# Error suppression in multicomponent cat codes with photon subtraction and
teleamplification

Saurabh U. Shringarpure1  Yong Siah Teo2  and Hyunseok Jeong3  Department of
Physics $\&$ Astronomy, Seoul National University, 1 Gwanak-ro, Gwanak-gu,
Seoul 08826, Republic of Korea<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

It is known that multiphoton states can be protected from decoherence due to a
passive loss channel by applying noiseless attenuation before and noiseless
amplification after the channel. In this work, we propose the combined use of
multiphoton subtraction on four-component cat codes and teleamplification to
effectively suppress errors under detection and environmental losses. The
back-action from multiphoton subtraction modifies the encoded qubit encoded on
cat states by suppressing the higher photon numbers, while simultaneously
ensuring that the original qubit can be recovered effectively through
teleamplification followed by error correction, thus preserving its quantum
information. With realistic photon subtraction and teleamplification-based
scheme followed by optimal error-correcting maps, one can achieve a worst-case
fidelity (over all encoded pure states) of over $93.5\%$ ($82\%$ with only
noisy teleamplification) at a minimum success probability of about $3.42\%$,
under a $10\%$ environmental-loss rate, $95\%$ detector efficiency and
sufficiently large cat states with the coherent-state amplitudes of 2. This
sets a promising standard for combating large passive losses in quantum-
information tasks in the noisy intermediate-scale quantum (NISQ) era, such as
direct quantum communication or the storage of encoded qubits on the photonic
platform.

##  1 Introduction

Multiphoton superposition states, crucial for various quantum applications,
are susceptible to decoherence because of photon loss, particularly affecting
higher energy terms. Noiseless attenuation and noiseless amplification have
been shown to be effective in suppressing decoherence due to photon loss [1].
These nondeterministic techniques work by strategically reducing the
probability amplitudes of higher photon number terms suffering from loss.

Bosonic error-correcting codes offer an alternative for protecting qubits from
the impact of photon loss. Quantum error correction using these codes has
recently garnered attention due to their physical-resource efficiency, as
opposed to discrete-variable counterparts that require coherence over a large
number of multiple physical modes [2, 3, 4, 5]. Numerically optimized bosonic
codes tailored for loss channels have been explored in previous studies [6, 2]
where entanglement fidelity of the channels is used as a figure of merit.
Reference [7] extends this analysis to include general loss-dephasing
channels, where a key takeaway was that for large values of noise, encodings
with higher average energy may not be suitable.

In this context, we explore the interplay between reducing average energy and
ensuring that an encoded qubit can be recovered by error correction. We also
consider the influence of additional noise introduced during realistic error
suppression on the encoded states. Optimized encodings for a given channel
mentioned earlier may not be easy to realize in practice.

For example, Gottesman–Kitaev–Preskill (GKP) [8] codes with high squeezing,
robust against loss channels pose significant experimental challenges.
Nevertheless, there has recently been some progress in generating them on
photonic platforms [9, 10]. This also motivates error suppression with
currently feasible encodings. This paper focuses on multicomponent cat codes,
particularly four-component cat codes that are resilient to errors induced by
single-photon annihilation [11, 12, 13, 14, 15] and restricts to the
amplitude-damping or loss channel, a predominant noise source across various
platforms. We study this on photonic platforms, where noiseless attenuation is
generally implemented using zero-photon subtraction (ZPS) [16, 17, 18] and has
been shown to preserve the coherence of two-component cat states [19].

Subtraction of an arbitrary number of photons from a cat state results in
another cat state, while also implementing noiseless attenuation like ZPS.
Subtraction of a larger number of photons also gives a higher success
probability in many situations. We discuss in some detail the consequences of
this for fidelity and success rates. Alternatives to error suppression of the
current discussion, where deterministic squeezing is considered instead of
multiphoton subtraction, have been explored [20, 21, 22, 23]. Recent
demonstrations, such as those in Ref. [24] on the circuit-QED platform,
encourage interest in active preemptive error suppression. Error filtration
[25, 26, 27] and channel correction [28] also seek to protect quantum
information by suppressing errors.

In this work, we similarly analyze a continuous-variable (CV) system that
encodes a single qubit and exploit the rotation symmetry of the cat codes.
Recently, quantum error mitigation with rotationally symmetric bosonic codes
was explored in Ref. [29] which also supports this work. However, we focus on
implementing the noiseless loss suppression scheme for photonic platforms and
take into account the additional sources of noise that would be involved in
its realization. Error mitigation techniques that suppress errors in
expectation values during data postprocessing have gained recent attention
[30, 31, 32], but these are different from active error suppression methods
that herald event-ready states of interest here. In these techniques, the
quantum state is destroyed during measurements, whereas we would like to
preserve the quantum information for later applications.

Deterministic phase-preserving linear amplification always adds noise [33,
34]. Efficient and noiseless amplification of two-component cat codes become
feasible, albeit nondeterministically, with multiple copies of the state as a
resource [35, 36, 37]. Quantum scissors have been considered for the
approximate noiseless amplification of low-amplitude states [38, 39].
Subsequently, teleportation-based noiseless linear amplifiers were introduced
for multicomponent cat codes that work exactly [40, 41, 42]. We adopt this
approach to amplify the amplitudes of the cat states specifically before
recovery and generalize the standard teleamplification to herald on a larger
set of photon number detections along with multiphoton subtraction for the
noiseless attenuation. The recovery for each compatible outcome is
appropriately adapted as shown in Fig. 1(e).

In our analyses, we prefer to use the worst-case fidelity across all input
pure states to characterize the performance. We find that a worst-case
fidelity of over $93.5\%$ can be achieved, at a minimum success probability of
about $3.42\%$, under a $10\%$ environmental-loss rate, a realistic $95\%$
detector efficiency, for cat states with the coherent-state amplitudes of as
large as 2. The ability to protect such large cat states is quite appealing
for quantum information applications in the near term, wherein large passive
losses before error correction are unavoidable. Storing photonic qubits in a
cavity or long fiber loops, or direct communication of the qubits over long
distances provide some near-term examples. These contribute to the development
of a distributed quantum network [43].

The rest of the paper is organized as follows. The preliminaries of photon
subtraction, teleamplification, photon losses, and multicomponent cats with
some of the notations used throughout are presented in Sec. 2. Section 3
discusses the error suppression for four-component cat codes. Optimal
recoveries in Sec. 4 and the results are presented in Sec. 5. The paper ends
with a discussion in Sec. 6.

##  2 Preliminaries

Passive photon loss affects the superpositions of the input Fock states
$\left|\psi_{\rm
in}\right\rangle=\sum_{n=0}^{\infty}{c_{n}\left|n\right\rangle}$ by decohering
them into mixtures $\hat{\rho}_{\rm out}$, as shown in Fig. 1(a). CV error-
correcting codes are multiphoton states with superpositions containing a large
number of photons. The purpose of these codes is to encode small logical
quantum information, such as a single (or a few) qubit(s), on a large number
of subsystems. Therefore, the infinitely many Fock states are well suited to
this end, and this is the reason behind the physical-resource efficiency of CV
error correction. As the photon losses occur, they cause errors on the encoded
information. However, error-correcting codes allow the detection and
correction of such errors through a recovery channel $\mathcal{R}$. This
includes measuring the syndrome such as the generalized photon number parity
of the affected state without destroying the logical superposition, as shown
in Fig. 1(b).

Figure 1: (a) A pure photonic state passing through a photon loss channel
becomes a mixture. Higher photon number terms in the initial state are more
likely to be lost to the environment. (b) The quantum information encoded in
the error-correcting codes is recovered after loss. The encoded states are
typically a large average photon number. (c) Quantum information is made
resilient to noise from photon loss by noiseless loss suppression. Operations
with $\nu<1$ and $g>1$ constitute noiseless attenuation and amplification
processes, respectively. It works by suppressing large energy terms. (d) The
two approaches are combined, where the states encoded in the error-correcting
codes are passed through a loss suppression procedure, which also suppresses
uncorrectable logical errors on the encoded states. (e) The procedure is
extended to include arbitrary photon annihilations during noiseless
attenuation and amplification sections. Error correction is suitably adapted
to outcomes obtained during the general photon subtraction and
teleamplification.

Different encodings are suited to different kinds of noise, but no realistic
encoding can detect and correct for any arbitrary number of photon losses, and
therefore we naturally end up in a mixed state in the computational basis. The
challenge of error correction is to detect and correct errors faster than they
can grow, and there has been great focus in this direction.

One way to protect the purity of states is the relatively less known method of
noiseless loss suppression, shown in Fig. 1(c). In this scheme, noiseless
attenuation is applied to the initial pure state which is an operator
$\nu^{\hat{n}}$ with the constant factor $\nu<1$. It results in a state
$\propto\sum_{n=0}^{\infty}c_{n}\nu^{n}\left|n\right\rangle$. Larger $n$ terms
are thus exponentially suppressed, and thus the state is greatly protected
against the occurrence of photon losses.

Noiseless attenuation is a nonunitary operation and can only be performed
nondeterministically. Although it makes the states resilient to possible
losses, the redistribution of the probability amplitudes associated with the
constituent Fock states can drastically distort the input. The state can be
recovered by an effective inverse under noiseless amplification which is
simply the operation $g^{\hat{n}}$ with the factor $g>1$. If $g=1/\nu$, then
$(\nu g)^{\hat{n}}=\hat{I}$.

In this work, we combine the two approaches to protecting fragile quantum
information against decoherence from photon loss, as shown in Fig. 1(d). We
are interested in noiselessly attenuating the physical states of arbitrary
computational information encoded in CV encodings before it experiences loss.
This will naturally distort the state and take it out of the codespace. It is
not obvious at the outset if the encoded states are resilient to such
nonunitary distortion, photon loss, and after noiseless amplification, still
be able to detect and correct logical errors during recovery.

In this work, we find that cat encodings due to their rotational symmetric
structure are indeed robust under this procedure and can utilize the
advantages of error correction and noiseless loss suppression. Moreover, the
particular implementations of noiseless loss suppression for cat states, with
photon subtraction and teleamplification, also exhibit another advantage in
that an arbitrary number of photon annihilations over these operations,
$\nu^{\hat{n}}$ and $g^{\hat{n}}$, do not affect the encoding. This will be
explained in detail in Sec. 3.

These photon annihilations are typically an undesirable consequence of
realizing noiseless loss suppression and lead to many failed events, for
example, when noiseless attenuation is heralded only on the detection of zero
photons. However, due to this feature of the cat encoding, we can adapt the
recovery channel $\mathcal{R}$ over a larger set of events than would be
possible if we restricted ourselves to ZPS and the standard teleamplificaiton,
as illustrated by Fig. 1(e).

In the following, preliminaries of photon subtraction, teleamplification,
photon loss channel, and multicomponent cat states are presented. This section
also introduces the notation we use throughout the paper.

###  2.1 Multiphoton subtraction

Unless otherwise mentioned a beamsplitter of transmittance $T$ (reflectance
$R$) mixes incident annihilation operators in a Bogoliubov transform given by
$\hat{a}_{1}\mapsto\sqrt{T}\hat{a}_{1}-\sqrt{R}\hat{a}_{2}$ and
$\hat{a}_{2}\mapsto\sqrt{R}\hat{a}_{1}+\sqrt{T}\hat{a}_{2}$. A general
$n$-photon subtraction in optical systems is performed by splitting the beam
and heralding the output in the transmitted signal on the detection of $n$
photons in the reflected ancilla.

It can be easily seen that when $n$ photons are detected, it implements an
operation $\propto\sqrt{T}^{\hat{n}}\hat{a}^{n}$ where $\hat{a}$ and $\hat{n}$
are the photon-annihilation and number operators, respectively. This is the
same as noiseless attenuation when $n=0$ photons are detected [1, 16] for any
input state,
$\left|\psi\right\rangle=\sum_{n=0}^{\infty}c_{n}\left|n\right\rangle$ as
$T<1$. Also note that the same $\sqrt{T}^{\hat{n}}$ that is responsible for
noiseless attenuation remains when $n>0$.

###  2.2 Teleamplification

Standard teleamplification implements the operation $\propto g^{\hat{n}}$ just
as photon subtraction implements noiseless attenuation. However, it only works
for inputs that are superpositions of coherent states having the same absolute
value of the amplitude, i.e. they lie on a ring. These states are optical cat
states.

Teleamplification works by heralding the output state on the detection of
several photons in various ancillas. In general, it implements $\propto
g^{\hat{n}}f(\hat{a})$ where $f(\hat{a})$ is some function of the annihilation
operators. The input is a superposition that can be written as
$\left|\psi\right\rangle=\sum_{k}c_{k}\left|\alpha u^{k}\right\rangle$ for
fixed magnitude $\alpha$ and phase $u$ [40] for arbitrary coefficients
$c_{k}$. Note that when inputs are restricted to this form, operations
$f(\hat{a})$ and $g^{\hat{n}}$ commute to an irrelevant global factor that
depends on the magnitude $\alpha$, because coherent states are eigenstates of
the photon-annihilation operator.

When $n$-photon subtraction and teleamplification are put together (for such
input cat states), we get the operation
$\propto\hat{a}^{n}f(\hat{a})(\sqrt{T}g)^{\hat{n}}$. If $g=1/\sqrt{T}$ we get
an operation that does not change the magnitude of the coherent state in the
superposition.

###  2.3 Photon loss

In a typical communication channel such as a fiber optic cable of length $l$
and attenuation length $L_{{\rm{attn.}}}$, the noise experienced during travel
time $t$ is modeled, in the rotating frame, by the LGKS master equation [44,
45] $\partial_{t}\hat{\rho}=\kappa\mathcal{D}[\hat{a}](\hat{\rho})$ with the
dissipation rate $\kappa$ and the dissipator
$\mathcal{D}[\hat{A}]=\hat{A}\bullet\hat{A}^{\dagger}-\\{\hat{A}^{\dagger}\hat{A},\bullet\\}/2$
for a jump operator $\hat{A}$.

The solution to this is the loss or the amplitude-damping quantum channel [46]
with a Kraus representation [47],
$\mathcal{E}_{\gamma}(\hat{\rho})=\sum_{k=0}^{\infty}\hat{E}_{k}\hat{\rho}\hat{E}_{k}^{\dagger},$
where
$\hat{E}_{k}=\sqrt{\left(1-\gamma\right)^{k}/k!}\sqrt{\gamma}^{\hat{n}}\hat{a}^{k}$
with the channel transmittance, $\gamma=e^{-\kappa
t}=e^{-l/L_{{\rm{attn.}}}}$. The set of operators
$\\{\hat{E}_{k}^{\dagger}\hat{E}_{k}\\}$ forms a POVM, with the completeness
or the trace-preserving condition,
$\sum_{k=0}^{\infty}{\hat{E}_{k}^{\dagger}\hat{E}_{k}}=\hat{I}$ [48].

###  2.4 Multicomponent cat codes

Even and odd coherent states are defined as [49]

$|\overline{0}\rangle=\mathcal{N}_{+}\left(\left|\alpha\right\rangle+\left|-\alpha\right\rangle\right)\
{\rm{and}}\
|\overline{1}\rangle=\mathcal{N}_{-}\left(\left|\alpha\right\rangle-\left|-\alpha\right\rangle\right)$
(1)

where the normalization is $\mathcal{N}_{\pm}=1/\sqrt{2\left(1\pm
e^{-2\alpha^{2}}\right)}$, the overline denoting the logical basis, and
assuming real $\alpha$. These are superpositions of coherent states that are
symmetrically spaced on a ring in the phase space and only of even and odd
Fock states. These states are now commonly known as the two-component cat
codes.

They can be used for efficient quantum computation [50, 51] and generated
using single-photon sources and simple all-optical operations [35, 52] or weak
nonlinearity [53]. Although they cannot correct logicl errors due to photon-
losses by themseleves, they are shown to be useful in increasing the fault
tolerance threshold in the biased noise regime in some platforms where
dephasing errors are more likely than relaxation errors [54, 55]. When an odd
number of photons is annihilated, it flips their photon-number parity, whereas
when the annihilation of an even number of photons leaves it unchanged.

Another set of cat codes, the four-component cat codes [11] given by

$\left|\overline{\mu}\right\rangle=\mathcal{N}_{+}\left(\left|\mathrm{i}^{\mu}\alpha\right\rangle+\left|-\mathrm{i}^{\mu}\alpha\right\rangle\right),$
(2)

with $\mu=0,1$, have recently achieved break-even with the bare encoding in
circuit-QED [12]. These codes are resilient to single-photon losses, as they
can be detected and corrected with this code. Single-photon annihilation on
these code states produce

$\left|\overline{\mu}^{\prime}\right\rangle=\mathcal{N}_{-}\left(\left|\mathrm{i}^{\mu}\alpha\right\rangle-\left|-\mathrm{i}^{\mu}\alpha\right\rangle\right),$
(3)

which have odd parity.

A qubit may be encoded on the basis given by Eq. (2), or alternatively on

$|\overline{\pm}\rangle={\mathcal{M}_{\pm}}\left[\left|\alpha\right\rangle+\left|-\alpha\right\rangle\pm\left(\left|\mathrm{i}\alpha\right\rangle+\left|-\mathrm{i}\alpha\right\rangle\right)\right]$
(4)

with
${\mathcal{M}}_{\pm}={1}/{\left[2\sqrt{2e^{-{\alpha}^{2}}\left(\cosh{\alpha^{2}}\pm\cos{\alpha^{2}}\right)}\right]},$

The corresponding odd parity states that form yet another orthonormal basis
are

$|\overline{\pm}^{\prime}\rangle=\mathcal{M}^{{}^{\prime}}_{\pm}\left[\left|\alpha\right\rangle-\left|-\alpha\right\rangle\pm\mathrm{i}\left(\left|\mathrm{i}\alpha\right\rangle-\left|-\mathrm{i}\alpha\right\rangle\right)\right]$
(5)

with
$\mathcal{M}^{{}^{\prime}}_{\pm}={1}/{\left[2\sqrt{2e^{-{\alpha}^{2}}\left(\sinh{\alpha^{2}}\mp\sin{\alpha^{2}}\right)}\right]}.$

##  3 Error suppression on encoded states

###  3.1 With perfect detectors

Consider four-component cat code given by Eq. (2) and a qubit encoded on them
as $|\overline{\psi}\rangle=\sum_{k=0}^{3}c_{k}\left|\alpha_{k}\right\rangle$
defining $\alpha_{k}=\alpha u^{k}$ with
$u=e^{\frac{2\pi\mathrm{i}}{4}}=\mathrm{i}$, where we note that $u$ takes on a
general phase factor for other multicomponent cat codes [40]. An ideal
$n$-photon subtraction with a beamsplitter of transmittance $T$ effectively
implements an operation
$\propto\sqrt{T}^{\hat{a}^{\dagger}\hat{a}}\hat{a}^{n}$. When $n=0$, it is the
same as noiseless attenuation.

An arbitrary encoded state
$|\overline{\psi}\rangle\propto{a|\overline{0}\rangle+b|\overline{1}\rangle}.$
where the logical 0 and 1 states are two-component even cats having identical
normalizations, $\mathcal{N}_{+}$, so the state is proportional to
${a\left(\left|\alpha\right\rangle+\left|-\alpha\right\rangle\right)+b\left(\left|\mathrm{i}\alpha\right\rangle+\left|-\mathrm{i}\alpha\right\rangle\right)}.$
As an example, after an ideal zero-photon subtraction (ZPS) with beamsplitter
of transmissivity $\sqrt{T}$, the coherent-state amplitudes are noiselessly
attenuated by the same factor giving the normalized state
$|\overline{\psi}\rangle=\frac{a|\widetilde{0}\rangle+b|\widetilde{1}\rangle}{\sqrt{1+2{\rm
Re}\left(a^{*}\langle\widetilde{0}|\widetilde{1}\rangle b\right)}},$ where all
quantities with a tilde correspond to reduced amplitudes. This state is
heralded by the detection of zero photons with a success probability,

$\displaystyle
P_{\rm{s}}=\left(\frac{\mathcal{N}_{+}}{\widetilde{\mathcal{N}}_{+}}\right)^{2}\frac{1+2{\rm
Re}\left(a^{*}\langle\widetilde{0}|\widetilde{1}\rangle b\right)}{1+2{\rm
Re}\left(a^{*}\langle\overline{0}|\overline{1}\rangle
b\right)}e^{-R\alpha^{2}}.$ (6)

Figure 2:  Setup for error suppression in four-component cat codes. Here,
zero-photon subtraction implements noiseless attenuation and the standard
teleamplification implements noiseless amplification. The standard
teleamplification requires $|\Phi\rangle$ a resource of a particular large-
amplitude cat state and coincident detection of single photons. Nonzero photon
subtraction and a more general heralding for teleamplification is also
considered with all $x_{k}\neq 0$. A vacuum state is heralded along with the
desired state in one of the outputs, here it is the mode $A$. The unmarked
beamsplitters are balanced $50:50$, and the triangle corresponds to the phase
shift by $\pi/2$. The output from the amplification is sent for recovery by
error correction. Figure 3: The success probability versus input coherent-
state amplitude for (a) standalone ZPS, with beamsplitter transmittances:
$T=95\%$ (solid), $85\%$ (dashed), and $50\%$ (dot-dashed), and for (b)
standalone standard teleamplification ($x_{j}=1$) with gains: $g=1.25$
(dashed) and $g=2.5$ (solid), for the inputs with $|\overline{+}\rangle$ (red)
and $|\overline{\vphantom{+}-}\rangle$ (blue). Perfect detectors have been
assumed.

A teleportation-based approach [40] is employed to noiselessly amplify the
coherent-state amplitudes as shown by Fig. 2. A resource cat state with a
large coherent-state amplitude, say $\beta$, is required. This state is chosen
as
$\left|\Phi\right\rangle\propto|\overline{0}^{\prime}\rangle+\mathrm{i}|\overline{1}^{\prime}\rangle=\mathcal{M}_{+}^{{}^{\prime}}\sum_{k=0}^{3}{u^{-3k}\left|\beta_{k}\right\rangle},$
where $\beta_{k}=u^{k}\beta$ and $\mathcal{M}_{+}^{{}^{\prime}}$ is a function
of the amplitude $\beta$. If $\beta=\alpha/\sqrt{R_{B}}$ and
$g=\sqrt{T_{B}/R_{B}}$ then the output state is given by [40]

$\displaystyle\begin{split}\left|\psi\right\rangle=\mathcal{M}_{+}^{{}^{\prime}}\sum_{j,k=0}^{3}{c_{j}}u^{-3k}&\left|g\alpha_{k}\right\rangle_{B}\left|\frac{\alpha}{2}(u^{j}-u^{k})\right\rangle_{A}\left|\frac{\alpha}{2\sqrt{2}}\left(u^{j}-u^{j+1}+u^{k}+u^{k+1}\right)\right\rangle_{A^{\prime}}\\\
&\left|-\frac{\alpha}{2}(u^{j}+u^{k})\right\rangle_{C}\left|\frac{\alpha}{2\sqrt{2}}\left(u^{j}+u^{j+1}+u^{k}-u^{k+1}\right)\right\rangle_{C^{\prime}}.\end{split}$
(7)

It can be verified that for any given pair $(j,k)$ there is one mode
guaranteed to be in the vacuum.

In our case, this is the mode $A$ whenever $j=k$. Therefore, conditionally
measuring strictly-nonzero Fock states $(x_{1},x_{2},x_{3})$ coincident in
modes $A^{\prime}$, $C$, and $C^{\prime}$ heralds a vacuum in mode $A$.

Assuming a real-valued $\alpha$ under such a heralding, we get the state,

$\mathcal{M}_{+}^{{}^{\prime}}\frac{e^{-\alpha^{2}}\alpha^{x_{1}+x_{2}+x_{3}}}{\sqrt{x_{1}!x_{2}!x_{3}!2^{x_{2}+x_{3}}}}\sum_{k=0}^{3}{c_{k}u^{k(x_{1}+x_{2}+x_{3}-3)}\left|g\alpha_{k}\right\rangle}_{B}.$
(8)

The success probability of heralding the desired state is given by

$P_{\rm{s}}=\left(\mathcal{M}_{+}^{{}^{\prime}}\frac{e^{-\alpha^{2}}\alpha^{x_{1}+x_{2}+x_{3}}}{\sqrt{x_{1}!x_{2}!x_{3}!2^{x_{2}+x_{3}}}}\right)^{2}\sum_{j,k=0}^{3}{c_{j}^{*}u^{(-j+k)(x_{1}+x_{2}+x_{3}-3)}c_{k}e^{-g^{2}\alpha^{2}\left(1-u^{-j+k}\right)}}.$
(9)

An ideal teleamplification thus implements the operation
$\propto\hat{a}^{x_{1}+x_{2}+x_{3}-3}g^{\hat{a}^{\dagger}\hat{a}}$ where
$(x_{1},x_{2},x_{3})$ are the numbers of (nonzero) photons detected in modes
$C$, $A^{\prime}$ and $C^{\prime}$ as shown in Fig. 2. This is valid only
because the input is restricted to cat states where the magnitudes of
coherent-state amplitudes are all identical and they have a discrete rotation
symmetry [56]. It typically does not hold for any arbitrary state in the phase
space. When $(x_{1},x_{2},x_{3})=(1,1,1)$ we get back the standard
teleamplification scheme from Ref. [40].

The combined operation of $n$-photon subtraction followed by general
teleamplification, in the absence of any noise and when $g=1/\sqrt{T}$, is
$\propto\hat{a}^{(n+x_{1}+x_{2}+x_{3}-3)}$. This is simply a subtraction of
the $(n+x_{1}+x_{2}+x_{3}-3)$ number of photons from the input cat states.

Any arbitrary number of photon subtractions leads to a finite number of
possible outcomes when the qubit is encoded on cat states. We have
$\hat{a}^{n^{\prime}}\mapsto\hat{a}^{n^{\prime}({\rm mod}\ 4)}$ for any
$n^{\prime}$ for four-component cats. Therefore, the final state is only one
of four possible mixtures, those corresponding to
$n^{\prime}=n+x_{1}+x_{2}+x_{3}-3=0,\,1,\,2,\,{\rm and}\ 3~{}({\rm mod}~{}4)$
subtracted photons rather than infinitely many. All combinations of detected
photon numbers $(n,x_{1},x_{2},x_{3})$ such that $x_{j}\neq 0$ herald the same
mixture $\hat{\rho}_{n^{\prime}({\rm mod}\,4){\rm PS}}$, as depicted by Fig.
4.

Figure 4: The result of multiphoton subtraction and the general
teleamplification together on arbitrary encoded states give a finite number of
possible outcomes and corresponds to an effective $n^{\prime}({\rm mod}\,4)$
photon subtraction. The Wigner distributions of
$|\overline{+}\rangle=\mathcal{M}_{+}\sum_{k=0}^{3}\left|\mathrm{i}^{k}\alpha\right\rangle$
under different numbers of subtracted photons are shown here. It cycles
between 4 outcomes due to the discrete rotation symmetry of the code along
with the coherent state being the eigenstates of the photon-annihilation
operator. The final state is one of the four
$\hat{\rho}_{n^{\prime}({\rm{mod}}\ 4){\rm PS}}$ mixtures when losses are
considered. Error correction procedure is adapted to each of these outcomes
separately.

An amplitude-damping channel acting on the state $\propto
a|\overline{0}\rangle+b|\overline{1}\rangle$ modified by noiseless attenuation
by a factor $\sqrt{T}$ gives the mixture (using Eq. (13) of Ref. [57]),

$\displaystyle\begin{split}\widetilde{\rho}=&\,\widetilde{p}_{0}N_{0}^{(\sqrt{\gamma
T}\alpha)}\left(a|\widetilde{0}\rangle+b|\widetilde{1}\rangle\right)\times{\rm{H.c.}}+\widetilde{p}_{1}N_{1}^{(\sqrt{\gamma
T}\alpha)}\left(a|\widetilde{0}^{\prime}\rangle+\mathrm{i}b|\widetilde{1}^{\prime}\rangle\right)\times{\rm{H.c.}}\\\
+&\,\widetilde{p}_{2}N_{2}^{(\sqrt{\gamma
T}\alpha)}\left(a|\widetilde{0}\rangle-b|\widetilde{1}\rangle\right)\times{\rm{H.c.}}+\widetilde{p}_{3}N_{3}^{(\sqrt{\gamma
T}\alpha)}\left(a|\widetilde{0}^{\prime}\rangle-\mathrm{i}b|\widetilde{1}^{\prime}\rangle\right)\times{\rm{H.c.}}\end{split}$
(10)

where the states with tilde correspond to the final amplitude $\sqrt{\gamma
T}\alpha$ after the amplitude damping channel and we have defined the
normalizations,

$\displaystyle\begin{split}N_{0}^{(\alpha)}&=\frac{1}{\sqrt{1+2{\rm
Re}\left(ab^{*}\right){\cos{\left(\alpha^{2}\right)}}/{\cosh{\left(\alpha^{2}\right)}}}},N_{1}^{(\alpha)}=\frac{1}{\sqrt{1-2{\rm
Re}\left(ab^{*}\right){\sin{\left(\alpha^{2}\right)}}/{\sinh{\left(\alpha^{2}\right)}}}},\\\
N_{2}^{(\alpha)}&=\frac{1}{\sqrt{1-2{\rm
Re}\left(ab^{*}\right){\cos{\left(\alpha^{2}\right)}}/{\cosh{\left(\alpha^{2}\right)}}}},N_{3}^{(\alpha)}=\frac{1}{\sqrt{1+2{\rm
Re}\left(ab^{*}\right){\sin{\left(\alpha^{2}\right)}}/{\sinh{\left(\alpha^{2}\right)}}}}.\end{split}$
(11)

The weights of the mixture are given by

$\widetilde{p}_{k}=p_{k}N_{0}^{(\sqrt{T}\alpha)}/N_{k}^{(\sqrt{\gamma
T}\alpha)}$ (12)

for $k=0,1,2,3$, where

$\displaystyle\begin{split}p_{0}={\cosh\left(\gamma
T\alpha^{2}\right)}&\left\\{\cosh\left[\left(1-\gamma\right)T\alpha^{2}\right]+\cos\left[\left(1-\gamma\right)T\alpha^{2}\right]\right\\}/{2\cosh{\left(T\alpha^{2}\right)}},\\\
p_{1}={\sinh\left(\gamma
T\alpha^{2}\right)}&\left\\{\sinh\left[\left(1-\gamma\right)T\alpha^{2}\right]+\sin\left[\left(1-\gamma\right)T\alpha^{2}\right]\right\\}/{2\cosh{\left(T\alpha^{2}\right)}},\\\
p_{2}={\cosh\left(\gamma
T\alpha^{2}\right)}&\left\\{\cosh\left[\left(1-\gamma\right)T\alpha^{2}\right]-\cos\left[\left(1-\gamma\right)T\alpha^{2}\right]\right\\}/{2\cosh{\left(T\alpha^{2}\right)}},\\\
p_{3}={\sinh\left(\gamma
T\alpha^{2}\right)}&\left\\{\sinh\left[\left(1-\gamma\right)T\alpha^{2}\right]-\cos\left[\left(1-\gamma\right)T\alpha^{2}\right]\right\\}/{2\cosh{\left(T\alpha^{2}\right)}}.\end{split}$
(13)

To recover the quantum information by error correction, the following map is
provided by Ref. [57]. The states associated with even photon number parity,
those corresponding to the probabilities $\widetilde{p}_{0}$ and
$\widetilde{p}_{2}$, are mapped to the codespace as is, identically, and the
states associated with odd photon number parity, those corresponding to the
probabilities $\widetilde{p}_{1}$ and $\widetilde{p}_{3}$, are mapped to the
codespace after a parity flip and a phase rotation of $\pi/2$. The worst-case
fidelity over all input states was shown to be $\mathcal{F}_{\rm
w}=\min_{|a|,|b|=1/\sqrt{2}}(\widetilde{p}_{0}+\widetilde{p}_{1})$ [57].
However, this mapping defines a particular recovery, which may generally be
suboptimal. Optimal recoveries are discussed in Sec. 4.

###  3.2 With imperfect detections

Detector efficiency plays a significant role in any conditional measurement.
In practice, superconducting-nanowire single-photon detectors with system
detection efficiency $>98\%$ and a small dark count rate of $10^{-4}$ Hz have
been realized [58]. More ubiquitous single-photon avalanche diodes offer
modest detection with efficiencies $\sim 50\%$, for reasonably low dark count
rates of $\sim 10^{2}$ Hz [59, 16]. In all of our discussions in this paper,
we assume a negligible dark count rate for convenience and simplicity.

The composition of encoding ($\mathcal{C}$), photon subtraction
($\mathcal{Z}$), loss ($\mathcal{E}_{\gamma}$), teleamplification
($\mathcal{T}$), recovery ($\mathcal{R}$), and decoding ($\mathcal{C}^{-1}$)
is the channel

$\mathcal{E}=\mathcal{C}^{-1}\circ\mathcal{R}\circ\left(\mathcal{T}\circ\mathcal{E}_{\gamma}\circ\mathcal{Z}\right)\circ\mathcal{C}.$
(14)

Assume that the encoding, recovery, and decoding are ideal and are implemented
instantaneously.

To simplify the analysis, the combined loss suppression setup can be recast
into an equivalent one as follows: (1) noiseless amplification by a factor
$T_{\rm{eff.}}$ followed by (2) ideal subtraction of $(n+x_{1}+x_{2}+x_{3}-3)$
photons, and finally (3) effective loss $(1-\gamma_{\rm{eff.}})$. The losses
arising from detector inefficiencies and the loss channel itself are combined
into an effective loss channel ($\mathcal{E}_{\gamma_{\rm{eff.}}}$) and,
similarly, the attenuation that comes from photon subtraction and
amplification from the teleamplification is combined into an effective
noiseless rescaling ($\mathcal{T_{\rm{eff.}}}$) so that we have

$\mathcal{E}=\mathcal{C}^{-1}\circ\mathcal{R}\circ\left(\mathcal{E}_{\gamma_{\rm{eff.}}}\circ\mathcal{T_{\rm{eff.}}}\right)\circ\mathcal{C}.$
(15)

The effective channels are described by the parameters given in Eq. (33). This
leads to the same state as the original scheme (Appendix A). However, the
success probabilities of heralding in the original setup cannot be calculated
with this equivalent setup.

##  4 Optimizing the recovery channel

Here we present the procedure for optimizing the worst-case-fidelities of the
recovered qubit over completely-positive trace-preserving (CPTP) recovery
maps. Instead of encoding on the $(\overline{0},\overline{1})$ basis presented
in Eq. (2) it is more convenient to use the physical states
$(\overline{+},\overline{\vphantom{+}-})$ to encode the states of the
computational basis $\left|\mu_{\rm{comp.}}\right\rangle$ using a channel
$\mathcal{C}=C\bullet C^{\dagger}$ where
$C=\left|\overline{+}_{\rm{cat}}\middle\rangle\middle\langle
0_{\rm{comp.}}\right|+\left|\overline{\vphantom{+}-}_{\rm{cat}}\middle\rangle\middle\langle
1_{\rm{comp.}}\right|.$ The subscripts will be suppressed going forward.

The states $|\overline{0}\rangle$ and $|\overline{1}\rangle$ and similarly
$|\overline{0}^{\prime}\rangle$ and $|\overline{1}^{\prime}\rangle$ are not
orthogonal to each other, while the states
$\propto|\overline{0}\rangle\pm|\overline{1}\rangle$ and
$\propto|\overline{0}^{\prime}\rangle\pm|\overline{1}^{\prime}\rangle$ are all
orthogonal, making them attractive choices for a basis. However, the qubits
encoded on these suffer from distortions under noiseless attenuation or
amplification. Consider the encoded state
$|\overline{\psi}\rangle=a|\overline{+}\rangle+b|\overline{\vphantom{+}-}\rangle$
which, after noiseless attenuation (with ZPS) or amplification, turns into

$\propto
a\frac{\mathcal{M}_{+}}{\widetilde{\mathcal{M}}_{+}}|\widetilde{+}\rangle+b\frac{\mathcal{M}_{-}}{\widetilde{\mathcal{M}}_{-}}|\widetilde{\vphantom{+}-}\rangle,$
(16)

where the quantities with tilde correspond to the modified coherent-state
amplitudes.

This shows a redistribution of the coefficients that distorts the qubits. The
final state is pure, assuming that all operations are ideal, and simply
differs from the original one by a rotation on the Bloch sphere. If the qubit
is perfectly known initially, then this distortion can be undone. However, for
an unknown arbitrary input qubit, this is a nonunitary transformation.

This problem is not unique to codes modified by noiseless attenuation. When
the encoded qubits pass through some lossy channel, their coherent-state
amplitudes decrease, also giving rise to this problem. This needs to be
addressed by a suitable amplification that compensates for this distortion, as
explained in Sec. 2.2.

Now we describe constructing the basis operators and the corresponding process
and Choi matrices for the case of $n^{\prime}(\textrm{mod}\,4)=0,2$, which are
effective ZPS and 2PS, respectively, during the operation $\mathcal{T}_{\rm
eff.}$. Similar constructions are carried out for the other two cases.

###  4.1 Orthonormal basis operators for the noise and recovery channels

The choice of states of $(\overline{+},\overline{\vphantom{+}-})$ as a basis
can be used to construct an orthonormal set of basis operators
$\\{\hat{B}_{i}\\}$ for the Kraus operators of the recovery channel as

$\displaystyle\hat{B}_{0e}=\Big{(}$
$\displaystyle|\overline{+}\rangle\langle\widetilde{+}|$ $\displaystyle+$
$\displaystyle|\overline{\vphantom{+}-}\rangle\langle\widetilde{\vphantom{+}-}|~{}\Big{)}\Big{/}{\sqrt{2}}\,,\,$
$\displaystyle\hat{B}_{1e}=\Big{(}$
$\displaystyle|\overline{\vphantom{+}-}\rangle\langle\widetilde{+}|$
$\displaystyle+$
$\displaystyle|\overline{+}\rangle\langle\widetilde{\vphantom{+}-}|~{}\Big{)}\Big{/}{\sqrt{2}}\,,$
$\displaystyle\hat{B}_{2e}=\mathrm{i}\Big{(}$
$\displaystyle|\overline{\vphantom{+}-}\rangle\langle\widetilde{+}|$
$\displaystyle-$
$\displaystyle|\overline{+}\rangle\langle\widetilde{\vphantom{+}-}|~{}\Big{)}\Big{/}\sqrt{2}\,,\quad$
$\displaystyle\hat{B}_{3e}=\Big{(}$
$\displaystyle|\overline{+}\rangle\langle\widetilde{+}|$ $\displaystyle-$
$\displaystyle|\overline{\vphantom{+}-}\rangle\langle\widetilde{\vphantom{+}-}|~{}\Big{)}\Big{/}\sqrt{2}\,,$
(17)

that project states in the errorspace formed as a consequence of even-numbered
photon annihilations after the effective loss channel to the codespace and
similarly defined operators $\\{\hat{B}_{jo}\\}$ using the states
$|\widetilde{\pm}^{\prime}\rangle$, rather than $|\widetilde{\pm}\rangle$, do
the same for the states after odd-numbered photon annihilations.

The quantities with tilde represent the coherent-state amplitude just after
the effective loss channel, whereas those with a bar represent the restored
amplitudes. The basis operators for the loss channel can have the same form as
the Hermitian conjugates to these operators, but with the tilde representing
the coherent-state amplitude after the loss channel and the bar representing
the amplitude after effective noiseless rescaling, $\mathcal{T}_{\rm eff.}$.

With this operator basis, we can now create the process [60] and Choi [61]
matrices of the loss and recovery channels.

###  4.2 Process and Choi matrices

Let ${\hat{R}_{r}}$ and ${\hat{E}_{k}}$ be the Kraus operators of the recovery
and effective loss channels respectively. The recovery process matrix is
defined by elements $X_{ij}=\sum_{r}{\chi_{ri}\chi^{*}_{rj}}$ such that
$\hat{R}_{r}=\sum_{l}\chi_{rl}\hat{B}_{l}$ for the set of basis operators
$\\{\hat{B}_{l}\\}$, and the process matrix for the effective loss channel is
defined by elements $W_{ij}=\sum_{k}\omega_{ki}\omega_{kj}^{*}$ such that
$\hat{E}_{k}=\sum_{l}\omega_{kl}\hat{B}_{l}^{\dagger}$. Both $X$ and $W$ are
Hermitian matrices and the index $k$ runs over all the Kraus operators of the
channel, corresponding to different numbers of photon losses, whereas the
index $i$ runs over all the basis operators.

We can conveniently arrange all the terms, $\omega_{ki}$, into different
matrices of a block-diagonal form $w_{n}={\rm diag}(\omega_{e},\omega_{o}),$
for $n=0,1,2,\dots$, such that the rows correspond to only four Kraus
operators for any given $n$,
$\\{\hat{E}_{4n},\hat{E}_{4n+2},\hat{E}_{4n+1},\hat{E}_{4n+3}\\}$, and the
columns correspond to all the basis operators, $\\{\hat{B}_{i}^{\dagger}\\}$,
forming the matrix elements, $[w_{n}]_{ki}={\rm
tr}\\{\hat{E}_{k}\hat{B}_{i}\\}$.

Such a construction gives

$\displaystyle\omega_{e}=\begin{pmatrix}\omega_{4n,0e}&0&0&\omega_{4n,3e}\\\
0&\omega_{4n+2,1e}&\omega_{4n+2,2e}&0\end{pmatrix},$ (18)

with

$\displaystyle\begin{split}&\omega_{4n,0e}=\frac{1}{\sqrt{2}}\langle
4n|\sqrt{1-\gamma}\alpha\rangle\left(\frac{\mathcal{M}_{+}}{\widetilde{\mathcal{M}}_{+}}+\frac{\mathcal{M}_{-}}{\widetilde{\mathcal{M}}_{-}}\right),\,\omega_{4n+2,1e}=\frac{1}{\sqrt{2}}\langle
4n+2|\sqrt{1-\gamma}\alpha\rangle\left(\frac{\mathcal{M}_{-}}{\widetilde{\mathcal{M}}_{+}}+\frac{\mathcal{M}_{+}}{\widetilde{\mathcal{M}}_{-}}\right),\\\
&\omega_{4n,3e}=\frac{1}{\sqrt{2}}\langle
4n|\sqrt{1-\gamma}\alpha\rangle\left(\frac{\mathcal{M}_{+}}{\widetilde{\mathcal{M}}_{+}}-\frac{\mathcal{M}_{-}}{\widetilde{\mathcal{M}}_{-}}\right),\,\omega_{4n+2,2e}=\frac{1}{\sqrt{2}}\langle
4n+2|\sqrt{1-\gamma}\alpha\rangle\left(\mathrm{i}\frac{\mathcal{M}_{-}}{\widetilde{\mathcal{M}}_{+}}-\mathrm{i}\frac{\mathcal{M}_{+}}{\widetilde{\mathcal{M}}_{-}}\right).\end{split}$
(19)

The matrix $\omega_{o}$ is similarly

$\displaystyle\omega_{o}=\begin{pmatrix}\omega_{4n+1,0o}&0&0&\omega_{4n+1,3o}\\\
0&\omega_{4n+3,1o}&\omega_{4n+3,2o}&0\end{pmatrix},$ (20)

with $\omega_{k,jo}$ containing $\widetilde{\mathcal{M}}^{{}^{\prime}}_{\pm}$
rather than $\widetilde{\mathcal{M}}_{\pm}$ in the denominators.

The process matrix is given by the elements

$\displaystyle
W_{ij}=\sum_{k=0}^{\infty}{\omega_{k,i}\omega^{*}_{k,j}}=\left(\sum_{n=0}^{\infty}w^{\intercal}_{n}w^{*}_{n}\right)_{ij}=\left(\sum_{n=0}^{\infty}w^{\dagger}_{n}w_{n}\right)_{ji}\cdotp$
(21)

We may truncate the sum to some reasonable value as an approximation while
using the Fock basis. However, the current problem has a closed analytical
form. The accompanying mathematica files contain the expressions.

The process matrix $X$ can be easily transformed into the corresponding Choi
matrix representation $J_{X}$ as[62],

$J_{X}=\sum_{kk^{\prime}}|{B}_{k}\rangle\negthickspace\rangle
X_{kk^{\prime}}\langle\negthickspace\langle{B}_{k^{\prime}}|,$ (22)

where $|\bullet\rangle\mkern-5.0mu\rangle$ (or ${\rm{vec}}[\bullet]$) is the
vectorization of matrices [63, 64] in the computational basis with the column-
stacking convention being used here. The Choi matrix $J_{W}$ is constructed in
a similar manner.

###  4.3 The algorithm

We now outline a gradient ascent algorithm to find the optimal worst-case
fidelity,

$\mathcal{F}_{\rm w}^{\rm
max}=\max_{\mathcal{R}}\min_{\left|\psi\right\rangle}{\rm{tr}}\left\\{|\psi\rangle\langle\psi|\mathcal{E}\left[|\psi\rangle\langle\psi|\right]\right\\},$
(23)

and the corresponding recovery with the tools prepared in the previous
sections as follows.

First, the optimal Choi matrix, $\mathcal{R}$, is initiated as a random CPTP
matrix (Appendix C.1) and then:

  1. 1.

The pure state $\left|\psi\right\rangle$ that minimizes the fidelity is found.
In matlab, this is easily done using the fmincon function.

  2. 2.

The gradient in $\mathcal{R}$ is calculated for the state
$\left|\psi\right\rangle$ (Appendix C.2).

  3. 3.

A small CPTP-constrained step $\delta\mathcal{R}$ (Appendix C.3) is taken in
the direction of the steepest gradient ascent (Appendix C.4).

These steps are repeated with a decreasing stepsize until the fidelity becomes
constant up to some tolerance or for a sufficiently large number of steps.

The domain is the surface of the Bloch sphere, which makes the nonconvex and
precludes the use of semi-defintie programs [2, 23] in this context. The
nonconvex optimization for a global optimal value is performed by repeating
the process with different initial conditions and taking the best values among
the results.

##  5 Results

This section compiles the results from the methods discussed in the previous
sections, while highlighting the key findings of the work.

The worst-case fidelity for the qubits encoded on
$(\overline{0},\overline{1})$ basis, $\mathcal{F}_{\rm w}$ due to Ref. [57],
which is modified under loss suppression is compared to the corresponding
worst-case fidelity under teleamplification alone in Fig. 5. It clearly
illustrates the benefit of the noiseless loss suppression scheme with ZPS and
standard teleamplification as a viable method of error suppression for various
system parameters and across the different values of the loss rates for a
realistic value of detector efficiency.

Figure 5: The performance of four-component cat code through a loss channel
followed by amplitude restoration by realistic, noisy teleamplification, with
(blue) and without (red) noisy ZPS. The back action of ZPS suppresses
uncorrectable errors in the encoded qubit for a wide range of parameters. For
each graph, only one parameter is varied while the fixed parameters take the
values: $\alpha=2$, $T=50\%$. The detection efficiency is fixed at
$\eta=95\%$. Figure 6: (a) Worst-case fidelity (over all pure encoded states)
for $(\overline{0},\overline{1})$ encoding with the recovery map from Ref.
[57] (dotted) and $(\overline{+},\overline{\vphantom{+}-})$ encoding with
numerically optimized recovery maps (dashed) versus the loss rate for
$T=35\%$. The choice of $T$ is motivated by (b) optimized worst-case fidelity
(blue) and the minimum success probability (red) plotted against $T$ for the
loss rate of $10\%$ with $\eta=90\%$ (dotted), the loss rate of $5\%$ with
$\eta=90\%$ (dashed), and the loss rate of $10\%$ with $\eta=95\%$ (solid) and
where the solid gray line represents the worst-case fidelity benchmark of
$93.5\%$ resulting from Ref. [57] wherein ideal amplification using
teleportation after recovery is considered instead of noisy teleamplification
before recovery. Figure 7: Optimized worst-case fidelities for (a)
$\alpha=\sqrt{2}$ and (b) $\alpha=2$. The worst-case fidelity due to Ref. [57]
is shown by the horizontal gray line. Minimum success probabilities for (c)
$\alpha=\sqrt{2}$ and (d) $\alpha=2$. The overall minimum success probability
of accepting up to 10 photons in each of the relevant output modes and
adapting the recoveries appropriately is depicted in black. These are plotted
against the beamsplitter transmittance $T$ for a loss rate of $10\%$ and a
detector efficiency of $\eta=95\%$. The results for effective ZPS, 1PS, 2PS,
and 3PS are shown in blue, green, orange, and red, respectively. For
$\alpha=\sqrt{2}$, ZPS outperforms the others in terms of the fidelity. On the
other hand, sufficiently large initial and final amplitudes such as
$\alpha\gtrsim 2$, the values for any multiphoton subtraction lie close to
each other. Figure 8: The minimum success probability considering subtractions
of up to 10 photons for noiseless attenuation and up to 10 photons in each of
the relevant ancilla modes for teleamplification is plotted in green (dashed).
The solid black curve is the same as in Fig. 7(d) and corresponds to accepting
up to 10 photons for noiseless attenuation but with teleamplification heralded
only on the coincident measurement of single photons in the relevant modes.
All parameters are the same as in Fig. 7(b,d). The large improvement in the
probability of success is because we accept a larger number of compatible
outcomes to herald the outcome and adapt the recoveries accordingly.

Attenuating the coherent-state amplitudes in encoded qubits also reduces
uncorrectable errors caused by detector inefficiencies during
teleamplification. For smaller coherent-state amplitudes, the fidelity without
suppression has higher values, as expected. However, a smaller amplitude does
not benefit quantum information tasks in later applications. Noiseless
attenuation is directly controlled by the beamsplitter transmittance, and
lower values lead to a dramatic increase in fidelity even if start with a
large amplitude. Any nonzero amplitude can be amplified during the noiseless
amplification stage of the scheme to produce a well-defined outcome.

The recovery operations used in Fig. 5 are suboptimal when channel losses are
large. This is seen by the large difference among the plots (dashed-red and
dotted-red) for when noiseless attenuation is absent, in Fig. 6(b). However,
noiseless attenuation before loss still shows a significant improvement for
realistic system parameters for such large channel losses, even after
considering optimal recoveries. Note that for optimizing the recoveries, we
transition to using the $(\overline{+},\overline{\vphantom{+}-})$ basis for
its operational convenience, as explained in Sec. 4.

As we saw earlier, the greatest improvement in fidelity occurs when the
beamsplitter transmittance is chosen near zero. However, Fig. 6(b) shows that
it comes at the expense of the probability of success. A benchmark in fidelity
of about $93.5\%$ is set by the setup without any noisy teleamplification
while also considering perfect amplification after recovery as in Ref. [57].

To go beyond this benchmark, the minimum success probability (over all input
pure states) takes the value of $0.08\%$ for cat codes with an initial
coherent-state amplitude $\alpha=2$, detector efficiency of $95\%$, and the
environmental-loss rate of $10\%$. This occurs for beamsplitter transmittance
of about $T=35\%$ used for the photon subtraction.

The worst-case fidelity naturally does not overcome such a threshold even if
it is combined with a smaller overall channel loss for sufficiently small
detector efficiency. This is explained by the high additional noise from
inefficiencies while trying to address the noise from the channel loss.

The success probability of the procedure is improved by accepting the
subtractions of arbitrary $n$ photons from the encoded qubits as shown by the
Figs. 7(c, d). In the $(\overline{+},\overline{\vphantom{+}-})$ basis, the
distortion of the qubit due to redistribution of the probability amplitudes is
undone when we use ZPS with standard teleamplification. However, when we
accept any arbitrary number of photon subtraction, it can cause a nonunitary
distortion even after restoring the coherent-state amplitudes. This becomes
prominent when the initial coherent-state amplitude is small and the
differences in the worst-case fidelities are seen clearly in Fig. 7(a) for
$\alpha=\sqrt{2}$. The optimized CPTP recovery channels cannot restore a
distorted qubit with high fidelity when $n\,(\textrm{mod}\,4)\neq 0$.

On the other hand, for sufficiently large initial (and therefore also the
final) coherent-state amplitudes $\alpha\gtrsim 2$, the fidelities under
different amounts of photon subtraction are roughly the same, as the
distortions are neglible, as seen by comparing the curves in Fig. 7(b).
Therefore, all arbitrary photon subtraction can be accepted to increase the
success probability to about $0.94\%$.

The results discussed have been restricted to standard teleamplification so
far. However, all combinations of $x_{k}\neq 0$ can be accepted for large cat
states with large coherent-state amplitudes and recovery is appropriately
adapted to further increase the success rate. These combinations also herald a
vacuum in mode $A$, and due to the symmetry of the encoding, we obtain one of
the same four plots for the worst-case fidelities for all four cases
corresponding to $n^{\prime}=(n+x_{1}+x_{2}+x_{3}-3)(\textrm{mod}\,4)$ shown
by Fig. 7.

Accepting up to 10 photons in each relevant mode such that $x_{k}\neq 0$
increases the previously discussed success probability to about $3.42\%$, as
illustrated by Fig. 8. With this sufficiently large success rate the procedure
becomes a promising alternative for error suppression for passive photon
losses.

Note that all the outcomes with up to 7 photons detected, including in the
mode $A$, constitute more than $99.6\%$ of all possible outcomes for the
$|\overline{+}\rangle$ state as an example, making it a good approximation to
cutoff at 10 photons.

##  6 Discussion

In this paper, we analyze noiseless loss suppression in the context of error-
correcting codes. In particular, we examine the four-component cat codes and
how well they can be protected against uncorrectable errors resulting from
decoherence from an amplitude-damping channel prior to error correction. This
is typically known as error suppression.

Sufficiently large amplitude cat states such as those with the coherent-state
amplitude of 2, can be error corrected using noiseless recovery maps to a
worst-case fidelity of about $93.5\%$ under a significantly large amplitude-
damping channel of $10\%$. We show worst-case fidelities beyond that which can
be achieved solely by error correction, using realistic and therefore noisy
detectors at the cost of success probability of about $3\%$.

Qubits encoded on the cat codes require $\alpha\gtrsim 2$ coherent-state
amplitude for efficient quantum computation tasks and the amplitude-damping
channel reduces this amplitude making noise-free amplification essential to
amplitude restoration. However, large-amplitude coherent states at the input
are disturbed more by the lossy channel as they carry more energy, and so it
is reasonable that noiselessly attenuating the signal before any transmission
through the lossy channel reduces the effects of decoherence and therefore,
also suppresses uncorrectable logical errors on the encoded states.

However, noiseless attenuation and amplification required for such loss
suppression are implemented nondeterministically with photon subtraction and
teleamplification, respectively, for the cat states. These processes require
conditional measurements with photon-number-resolving detectors which
introduces further noise into the system. Therefore, the impact of these on
the effectiveness of the overall scheme needs to be assessed with realistic
values of detector efficiency. We found that with currently available
superconducting-nanowire detectors, noiselesss loss suppression can lead to
substantial gains in optimized worst-case fidelity over all the input pure
states.

The ability to protect large cat states under large losses is quite useful for
quantum information applications in the near term when such losses before
error correction are unavoidable in a physical realization even if we assume
noise-free encoding, error correction, and decoding procedures. Near-term
applications include storing photonic qubits in a cavity or long fiber loops,
and for direct communication of encoded qubits over long distances as a
stepping-stone towards distributed quantum computing [43].

For the future outlook, it may be interesting to extend the analysis presented
here to cat codes with more than four components, such as in Refs. [57, 42],
arbitrary rotation symmetric codes [56, 29] and translation symmetric codes
such as GKP codes [8]. Similar preemptive measures could be sought for other
relevant noise models, like the dephasing channel, such as in Refs. [25, 26]
that can affect the photonic qubits.

It may be interesting to study loss suppression on arbitrary nongaussian,
nonclassical states with other nongaussian conditional measurements photon
catalysis, for example, that have been useful in quantum key distribution [65,
66, 67, 68, 69]. More generally, better error suppression techniques can be
explored for arbitrary noise channels on diverse platforms [29]. Finally, the
nonconvex optimization algorithms used to find the maximum worst-case fidelity
may be improved.

Acknowledgments S.U. Shringarpure acknowledges insightful conversations with
Jaehak Lee and Soumyakanti Bose. The authors acknowledge support from the
National Research Foundation of Korea (NRF) grants funded by the Korean
government (Grant Nos. NRF2020R1A2C1008609, 2023R1A2C1006115,
NRF2022M3E4A1076099 and RS5 2023-00237959) via the Institute of Applied
Physics at Seoul National University, the Institute of Information &
Communications Technology Planning & Evaluation (IITP) grant funded by the
Korea government (MSIT) (IITP-2021-0-01059 and IITP-20232020-0-01606), and the
Brain Korea21 FOUR Project grant funded by the Korean Ministry of Education.

Disclosures The authors declare no conflicts of interest.

Data availability No data were generated or analyzed in the presented
research. The results in this paper can be reproduced using the files in
github.com/saurabh-shringarpure/nps-teleamp.

##  Appendix A Equivalent setups for noisy multiphoton subtraction,
teleamplification and their combination

###  A.1 Multiphoton subtraction

Noisy $n$-photon subtraction with a beamsplitter of transmittance $T$
generates,

$\displaystyle\begin{split}&e^{-\frac{\left|\alpha\right|^{2}\eta
R}{2}}\frac{\left(\sqrt{\eta
R}\alpha\right)^{n}}{\sqrt{n!}}\sum_{k=0}^{3}{c_{k}u^{nk}\left|\sqrt{T}\alpha_{k}\right\rangle\left|\sqrt{(1-\eta)R}\alpha_{k}\right\rangle_{\rm{env.}}}.\end{split}$
(24)

Alternatively, consider a setup where the encoded state first experiences
noiseless amplification by $\mu^{\hat{n}}$ followed by an ideal $n$-photon
subtraction with the same beamsplitter and lastly a loss of
$(1-\eta^{\prime})$ to give

$e^{-\frac{\left|\alpha\right|^{2}(1-\mu^{2}T)}{2}}\frac{\left(\sqrt{R\mu^{2}}\alpha\right)^{n}}{\sqrt{n!}}\sum_{k=0}^{3}c_{k}u^{nk}\left|\sqrt{\eta^{\prime}T\mu^{2}}\alpha_{k}\right\rangle\left|\sqrt{(1-\eta^{\prime})T\mu^{2}}\alpha_{k}\right\rangle_{\rm{env.}}.$
(25)

Inspection shows that the two output states become equivalent, up to a factor,
when $\mu={1}/{\sqrt{\eta^{\prime}}}$ and $\eta^{\prime}={T}/\left(1-\eta
R\right)$. The attenuation is the same as for ZPS. The difference is only in
the prefactor and the phases $u^{nk}$ attached to the various coherent-state
amplitudes.

If the output state of a noisy ZPS passes through an amplitude-damping channel
with transmittance $\gamma$ the effective transmission becomes the product
$\eta^{\prime}\gamma$. Therefore, we can analyze the performance of cat codes
modified by noisy ZPS by using Eq. (10) and these modified values of the
initial amplitude and channel transmittance.

###  A.2 Teleamplification

The result of non-ideal teleamplification assuming that we have detectors with
efficiency $\eta$ that measure, say, $(x_{1},x_{2},x_{3})$ number of photons
in modes $C$, $A^{\prime}$ and $C^{\prime}$, and a loss $R_{1}$ after the
preparation of the input state, as shown in Fig. 9(c), is straightforward with
an unchanged $g$ and $\beta=\sqrt{{(1-R_{1})}/{R_{B}}}\alpha.$

Equation (7) when $j=k$ is extended as

$\displaystyle\begin{split}&\mathcal{M}_{+}^{{}^{\prime}}\left|0\right\rangle_{A}\sum_{k=0}^{3}c_{k}u^{-3k}\left|g\sqrt{1-R_{1}}\alpha_{k}\right\rangle_{B}\left|-\sqrt{R_{1}}\alpha_{k}\right\rangle_{1}\left|-\sqrt{\eta(1-R_{1})}\alpha_{k}\right\rangle_{C}\left|\sqrt{(1-\eta)(1-R_{1})}\alpha_{k}\right\rangle_{c}\\\
&\left|\sqrt{\frac{\eta}{2}(1-R_{1})}\alpha_{k}\right\rangle_{A^{\prime}}\left|-\sqrt{\frac{1-\eta}{2}(1-R_{1})}\alpha_{k}\right\rangle_{a^{\prime}}\left|\sqrt{\frac{\eta}{2}(1-R_{1})}\alpha_{k}\right\rangle_{C^{\prime}}\left|-\sqrt{\frac{1-\eta}{2}(1-R_{1})}\alpha_{k}\right\rangle_{c^{\prime}},\end{split}$
(26)

where the additional modes $1$, $c$, $a^{\prime}$ and $c^{\prime}$ are lost to
the environment and ideal projections onto $(x_{1},x_{2},x_{3})$ Fock states
in the modes $C$, $A^{\prime}$ and $C^{\prime}$ can be assumed.

We can also combine all losses into a single mode, $2$, as shown in Fig. 9(d),
and measure the modes $A$, $C$, $A^{\prime}$, and $C^{\prime}$, such that the
unnormalized state is given by

$\displaystyle\begin{split}&\mathcal{M}_{+}^{{}^{\prime}}e^{-\eta(1-R_{1})\alpha^{2}}\frac{\left(\sqrt{\eta(1-R_{1})}\alpha\right)^{x_{1}+x_{2}+x_{3}}}{\sqrt{x_{1}!x_{2}!x_{3}!2^{x_{2}+x_{3}}}}\times\\\
&\sum_{k=0}^{3}c_{k}u^{k(x_{1}+x_{2}+x_{3}-3)}\left|-\sqrt{R_{1}}\alpha_{k}\right\rangle_{1}\left|\sqrt{1-R_{2}}g^{\prime}\sqrt{1-R_{1}}\alpha_{k}\right\rangle_{B}\left|-\sqrt{R_{2}}g^{\prime}\sqrt{1-R_{1}}\alpha_{k}\right\rangle_{2},\end{split}$
(27)

where

$\displaystyle g^{\prime}=\frac{g}{\sqrt{1-R_{2}}}\ {\rm{and}}\
R_{2}=\frac{2(1-\eta)}{g^{2}+2(1-\eta)}.$ (28)

Figure 9: An inefficient detector is modeled by placing a beamsplitter with
transmittance $\eta$ before a perfect detector. (a) Noisy $n$-photon
subtraction with a photon-number-resolving detector of efficiency $\eta$ is
equivalent, up to a factor, to (b) a noiseless preamplification by
$1/\sqrt{\eta^{\prime}}$, noiseless attenuation and $n$-photon subtraction
followed by a loss of $1-\eta^{\prime}$. Similarly, (c) noisy
teleamplification is equivalent, up to a constant factor, to (d) a noiseless
amplification by $g^{\prime}=g/\sqrt{1-R_{2}}$ followed by some loss $R_{2}$
and the subtraction of $(x_{1}+x_{2}+x_{3}-3)$ photons. The input loss denoted
by $R_{1}$ is unchanged. (e) The setup with input losses before amplification
can be made again equivalent to photon subtraction followed by amplification
by $g^{\prime\prime}$, and the losses $R_{1}^{\prime}$ and $R_{2}$. Such
manipulations are possible due to cat encoding of the input states.

###  A.3 The combined setup

The output of the ideal attenuation, from Fig. 9(b), can be taken as input to
the teleamplification in Fig. 9(c), and the effective losses during and after
$n$-photon subtraction can be represented by input losses with substitution
$R_{1}\mapsto 1-\eta^{\prime}\gamma.$

Furthermore, consider a situation where these input losses and the noiseless
amplification are switched in the order shown in Fig. 9(e) for which we need
solutions to

$\displaystyle\begin{split}\sqrt{R_{1}}=g^{\prime\prime}\sqrt{R_{1}^{\prime}}\
{\rm and}\
g^{\prime}\sqrt{1-R_{1}}=g^{\prime\prime}\sqrt{1-R_{1}^{\prime}},\end{split}$
(29)

which are

$\displaystyle\begin{split}g^{\prime\prime}=\sqrt{(1-R_{1})g^{\prime
2}+R_{1}}\ {\rm and}\ R_{1}^{\prime}=\frac{R_{1}}{(1-R_{1})g^{\prime
2}+R_{1}}.\end{split}$ (30)

We have a constraint,

$\sqrt{{T}/{\eta^{\prime}}}(g^{\prime\prime}){\sqrt{(1-R_{1}^{\prime})(1-R_{2})}}=1,$
(31)

to restore the coherent-state amplitude of the encoded qubit which gives the
equation,

$\displaystyle\begin{split}g^{6}&(\gamma T)^{2}+g^{4}\gamma T\left\\{\gamma
T-\eta\left[(2\gamma-1)T+1\right]\right\\}+\\\
g^{2}&\left\\{\eta-T\left[\eta+\gamma(1-2\eta)\right]-1\right\\}+2\left[1-\eta\right]\left[\gamma
T+\eta(1-T)-1\right]=0.\end{split}$ (32)

It is easily solved for real-valued $g\geq 1$ in mathematica. The final state
is then simply given by Eq. (10) with substitutions

$T\mapsto T_{\rm{eff.}}={Tg^{\prime\prime
2}}/{\eta^{\prime}},\,\gamma\mapsto\gamma_{\rm{eff.}}=(1-R_{1}^{\prime})(1-R_{2}),$
(33)

and the attenuation with $n$-photon subtraction and and a general
teleamplification with heralding on $(x_{1},x_{2},x_{3})$ in the modes $C$,
$A^{\prime}$, and $C^{\prime}$ can be put together in a complete setup.

The success probability, for an input
$\sum_{k=0}^{3}c_{k}\left|\alpha_{k}\right\rangle$ becomes

$\displaystyle\begin{split}P_{\rm{s}}=&\left[\mathcal{M}_{+}^{{}^{\prime}}e^{-\eta\left(\frac{R}{2}+\gamma
T\right)\alpha^{2}}\frac{\left(\sqrt{\eta
R}\alpha\right)^{n}\left(\sqrt{\eta\gamma
T}\alpha\right)^{x_{1}+x_{2}+x_{3}}}{\sqrt{n!x_{1}!x_{2}!x_{3}!}\sqrt{2}^{x_{2}+x_{3}}}\right]^{2}\times\\\
&\sum_{j,k=0}^{3}c_{j}^{*}u^{(-j+k)(n+x_{1}+x_{2}+x_{3}-3)}c_{k}e^{-\alpha^{2}\left[1-\eta
R+\gamma
T\left(1+g^{2}-2\eta\right)\right]\left(1-u^{-j+k}\right)},\end{split}$ (34)

where $\mathcal{M}_{+}^{{}^{\prime}}$ is a function of the coherent-state
amplitude $\beta=\sqrt{(1+g^{2})\gamma T}\alpha$.

##  Appendix B Alternate setup for amplitude restoration

In the main text, we considered the use of teleamplification to restore the
coherent-state amplitude after the loss channel. Alternatively, the loss
channel may be placed within the teleamplification setup, in the mode that
contains one part of the entangled resource state. This is the section of mode
$C$ in Fig. 2 between modes $A$ and $B$.

Figure 10: An alternate approach to restore the inevitable drop in coherent-
state amplitude due to the loss channel. Half of the entangled resource state
is sent through the loss channel to interact with the encoded qubit.

Assume perfect detectors and no ZPS for simplicity to compare the performance
of this alternate setup to the original one in Fig. 2. Losses occur only
through the loss channel with transmittance $\gamma$. Following the procedure
described before, we get the heralded state

$\displaystyle\begin{split}\sim&\sum_{k=0}^{3}c_{k}\left|g_{\rm{alt.}}\alpha_{k}\right\rangle_{B}\left|\sqrt{\frac{1-\gamma}{\gamma}}\alpha_{k}\right\rangle_{\rm{env.}},\end{split}$
(35)

where $g_{\rm{alt.}}=1=\sqrt{{T_{B}}/{R_{B}\gamma}}$ as opposed to the
heralded state in the original setup

$\displaystyle\begin{split}\sim\sum_{k=0}^{3}c_{k}\left|g_{\rm{ori.}}\sqrt{\gamma}\alpha_{k}\right\rangle_{B}\left|-\sqrt{1-\gamma}\alpha_{k}\right\rangle_{\rm{env.}}\end{split}$
(36)

where $g_{\rm{ori.}}={1}/{\sqrt{\gamma}}=\sqrt{{T_{B}}/{R_{B}}}.$

The alternate setup demands higher values of the gain from teleamplification
to restore the coherent-state amplitude as the losses grow. This is due to the
factor of $\gamma$ in the denominator.

It in turn leads to the demand for a larger coherent-state amplitude in the
resource state. This amplitude has values
$\beta_{\rm{alt.}}=\sqrt{{1}/{R_{B}\gamma}}\alpha$ and
$\beta_{\rm{ori.}}=\sqrt{{\gamma}/{R_{B}}}\alpha,$ in the two setups.

Compare the intensities lost to the environment in the two cases corresponding
to any $\alpha_{k}$. For the original setup, it is $(1-\gamma)\alpha^{2}$,
which grows linearly with the losses. On the other hand, it is
$(1-\gamma)\alpha^{2}/\gamma$ for the alternate one and blows up as the losses
increase.

Therefore, the original setup used in the main text is more resilient to the
loss channel. The loss of entanglement in the resource along with the
requirement of a larger coherent-state amplitude for the resource state
negatively impacts this setup more, reducing coherence.

##  Appendix C Numerical optimization

###  C.1 CP, TP, and CPTP projections in terms of the Choi matrix for
initiating the optimization

Initiating the optimization requires a random, completely-positive, trace-
preserving (CPTP) matrix. This is constructed from a random full-rank seed
matrix that is subjected to a projection onto the space of CPTP matrices [70,
71, 72, 73].

The CPTP projection involves alternating CP and TP projections according to
Ref. [72]. The CP projection, individually, is obtained by simply replacing
the negative eigenvalues of a matrix to zero in its eigendecomposition, i.e.

$\mathcal{CP}\left[\mathcal{R}\right]=V\max\left(D,0\right)V^{-1},$ (37)

where $\mathcal{R}=VDV^{-1}$ is the eigendecomposition.

The TP projection is given by

$\mathcal{TP}\left[\mathcal{R}\right]={\rm{vec}}^{-1}\left[|\mathcal{R}\rangle\negthickspace\rangle-\frac{1}{{\rm{dim}}\mathcal{K}}M^{\dagger}M|\mathcal{R}\rangle\negthickspace\rangle+\frac{1}{{\rm{dim}}\mathcal{K}}M^{\dagger}|I_{\mathcal{H}}\rangle\negthickspace\rangle\right],$
(38)

and

$M=\sum_{k}{I_{\mathcal{H}}\otimes\left\langle k\right|\otimes
I_{\mathcal{H}}\otimes\left\langle k\right|},$ (39)

where $\left|k\right\rangle$ are the appropriate orthonomal bases in
$\mathcal{K}$.

###  C.2 The gradient of fidelity

We are interested in the fidelity of the recovered qubit after the channel
$\mathcal{E}$ (Eq. (15)) to the input one, a pure state in the computational
basis given by

$\mathcal{F}={\rm{tr}}\left\\{|\psi\rangle\langle\psi|\mathcal{E}\left[|\psi\rangle\langle\psi|\right]\right\\}.$
(40)

Assuming ideal encoding, recovery, and decoding operations, it may be
rewritten in terms of the Choi matrix for the recovery operation as

$\mathcal{F}={\rm{tr}}_{\mathcal{K}}\left\\{\rho\;{\rm{tr}}_{\mathcal{H}}\left\\{\left(\rho^{\prime\intercal}\otimes
I_{\mathcal{K}}\right)\mathcal{R}\right\\}\right\\}={\rm{tr}}\left\\{\left(\rho^{\prime\intercal}\otimes\rho\right)\mathcal{R}\right\\},$
(41)

where $\rho$ is input, pure state after encoding and $\rho^{\prime}$ is the
state just before the recovery.

Therefore, the variation of fidelity becomes

$\displaystyle\delta\mathcal{F}={\rm{tr}}\left\\{\left(\rho^{\prime\intercal}\otimes\rho\right)\delta\mathcal{R}\right\\},$
(42)

and the gradient in $\mathcal{R}$,

$\mathcal{M}=\rho^{\prime\intercal}\otimes\rho.$ (43)

###  C.3 CPTP constrained variation of Choi matrices

Consider a CPTP channel represented by Choi matrix $\mathcal{R}$. A small
unrestricted variation on this channel under the constraint that it remains
CPTP, is given by [74]

$\mathcal{R}+\delta\mathcal{R}=\left(I+\delta
Z^{\dagger}\right)\mathcal{R}\left(I+\delta Z\right),$ (44)

where $\delta Z$ is an operator such that the TP constraint
${\rm{tr}}_{\mathcal{K}}\left\\{\mathcal{R}\right\\}=I_{\mathcal{H}}$ is
satisfied where $\mathcal{K}$ and $\mathcal{H}$ are the output and the input
Hilbert spaces of the channel. The matrix $\delta Z$ is further parametrized
by a small, arbitrary Hermitian operator, $\delta{A}$, such that the CP
constraint is upheld, which gives [74]

$I+\delta
Z=(I+\delta{A})\left[\sqrt{{\rm{tr}}_{\mathcal{K}}\left\\{\left(I+\delta{A}^{\dagger}\right)\mathcal{R}\left(I+\delta{A}\right)\right\\}}\otimes
I_{\mathcal{K}}\right]^{-1}.$ (45)

###  C.4 CPTP restricted steepest ascent

Consider a function $\mathcal{F}$ of a CPTP-constrained argument $\mathcal{R}$
whose variation is given by

$\delta{\mathcal{F}}={\rm{tr}}\left\\{\mathcal{M}\delta{\mathcal{R}}\right\\},$
(46)

where $\mathcal{M}$ is its gradient in $\mathcal{R}$.

A small CPTP restricted ascent in $\mathcal{F}$ in the direction of the
highest gradient can be constructed using Eqs. (44) and (45) with [74]

$\delta{A}=\delta{A}^{\dagger}=\frac{\epsilon}{2}\left(\mathcal{M}-\frac{1}{2}{\rm{tr}}_{\mathcal{K}}\left\\{\mathcal{M}\mathcal{R}+\mathcal{R}\mathcal{M}\right\\}\otimes
I_{\mathcal{K}}\right),$ (47)

for some small $\epsilon>0$.

###  C.5 A typical instance of numerical optimization

The results for an instance of numerical optimization for the results shown in
Fig. 6(a) are presented in Fig. 11. The final optimized recovery map is block-
diagonal, with blocks corresponding to the subspaces with even and odd photon-
number parities.

Figure 11: An instance of optimization for the optimal fidelity in Fig. 6(a)
with ZPS for $T=35\%$ and $10\%$ loss rate with $\eta=95\%$. (a) The operator
norm of the variation on the recovery map (red), $\delta\mathcal{R}$, and the
optimized fidelity (blue) are plotted against the step count. Fidelities
saturated within 100 steps for the initial $\epsilon=3$, which is decreased by
$0.994$ at each step. (b) Recovered qubit states (blue) resulting from the
optimized recovery map (after 1000 steps), derived from 50,000 uniformly
sampled initial pure states on a Bloch sphere (red). (c) Fidelities of the
optimally recovered qubits to 10000 uniformly sampled input states are shown
in blue. These are above the red line that represents the worst-case fidelity
$93.5\%$ for the optimal $\mathcal{R}$. This indicates that the optimized
solution is indeed the worst-case fidelity for a given recovery. (d) Worst-
case fidelity of $100$, full-rank, unoptimized, randomly chosen channels
$\mathcal{R}$ in blue. These lie below the red line, supporting that the
optimized recovery map performs better.

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