagger}(2m\overline{\omega}))\hat{b}_{\tau}^{\dagger}(2n\overline{\omega})\\\
-(\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{a}^{\dagger}(2m\overline{\omega}))\hat{b}^{\dagger}(2n\overline{\omega})\ket{\Omega}.$
(34)

Let us first consider the ideal case, where the spectral distribution is a
Dirac one $G_{2n}(\omega)=\delta(\omega-2n\overline{\omega})$. We point out
that
$(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega}))\ket{0}=(e^{2in\overline{\omega}\tau}+1)\ket{2n\overline{\omega}}$,
with $\tau=\pi/\overline{\omega}$, we have
$(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega}))\ket{0}=2\ket{2n\overline{\omega}}$.
We have also
$(\hat{b}_{\tau}^{\dagger}(2m\overline{\omega})-\hat{b}^{\dagger}(2m\overline{\omega})\ket{2n\overline{\omega}}\ket{0}=0$.
We can verify that the others terms are zero. It means that there is no
coincidence event which is the desired outcome.

We now rearrange and post-select only the bunching terms:

$\frac{1}{4}\sum_{n,m\in\mathds{Z}^{2}}c_{2n}\tilde{c}_{2m}(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega})\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})\\\
+\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})\hat{a}^{\dagger}(2m\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega})\hat{a}^{\dagger}(2m\overline{\omega})\\\
+\hat{b}^{\dagger}_{\tau}(2n\overline{\omega})\hat{b}^{\dagger}_{\tau}(2m\overline{\omega})-\hat{b}^{\dagger}(2n\overline{\omega})\hat{b}^{\dagger}_{\tau}(2m\overline{\omega})\\\
+\hat{b}^{\dagger}_{\tau}(2n\overline{\omega})\hat{b}^{\dagger}(2m\overline{\omega})-\hat{b}^{\dagger}(2n\overline{\omega})\hat{b}^{\dagger}(2m\overline{\omega}))\ket{\Omega}.$
(35)

In the ideal case, the terms in the spatial port $a$ remains, and the ones in
the spatial port $b$ interfere destructively.

For a time-frequency GKP state with a finite bandwidth, there is no longer a
perfect destructive (or constructive) interference effect that separates the
even and odd components of the comb perfectly. We analyze what happens to the
state $\ket{0\tilde{1}}_{aa}$. Post-selecting on the coincidence we have:

$\ket{0\tilde{1}}_{aa}\rightarrow\ket{0_{G}}_{a}\ket{\tilde{1}_{G}}_{a^{\prime}}+\ket{0_{E}}_{a^{\prime}}\ket{\tilde{1}_{E}}_{a}$
(36)

where we have defined:

$\displaystyle\ket{0_{G}}_{a}=\frac{N_{e}}{2}\sum_{n\in\mathds{Z}}c_{2n}\int
d\omega(e^{i\omega\tau}+1)G_{2n}^{\sigma}(\omega)\ket{\omega}_{a}$ (37)
$\displaystyle\ket{\tilde{1}_{G}}_{a^{\prime}}=\frac{N_{o}}{2}\sum_{n\in\mathds{Z}}c_{2n+1}\int
d\omega(e^{i\omega\tau}-1)G_{2m+1}^{\tilde{\sigma}}(\omega)\ket{\omega}_{a^{\prime}}$
(38)
$\displaystyle\ket{\tilde{1}_{E}}_{a}=\frac{N_{o}}{2}\sum_{n\in\mathds{Z}}c_{2n+1}\int
d\omega(e^{i\omega\tau}+1)G_{2m+1}^{\tilde{\sigma}}(\omega)\ket{\omega}_{a}$
(39)
$\displaystyle\ket{0_{E}}_{a^{\prime}}=\frac{N_{e}}{2}\sum_{n\in\mathds{Z}}c_{2n}\int
d\omega(e^{i\omega\tau}-1)G_{2n}^{\sigma}(\omega)\ket{\omega}_{a^{\prime}}.$
(40)

The post-selection on coincidence results in only those events where both the
even and odd components are in the correct (designated by $G$) and incorrect
(designated by $E$) spatial ports. It should be noted that the output state is
no longer in the GKP subspace and results in detection errors. These errors
can be corrected through the use of frequency filters or by using frequency-
resolved detection and setting a frequency threshold width to accept only
certain events [52]. Additionally, the imperfect spatial separation described
in Eq. (36) can also be interpreted as an attack to extract some information
of the quantum state of interest in a quantum communication protocol.

##  Appendix B Teleportation-based error correction protocol with physical
time-frequency GKP state

In the following, we will employ the GKP coherent picture [53]. The GKP qubit
defined by Eq. (12) is composed of an envelope of width $\kappa$ and each peak
has a width of $\sigma$ (resp. $\kappa_{1},\sigma_{1}$), while the frequency
widths of the EPR state will be noted $\tilde{\kappa}$ and $\tilde{\sigma}$.

As has been noted, when the time-frequency GKP state has a limited bandwidth,
the frequency qubit beam-splitter is not able to perfectly separate the odd
and even components of the comb. This results in some of the state leaking out
of each spatial port, and not conforming to a GKP state. Furthermore, the
state in the right port is also distorted. To address these two challenges, it
is necessary to employ frequency filters that enable projection back into the
GKP subspace and eliminate the undesirable components. Given that the state
being teleported and the EPR state have different frequency widths, a
frequency filter is placed prior to detection, which establishes the frequency
bin and aligns with the reference EPR state. The projector modeling for the
frequency filtering process has the following form:

$\displaystyle\hat{\Pi}_{a}=N^{2}_{e}(\sigma,\kappa)\sum_{n\in\mathds{Z}}\tilde{c}_{2n}\int
G_{2n}^{\tilde{\sigma}}(\omega)\ket{\omega}\bra{\omega}$ (41)
$\displaystyle\hat{\Pi}_{a^{\prime}}=N^{2}_{o}(\sigma,\kappa)\sum_{n\in\mathds{Z}}\tilde{c}_{2n+1}\int
G_{2n+1}^{\tilde{\sigma}}(\omega)\ket{\omega}\bra{\omega}$ (42)

The normalization constant is found by using $\hat{\Pi}^{2}=\mathds{I}$ and
$\text{Tr}(\hat{\Pi}^{2})=1$,
$N^{2}_{e}(\tilde{\sigma},\tilde{\kappa})=\sqrt{2\pi}/\tilde{\sigma}\sum_{n}\absolutevalue{\tilde{c}_{2n}}^{2}$.
In the large comb approximation, we have
$N_{e}(\tilde{\sigma},\tilde{\kappa})=N_{o}(\tilde{\sigma},\tilde{\kappa})$.

The coincidence destructive measurement is described by the positive operator
value measurement:

$\hat{\Pi}_{a,a^{\prime}}=\hat{\Pi}_{a}\hat{\Pi}_{a^{\prime}}\otimes\ket{1}\bra{1}.$
(43)

By assuming that the state is pure, the probability of coincidence in the
spatial port $a,a^{\prime}$,
$P_{aa^{\prime}}(0,1)=\text{Tr}(\hat{\Pi}_{a,a^{\prime}}\ket{\psi}\bra{\psi}\hat{\Pi}_{a,a^{\prime}}^{\dagger})$
is:

$P_{aa^{\prime}}(0,1)=\absolutevalue{\frac{1}{2\sqrt{2}}}^{2}\absolutevalue{a^{01}_{\sigma\tilde{\sigma}}+b^{01}_{\sigma\tilde{\sigma}}}^{2}\times\\\
(\int
d\omega(\absolutevalue{\alpha}^{2}\absolutevalue{\bra{\omega}\ket{\tilde{0}}}^{2}+\absolutevalue{\beta}^{2}\absolutevalue{\bra{\omega}\ket{\tilde{1}}}^{2}+2\text{Re}(\alpha^{*}\beta\bra{\omega}\ket{\tilde{0}}\bra{\omega}\ket{\tilde{1}}))$
(44)

where we have used
$\absolutevalue{a^{01}_{\sigma\tilde{\sigma}}+b^{01}_{\sigma\tilde{\sigma}}}=\absolutevalue{a^{10}_{\sigma\tilde{\sigma}}+b^{10}_{\sigma\tilde{\sigma}}}$
which is shown afterward. For an EPR with a sufficiently narrow distribution,
we assume that $\bra{\tilde{0}}\ket{\tilde{1}}=0$, $\int
d\omega\absolutevalue{\bra{\omega}\ket{\tilde{0}}}^{2}=\int
d\omega\absolutevalue{\bra{\omega}\ket{\tilde{1}}}^{2}=1$. These two
conditions are important, since the probability then does not depend on
$\alpha$ and $\beta$,

$P_{aa^{\prime}}(0,1)=\absolutevalue{\frac{1}{2\sqrt{2}}}^{2}\absolutevalue{a^{01}_{\sigma\tilde{\sigma}}+b^{01}_{\sigma\tilde{\sigma}}}^{2}$
(45)

since otherwise, information about the quantum state could be extracted during
the measurement.

The wavefunction of the state after the detection is:
$\ket{\psi}_{c}=\hat{\Pi}_{i,j}\ket{\psi}/\text{Tr}(\hat{\Pi}_{i,j}\ket{\psi}\bra{\psi}\hat{\Pi}_{i,j}^{\dagger})$,
where $i,j=a,b;a^{\prime},b^{\prime}$. When coincidence is detected at the
spatial port $a$ and $a^{\prime}$, the post-selected state is:

$\ket{\psi}_{c}=\frac{1}{\absolutevalue{a^{01}_{\sigma\tilde{\sigma}}+b^{01}_{\sigma\tilde{\sigma}}}}(\alpha(a^{1,\tilde{\sigma}}_{0,\sigma}+b^{1,\sigma}_{0,\tilde{\sigma}})\ket{\tilde{0}}_{c}\\\
+\beta(a^{1,\sigma}_{0,\tilde{\sigma}}+b^{1,\tilde{\sigma}}_{0,\sigma})\ket{\tilde{1}}_{c}).$
(46)

The expression is similar for the other coincidences events in the other
spatial ports, and extra Pauli operations have to be performed. The
coefficients $a^{10}_{\tilde{\sigma}\sigma}$ and
$b^{10}_{\sigma\tilde{\sigma}}$ have the expression:

$a^{10}_{\tilde{\sigma}\sigma}=\frac{1}{4}N_{e}(\sigma,\kappa)N_{o}(\tilde{\sigma},\tilde{\kappa})N_{e}(\tilde{\sigma},\tilde{\kappa})N_{o}(\tilde{\sigma},\tilde{\kappa})\\\
\times\sum_{n,m,k,k^{\prime}}c_{2n}\tilde{c}_{2m+1}\tilde{c}_{2k}\tilde{c}_{2k^{\prime}+1}\int
d\omega(e^{i\omega\tau}+1)G_{2n}^{\sigma}(\omega)G_{2k}^{\tilde{\sigma}}(\omega)\\\
\times\int
d\omega(e^{i\omega\tau}-1)G_{2m+1}^{\tilde{\sigma}}(\omega))G_{2k^{\prime}+1}^{\tilde{\sigma}}(\omega)$
(47)
$b^{10}_{\sigma\tilde{\sigma}}=\frac{1}{4}N_{e}(\tilde{\sigma},\tilde{\kappa})N_{o}(\sigma,\kappa)N_{e}(\tilde{\sigma},\tilde{\kappa})N_{o}(\tilde{\sigma},\tilde{\kappa})\\\
\times\sum_{n,m,k,k^{\prime}}\tilde{c}_{2n}c_{2m+1}\tilde{c}_{2k}\tilde{c}_{2k^{\prime}+1}\int
d\omega(e^{i\omega\tau}+1)G_{2m+1}^{\sigma}(\omega)G_{2k}^{\tilde{\sigma}}(\omega)\\\
\times\int
d\omega(e^{i\omega\tau}-1)G_{2n}^{\tilde{\sigma}}(\omega))G_{2k^{\prime}+1}^{\tilde{\sigma}}(\omega).$
(48)

The $b$ coefficient contains odd (resp. even) terms in a spatial port where
the frequency filters is centered at even (resp. odd) frequencies. In the
ideal case, namely if both the EPR and the teleported state are ideal time-
frequency GKP state we remind that $a^{01}_{\sigma\tilde{\sigma}}=2$ and
$b^{01}_{\sigma\tilde{\sigma}}=0$. After evaluation of the integrals, and
assuming that $2k=2n$, meaning that the temporal spreading only reach the next
bin, we find that:

$a^{10}_{\tilde{\sigma}\sigma}=\frac{\sqrt{\tilde{\sigma}\tilde{\sigma}}\sqrt{\sigma\tilde{\sigma}}}{4\sqrt{\tilde{\sigma}^{2}+\sigma^{2}}\sqrt{\tilde{\sigma}^{2}+\tilde{\sigma}^{2}}}(e^{-\frac{\pi^{2}\alpha^{2}}{2\overline{\omega}^{2}}}+1)(-e^{-\frac{\pi^{2}\tilde{\alpha}^{2}}{2\overline{\omega}^{2}}}-1)$
(49)
$b^{10}_{\sigma\tilde{\sigma}}=\frac{\sqrt{\tilde{\sigma}\tilde{\sigma}}\sqrt{\sigma\tilde{\sigma}}}{4\sqrt{\tilde{\sigma}^{2}+\sigma^{2}}\sqrt{\tilde{\sigma}^{2}+\tilde{\sigma}^{2}}}\frac{\sum_{n,m}c_{2m+1}c_{2m}c_{2n}c_{2n+1}}{\sum_{n,m}\absolutevalue{c_{2m+1}}^{2}\absolutevalue{c_{2n}}^{2}}\\\
\times(e^{-\frac{\pi^{2}\alpha^{2}}{2\overline{\omega}^{2}}}e^{-\frac{\overline{\omega}^{2}}{2(\tilde{\sigma}^{2}+\tilde{\sigma}^{2})}}e^{i\frac{\pi\tilde{\sigma}^{2}}{(\sigma^{2}+\tilde{\sigma}^{2})}}+1)\\\
\times(e^{-\frac{\pi^{2}\alpha^{2}}{2\overline{\omega}^{2}}}e^{-\frac{\overline{\omega}^{2}}{2(\tilde{\sigma}^{2}+\tilde{\sigma}^{2})}}e^{i\frac{\pi\tilde{\sigma}^{2}}{(\sigma^{2}+\tilde{\sigma}^{2})}}-1)$
(50)

where we have defined that
$\alpha^{2}=\sigma^{2}\tilde{\sigma}^{2}/(\sigma^{2}+\tilde{\sigma}^{2})$.
While $a^{10}_{\tilde{\sigma}\sigma}$ is real, $b^{10}_{\sigma\tilde{\sigma}}$
is a complex quantity. From these expressions, we point out that
$a^{1,\tilde{\sigma}}_{0,\sigma}=a^{0,\tilde{\sigma}}_{1,\sigma}$ and
$b^{1,\sigma}_{0,\tilde{\sigma}}=b^{0,\sigma}_{1,\tilde{\sigma}}$. When the
width of the frequency filter is $\tilde{\sigma}\rightarrow 0$, we have
$a^{01}_{\sigma\tilde{\sigma}}=2$ and $b^{01}_{\sigma\tilde{\sigma}}=0$ which
is as the ideal case and thus lead to a high fidelity of the state, but at the
cost of losing many photons. As the imperfect spatial separation of the two
logical states leads to a decreasing of the fidelity, it is a reminiscent fact
coming that the state possesses continuous variables.

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