#  Slowing quantum decoherence of oscillators by hybrid processing

Kimin Park  Center for Macroscopic Quantum States (bigQ), Department of
Physics, Technical University of Denmark, Building 307, Fysikvej, 2800 Kgs.
Lyngby, Denmark  Department of Optics, Palacky Univeristy, 77146 Olomouc,
Czech Republic  Jacob Hastrup  Center for Macroscopic Quantum States (bigQ),
Department of Physics, Technical University of Denmark, Building 307,
Fysikvej, 2800 Kgs. Lyngby, Denmark  Jonas Schou Neergaard-Nielsen  Center for
Macroscopic Quantum States (bigQ), Department of Physics, Technical University
of Denmark, Building 307, Fysikvej, 2800 Kgs. Lyngby, Denmark  Jonatan Bohr
Brask  Center for Macroscopic Quantum States (bigQ), Department of Physics,
Technical University of Denmark, Building 307, Fysikvej, 2800 Kgs. Lyngby,
Denmark  Radim Filip  Department of Optics, Palacky Univeristy, 77146 Olomouc,
Czech Republic  Ulrik L. Andersen  Center for Macroscopic Quantum States
(bigQ), Department of Physics, Technical University of Denmark, Building 307,
Fysikvej, 2800 Kgs. Lyngby, Denmark

(August 28, 2024)

###### Abstract

Quantum information encoded into superposition of coherent states is an
illustrative representative of practical applications of macroscopic quantum
coherence possessing. However, these states are very sensitive to energy loss,
losing their non-classical aspects of coherence very rapidly. An available
deterministic strategy to slow down this decoherence process is to apply a
Gaussian squeezing transformation prior to the loss as a protective step.
Here, we propose a deterministic hybrid protection scheme utilizing strong but
feasible interactions with two-level ancillas immune to spontaneous emission.
We verify robustness of the scheme against dephasing of qubit ancilla. Our
scheme is applicable to complex superpositions of coherent states in many
oscillators, and remarkably, the robustness to loss is _enhanced_ with the
amplitude of the coherent states. This scheme can be realized in experiments
with atoms, solid-state systems and superconducting circuits.

##  I Introduction

Quantum information processing based on continuous-variable (CV) resources
Braunstein and Loock (2005); Cerf et al. (2007); Weedbrook et al. (2012);
Andersen et al. (2015); Lvovsky et al. (2020) represents an interesting
alternative to a more common discrete-variable (DV) approach based on photons
Kok et al. (2007); O’brien et al. (2009); Flamini et al. (2018), or other
particles Bruzewicz et al. (2019); Kjaergaard et al. (2020); Vandersypen and
Eriksson (2019). In particular, non-Gaussian states represented by the
superposition or entanglement of a finite number of coherent states are useful
resources in various protocols in both approaches Sanders (2012). The interest
in these states has been fueled by various schemes for sensing Zurek (2001);
Joo et al. (2011), computing Ralph et al. (2003); Jeong and Kim (2002); Marek
and Fiurasek (2010); Mirrahimi et al. (2014) and communication van Loock et
al. (2006); Sangouard et al. (2011) with promising merits in scalability and
fault-tolerance Lund et al. (2008); Leghtas et al. (2013); Rosenblum et al.
(2018). There have been significant efforts into the generation of the
superposition of coherent states (SCSs) and entangled coherent states (ECSs),
both in the optical Gerrits et al. (2010); Neergaard-Nielsen et al. (2010);
Ourjoumtsev et al. (2007); Dong et al. (2014); Hacker et al. (2019), microwave
Wineland (2013); Kienzler et al. (2016); Deleglise et al. (2008); Haroche
(2013); Vlastakis1 et al. (2013); Pfaff et al. (2017); Liu et al. (2016) and
phononic domains Hoff et al. (2016); Khosla et al. (2018); Teh et al. (2018).
Moreover, simple CV quantum processing tasks have been carried out on optical
fields Tipsmark et al. (2011); Larsen et al. (2020), while more advanced
processing has been demonstrated in the microwave regimes including quantum
error correction Heeres et al. (2017); Ofek et al. (2016).

A formidable challenge associated with the faithful processing, transmission
and storage of SCSs lies in the inevitable decoherence by bosonic loss. While
non-Gaussian Aoki et al. (2009); Lassen et al. (2010) or correlated Lassen et
al. (2013) noise sources can be circumvented for any type of encoding in
bosonic modes by means of simple Gaussian transformations, the stationary
bosonic loss – the dominant decoherence source in most bosonic systems – is
non-trivial to correct for. To overcome the effect of loss in quantum
information processing, one could use the protocol of entanglement
distillation of resources Ralph (2011) in combination with deterministic
teleportation, or more advanced quantum error correction coding schemes in
which errors are corrected by encoding quantum information into special
bosonic codes such as Gottesman-Kitaev-Preskill (GKP) states Ofek et al.
(2016); Noh et al. (2020); Fabre et al. (2019); Heeres et al. (2017); Hastrup
et al. (2021); Michael et al. (2016); Noh and Chamberland (2020); Fluhmann et
al. (2019); Campagne-Ibarcq et al. (2020); Tzitrin et al. (2020); Terhal et
al. (2020), binomial states Michael et al. (2016); Hu et al. (2019); Noh
(2021) or SCSs Puri et al. (2020); Grimm et al. (2020); Chamberland et al.
(2020). However, those “ultimate” approaches requires a challenging large
resources to universal loss correction.

Bosonic losses can be also partially but deterministically compensated for by
transforming the encoded quantum state with a special symmetry into a state
that is more robust against losses. For example, by conditionally de-
amplifying a coherent state Ralph (2011); Müller et al. (2012); Haw et al.
(2016); Brewster et al. (2018), or unconditionally squeezing their
superposition states Jeannic et al. (2018); Filip (2001); Serafini et al.
(2004); Filip (2013) prior to a lossy bosonic channel, the coherence or non-
classicality can survive for longer time. In these two protective schemes
enabled by pre-processing, the quantum information has resided entirely in a
bosonic quantum state.

In this article, we propose a next step in deterministic hybrid protection
strategy for SCSs and ECSs, or a qubit _bypass_ strategy, where the quantum
information of these states does not reside fully in the lossy bosonic mode -
as in previous schemes - but is substantially converted into a lossless two-
level system. The quantum information is by large protected from bosonic loss
by such a hybrid strategy, albeit being traded for phase decoherence of the
two-level system. Our protection scheme is enabled by a strong coherent
coupling – the Rabi coupling – which can be implemented in superconducting
systems, ion systems, or photonic or molecular crystal systems Kockum et al.
(2019); Forn-Diaz et al. (2019); Mueller et al. (2020); Flühmann et al.
(2018); Langford et al. (2017); Lv et al. (2018) where the bosonic mode is
represented by a microwave, a phononic, or even an optical field. In this
scheme, the number of used two-level systems and Rabi couplings is kept at a
minimum, while the coupling strength is dictated by the amplitudes of the
coherent states of the SCSs and ECSs. Our approach is therefore complementary
to an approach where the available number of Rabi gates and ancillas are
unlimited which in principle would allow for a complete transfer of the
information from the bosonic mode to the two-level systems Hastrup et al. . We
analyze our protocol using different measures (i.e. fidelity, coherence in
phase space and entanglement), with respect to the simple protection scheme of
pre-squeezing Jeannic et al. (2018). We find that the robustness to losses is
significantly increased in all these measures, remarkably for large amplitude
SCSs. Our strategy can be extended to states with higher complexity, e.g. with
more coherent state components or modes, thereby indicating that further
extensions of the method to general CV states might be viable.

We proceed by introducing the hybrid protection protocol in section II, and
presenting the results in section III. In section IV we conclude the study.

##  II QUBIT Bypass protocol

Our qubit bypass protocol is based on the quantum Rabi model Kockum et al.
(2019); Forn-Diaz et al. (2019). This model incorporates a CV-DV hybrid non-
Gaussian interaction, where an electro-magnetic field or mechanical oscillator
strongly couples to a two-level system, e.g. an atom. The unitary evolution
opeator associated with the coupling in the model is written as
$\exp[it\hat{\sigma}_{\theta}\hat{X}_{\phi}]$ with strength $t$ (the strength
of the Hamiltonian is combined with the duration time), where a Pauli operator
$\hat{\sigma}_{\theta}$ with $\theta=\\{x,y,z\\}$ acts on the two-level
system, and the generalized quadrature operator
$\hat{X}_{\Phi}=\frac{\hat{a}e^{-i\Phi}+\hat{a}^{\dagger}e^{i\Phi}}{\sqrt{2}}$
with arbitrary angle $\Phi$ governs the field. The rotating-wave approximation
is not valid within this model, and the unitary operator contains both
rotating and counter-rotating terms. Strong Rabi coupling has been achieved
experimentally in cavity QED systems such as trapped ions Lv et al. (2018) or
superconducting systems Markovic et al. (2018) among many other systems, and
has been simulated in all-optical setup Park et al. (2020). It can be used to
create diverse classes of non-Gaussian effects such as nonlinear phase gates
Park et al. (2016, 2018), which are essential in CV quantum information
processing Weedbrook et al. (2012).

The unitary Rabi gate executes a controlled displacement operation, where the
conjugate generalized momentum, $\hat{P}_{\Phi}=\hat{X}_{\Phi+\pi/2}$, of an
oscillator is displaced depending on the state of an ancillary qubit encoded
in $\hat{\sigma}_{\theta}$-eigenstates. Here, we will drop the subscript for
$\Phi=0$ below for these operators. The Rabi gate is particularly suitable for
the generation of SCSs Forn-Diaz et al. (2019); Kockum et al. (2019). These
controlled displacement operations have been used in numerous experiments to
generate superpositions of two and four coherent states (for example, as in
Roszak et al. (2015)). In the current work, we will reverse the generation
process by mapping the SCSs onto a two-level ancilla, or more precisely,
partially converting the complex coefficients of SCSs onto the complex
coefficients of a qubit, addressing the possibility to manipulate the SCSs by
Rabi gates for the first time.

Two coherent-state superposition— Throughout the paper, we will refer the
superposition of $n$ coherent states as n-SCSs. We start with the simplest
SCSs, namely the arbitrary superposition of two coherent states (2-SCS) of
opposite amplitudes,
$\left|\psi_{0}\right\rangle=N(\mu\left|\alpha\right\rangle+\nu\left|-\alpha\right\rangle)$.
The normalization factor is given by
$N^{-2}=|\mu|^{2}+|\nu|^{2}+(\mu^{*}\nu+\mu\nu^{*})e^{-2\alpha^{2}}$ but will
be omitted below for simplicity. While the unknown complex coefficients of the
superposition $\mu,\nu\in\mathbb{C}$ are carrying encoded information, the
amplitude $\alpha\in\mathbb{R}$ is typically assumed to be known. We focus
primarily on nearly orthogonal coherence states with large $\alpha\gg 2$ when
this state can be considered as a qubit of basis
$\left|\pm\alpha\right\rangle$.

Before going to the details of our protocol, let us first consider the
decoherence of an unprotected 2-SCS undergoing bosonic loss $\Gamma_{l}$. Any
density matrix $\rho$ under $\Gamma_{l}$ evolves as
$\Gamma_{l}[\rho]=\sum_{l}\frac{(1-\eta)^{l}}{l!}\eta^{\hat{n}/2}\hat{a}^{l}\rho\hat{a}^{\dagger
l}\eta^{\hat{n}/2}$ with $\eta$ the loss parameter. 2-SCS evolves by this loss
simply as

$\Gamma_{l}[\left|\psi_{0}\right\rangle\left\langle\psi_{0}\right|]=|\mu|^{2}\left|\sqrt{\eta}\alpha\right\rangle\left\langle\sqrt{\eta}\alpha\right|+|\nu|^{2}\left|-\sqrt{\eta}\alpha\right\rangle\left\langle-\sqrt{\eta}\alpha\right|+\mu\nu^{*}e^{-2\alpha^{2}(1-\eta)}\left|\sqrt{\eta}\alpha\right\rangle\left\langle-\sqrt{\eta}\alpha\right|+\mu^{*}\nu
e^{-2\alpha^{2}(1-\eta)}\left|-\sqrt{\eta}\alpha\right\rangle\left\langle\sqrt{\eta}\alpha\right|.$
(1)

The off-diagonal elements carrying the quantum coherence in this nearly
orthogonal coherent-state basis, are rapidly decreasing for a large $\alpha$
by the factor $e^{-2\alpha^{2}(1-\eta)}$. This rapid decoherence can be
avoided if $\alpha$ is effectively reduced, e.g. by mapping the information
onto a two-level system.

Figure 1:  (a) Schematic of the protection protocol for a superposition of two
coherent states in a bosonic mode (thick), exploiting a single qubit bypass
(thin). The oscillator and the ancillary qubit interact by a unitary coupling
$\hat{U}_{T}$ made of two Rabi gates, i.e. controlled qubit rotation
$\hat{U}_{R}=\exp[i\epsilon\hat{\sigma}_{x}\hat{X}]=\hat{R}_{x}[\epsilon\hat{X}]$
where qubit rotation is denoted as
$\hat{R}_{x}[\vartheta]=\exp[i\vartheta\hat{\sigma}_{x}]$, and controlled
displacement
$\hat{U}_{D}=\exp[-i\sqrt{2}\alpha\hat{\sigma}_{y}\hat{P}]=\hat{D}[i\sqrt{2}\alpha\hat{\sigma}_{y}]$.
This bosonic channel undergoes linear damping in a lossy channel, whereas
lossless qubit channel degrades only by a phase damping; the bosonic damping
and phase damping are represented with trace-preserving maps $\Gamma_{l}$ and
$\Gamma_{d}$, respectively. The bypass is intrinsically protected against the
amplitude damping by the qubit encoding to two ground states. After
$\hat{U}_{T}$, the quantum information is transferred to the qubit
substantially, but keeping the CV channel is still beneficial. After
decoherence by the error channels, the input state is restored by the inverse
unitary process $\hat{U}_{T}^{\dagger}$. An extension of the protocol
exploiting two bypass qubits (b) to a single-mode superposition of four
coherent states, and (c) to bipartite entangled states where the protocol is
applied locally on each mode.

Our bypass protocol illustrated in Fig. 1 (a) consists of a set of two Rabi
gates $\hat{U}_{T}=\hat{U}_{D}\hat{U}_{R}$ applied before the lossy channel.
The first unitary Rabi gate,
$\hat{U}_{R}=\exp[i\epsilon\hat{\sigma}_{x}\hat{X}]$ with
$\epsilon=\pi/4\sqrt{2}\alpha$, rotates the initial ancillary qubit state
$\left|g\right\rangle$ controlled by the amplitudes in the coherent states to
have an entangled state as:

$\displaystyle\exp[i\epsilon\hat{\sigma}_{x}\hat{X}]\left|g\right\rangle(\mu\left|\alpha\right\rangle+\nu\left|-\alpha\right\rangle)=\mu\hat{D}[\alpha]\exp[i\epsilon\hat{\sigma}_{x}(\hat{X}+\sqrt{2}\alpha)]\left|g\right\rangle\left|0\right\rangle+\nu\hat{D}[-\alpha]\exp[i\epsilon\hat{\sigma}_{x}(\hat{X}-\sqrt{2}\alpha)]\left|g\right\rangle\left|0\right\rangle$
$\displaystyle=\frac{i}{2}\left\\{\mu\hat{D}[\alpha]\left(\left|-_{i}\right\rangle\left|\frac{i\epsilon}{\sqrt{2}}_{+}\right\rangle-i\left|+_{i}\right\rangle\left|\frac{i\epsilon}{\sqrt{2}}_{-}\right\rangle\right)-\nu\hat{D}[-\alpha]\left(\left|+_{i}\right\rangle\left|\frac{i\epsilon}{\sqrt{2}}_{+}\right\rangle+i\left|-_{i}\right\rangle\left|\frac{i\epsilon}{\sqrt{2}}_{-}\right\rangle\right)\right\\}$
$\displaystyle\stackrel{{\scriptstyle\alpha\gg
2}}{{\longrightarrow}}i\left(\mu\left|-_{i}\right\rangle\left|\alpha\right\rangle-\nu\left|+_{i}\right\rangle\left|-\alpha\right\rangle\right),$
(2)

where an ordering relation
$\hat{D}[\alpha]\hat{X}\hat{D}[-\alpha]=\hat{X}+\sqrt{2}\alpha$ was used on
the first equality. The approximation used on the last line is
$\left|\beta+i\varepsilon\right\rangle\approx\exp[i\beta\varepsilon]\left|\beta\right\rangle$,
for $\varepsilon\ll 1$. Here, the unnormalized coherent superposition states
are denoted as
$\left|\delta_{\pm}\right\rangle=\left|\delta\right\rangle\pm\left|-\delta\right\rangle$
and the qubit states $\left|\pm_{i}\right\rangle$ denote the eigenstates of
$\hat{\sigma}_{y}$ with eigenvalues $\pm 1$. In the limit of a large amplitude
$\alpha\gg 2$ Bayer et al. (2017); Yoshihara et al. (2017), the odd
superpositions states vanish
$\left|\frac{i\epsilon}{\sqrt{2}}_{-}\right\rangle\rightarrow 0$ and the even
superpositions approach the vacuum
$\left|\frac{i\epsilon}{\sqrt{2}}_{+}\right\rangle\rightarrow\left|0\right\rangle$.
In total, eq. (2) can be interpreted as entangling process of the 2-SCS and
the qubit ancilla.

The second Rabi gate
$\hat{U}_{D}=\exp[-i\sqrt{2}\alpha\hat{\sigma}_{y}\hat{P}]=\hat{D}[i\sqrt{2}\alpha\hat{\sigma}_{y}]$
acts as a controlled displacement, i.e. the ancilla qubit encoded in
$\left|\pm_{i}\right\rangle$ from (2) as the control and oscillator as the
target, which transfers the information to the qubit ancilla. Applying
$\hat{U}_{D}$ on (2), we obtain the state which will undergo the noise
channels:

$\displaystyle\exp\left[-i\sqrt{2}\alpha\hat{\sigma}_{y}\hat{P}\right]\exp[i\epsilon\hat{\sigma}_{x}\hat{X}]\left|g\right\rangle(\mu\left|\alpha\right\rangle+\nu\left|-\alpha\right\rangle)$
$\displaystyle=i\mu\left(\left|-_{i}\right\rangle\frac{\left|i\epsilon/\sqrt{2}\right\rangle+\left|-i\epsilon/\sqrt{2}\right\rangle}{2}+\left|+_{i}\right\rangle\frac{e^{-i\pi/4}\left|2\alpha+i\epsilon/\sqrt{2}\right\rangle-e^{i\pi/4}\left|2\alpha-i\epsilon/\sqrt{2}\right\rangle}{2i}\right)$
$\displaystyle-i\nu\left(\left|+_{i}\right\rangle\frac{\left|i\epsilon/\sqrt{2}\right\rangle+\left|-i\epsilon/\sqrt{2}\right\rangle}{2}-\left|-_{i}\right\rangle\frac{e^{i\pi/4}\left|-2\alpha+i\epsilon/\sqrt{2}\right\rangle-e^{-i\pi/4}\left|-2\alpha-i\epsilon/\sqrt{2}\right\rangle}{2i}\right)$
$\displaystyle\stackrel{{\scriptstyle\alpha\gg
2}}{{\longrightarrow}}i\left(\mu\left|-_{i}\right\rangle-\nu\left|+_{i}\right\rangle\right)\left|0\right\rangle.$
(3)

This state contains even superposition of coherent states $\left|\pm
i\epsilon/\sqrt{2}\right\rangle$ close to a vacuum state and erroneous states
$\left|\pm 2\alpha\pm i\epsilon/\sqrt{2}\right\rangle$ far from the phase
space origin. We note that only the erroneous terms are affected heavily by
loss due to their large amplitudes, and it can be still advantageous to keep
the bosonic channel.

We consider qubit ancilla to be lossless without amplitude damping, but
sensitive to small phase damping Bruzewicz et al. (2019); Kjaergaard et al.
(2020) as the main threat to robustness. We model the phase damping as
$\Gamma_{d}[\rho]=(1-\tfrac{p}{2})\rho+\tfrac{p}{2}\hat{\sigma}_{z}\rho\hat{\sigma}_{z}$
for any state $\rho$ with dephasing parameter $p$. This channel is identity
channel at $p=0$, and maximal dephasing channel at $p=1$. The same dephasing
channels act independently on all physical qubit ancillas involved through the
paper. Note that it is also possible to correct this dephasing error actively
by transferring the information into multiple ancillas for conventional DV
error correction methods Lidar and Brun T. (2013); Gaitan (2008), or by a
hybrid qubit protection scheme using the Rabi type of interactions as in the
Appendix D. In section III, we show that after the error channels $\Gamma_{l}$
and $\Gamma_{d}$, the original state is substantially restored by the inverse
operations $\hat{U}_{T}^{\dagger}$. We also note that in principle, the entire
protocol can be achieved by dispersive interaction and displacement Spiller et
al. (2006); van Loock et al. (2008) in travelling-wave microwave/optical
platforms for future experiments extending the scheme in Hacker et al. (2019).

Four-coherent state superposition— This methodology can be scaled up to cases
of more coherent state components in the superposition, e.g. to 4-SCS
$\mu_{1}\left|\alpha\right\rangle+\mu_{2}\left|\alpha^{*}\right\rangle+\mu_{3}\left|-\alpha\right\rangle+\mu_{4}\left|-\alpha^{*}\right\rangle$
for $\alpha=\alpha_{r}+i\alpha_{i}\in\mathbb{C}$ Hastrup et al. (2020a).
Gaussian operations such as squeezing cannot protect such states _at all_ , as
the broken symmetry in phase space by squeezing leads to a more severe
decoherence. Therefore, these states can be an advanced test bed for the
extension of the bypass strategy to general SCSs. We note that a single qubit
will not be sufficient for the bypass due to the existence of 4 unknown
coefficients $\mu_{j}$’s, and we need to extend the circuit to two ancilla
qubits for the minimal complexity.

To protect a general 4-SCS with our hybrid bypass scheme, two ancillas with
indices $j=1^{\prime},2^{\prime}$ couple respectively to the quadrature
variables $X$ and $P$ of oscillator by four Rabi gates forming
$\hat{U}_{T_{X,P}}$ as depicted in Fig. 1 (b). The two quadratures are
independently coupled to each of the ancillas by the operations
$\hat{U}_{R}^{(1^{\prime})}=\exp[i\epsilon_{r}\hat{\sigma}_{x}^{(1^{\prime})}\hat{X}]$
and
$\hat{U}_{R}^{(2^{\prime})}=\exp[i\epsilon_{i}\hat{\sigma}_{x}^{(2^{\prime})}\hat{P}]$
with a single Pauli operator acting on ancilla $j$ $\hat{\sigma}_{x}^{(j)}$,
approximated in the large amplitude limit $\alpha_{r,i}\gg 2$ as:

$\displaystyle\exp[i\epsilon_{r}\hat{\sigma}_{x}^{(1^{\prime})}\hat{X}]\left|g\right\rangle_{1^{\prime}}\left|\alpha_{r}+i\alpha_{i}\right\rangle\approx(\cos[\sqrt{2}\alpha_{r}\epsilon_{r}]\left|g\right\rangle_{1^{\prime}}+i\sin[\sqrt{2}\alpha_{r}\epsilon_{r}]\left|g^{\prime}\right\rangle_{1^{\prime}})\left|\alpha_{r}+i\alpha_{i}\right\rangle\approx\left|\mathrm{sgn}[\alpha_{r}]_{i}\right\rangle_{1^{\prime}}\left|\alpha_{r}+i\alpha_{i}\right\rangle,$
$\displaystyle\exp[i\epsilon_{i}\hat{\sigma}_{x}^{(2^{\prime})}\hat{P}]\left|g\right\rangle_{2^{\prime}}\left|\alpha_{r}+i\alpha_{i}\right\rangle\approx(\cos[\sqrt{2}\alpha_{i}\epsilon_{i}]\left|g\right\rangle_{2^{\prime}}+i\sin[\sqrt{2}\alpha_{i}\epsilon_{i}]\left|g^{\prime}\right\rangle_{2^{\prime}})\left|\alpha_{r}+i\alpha_{i}\right\rangle\approx\left|\mathrm{sgn}[\alpha_{i}]_{i}\right\rangle_{2^{\prime}}\left|\alpha_{r}+i\alpha_{i}\right\rangle.$
(4)

Here, $\mathrm{sgn}[x]$ denotes the sign of the argument $x$. Coupling
strengths are set respectively as $\epsilon_{r,i}=\pi/4\sqrt{2}\alpha_{r,i}$.
The coherent peaks are then shifted toward the phase space origin by the
controlled displacement $\hat{U}_{D}^{(j)}$ by the ancillary states before the
erroneous channels as

$\displaystyle\exp[i\sqrt{2}\alpha_{r}\hat{\sigma}_{y}^{(1^{\prime})}\hat{P}]\left|\mathrm{sgn}[\alpha_{r}]_{i}\right\rangle_{1^{\prime}}\left|\alpha_{r}+i\alpha_{i}\right\rangle\approx\left|\mathrm{sgn}[\alpha_{r}]_{i}\right\rangle_{1^{\prime}}\left|i\alpha_{i}\right\rangle,$
$\displaystyle\exp[-i\sqrt{2}\alpha_{i}\hat{\sigma}_{y}^{(2^{\prime})}\hat{X}]\left|\mathrm{sgn}[\alpha_{i}]_{i}\right\rangle_{2^{\prime}}\left|i\alpha_{i}\right\rangle\approx\left|\mathrm{sgn}[\alpha_{i}]_{i}\right\rangle_{2^{\prime}}\left|0\right\rangle.$
(5)

After the error channels, the original state can be substantially restored by
the inverse operations $\hat{U}_{T_{X,P}}^{\dagger}$. This example
demonstrates that complex n-SCSs can be protected by the qubit bypass in
similar ways; for other examples see Appendix F.

Two-mode entangled coherent states—Our scheme can be extended to multi-mode
ECSs in a straightforward way, for the simplest example to the two-mode state
$\left|\mathrm{ECS}^{\pm}\right\rangle=N_{\pm}(\left|\alpha\right\rangle_{1}\left|\alpha\right\rangle_{2}\pm\left|-\alpha\right\rangle_{1}\left|-\alpha\right\rangle_{2})$
with oscillator indices $J=1,2$ with a normalization factor $N_{\pm}=(2\pm
2e^{-4\alpha^{2}})^{-1/2}$. Without any protective scheme, the two-mode ECS
will decohere at approximately double decay rate of single mode superposition
states as

$\displaystyle\Gamma_{l}^{(2)}[\Gamma_{l}^{(1)}[\left|\mathrm{ECS}^{\pm}\right\rangle\left\langle\mathrm{ECS}^{\pm}\right|]]=$
$\displaystyle
N_{\pm}^{2}(\left|\alpha^{\prime}\right\rangle_{1}\left\langle\alpha^{\prime}\right|\otimes\left|\alpha^{\prime}\right\rangle_{2}\left\langle\alpha^{\prime}\right|+\left|-\alpha^{\prime}\right\rangle_{1}\left\langle-\alpha^{\prime}\right|\otimes\left|-\alpha^{\prime}\right\rangle_{2}\left\langle-\alpha^{\prime}\right|$
$\displaystyle\pm
e^{-4\alpha^{2}(1-\eta)}\left|\alpha^{\prime}\right\rangle_{1}\left\langle-\alpha^{\prime}\right|\otimes\left|\alpha^{\prime}\right\rangle_{2}\left\langle-\alpha^{\prime}\right|\pm
e^{-4\alpha^{2}(1-\eta)}\left|-\alpha^{\prime}\right\rangle_{1}\left\langle\alpha^{\prime}\right|\otimes\left|-\alpha^{\prime}\right\rangle_{2}\left\langle\alpha^{\prime}\right|),$
(6)

where $\alpha^{\prime}=\sqrt{\eta}\alpha$. For this state, our scheme utilizes
two _local_ bypasses of ancillary modes $1^{\prime},2^{\prime}$ acting on both
CV modes. The state $\left|\mathrm{ECS}^{\pm}\right\rangle$ then evolves by
$\hat{U}_{T_{1}}\hat{U}_{T_{2}}$ to a loss-robust state

$\displaystyle\hat{U}_{T_{1}}\hat{U}_{T_{2}}\left|\mathrm{ECS}^{\pm}\right\rangle=i\left|\phi_{+}\right\rangle_{1,1^{\prime}}\left|\phi_{+}\right\rangle_{2,2^{\prime}}\mp
i\left|\phi_{-}\right\rangle_{1,1^{\prime}}\left|\phi_{-}\right\rangle_{2,2^{\prime}}$
(7)

where
$\left|\phi_{+}\right\rangle=\left|-_{i}\right\rangle\frac{\left|i\epsilon/\sqrt{2}\right\rangle+\left|-i\epsilon/\sqrt{2}\right\rangle}{2}+\left|+_{i}\right\rangle\frac{e^{-i\pi/4}\left|2\alpha+i\epsilon/\sqrt{2}\right\rangle-e^{i\pi/4}\left|2\alpha-i\epsilon/\sqrt{2}\right\rangle}{2i}$
and
$\left|\phi_{-}\right\rangle=\left|+_{i}\right\rangle\frac{\left|i\epsilon/\sqrt{2}\right\rangle+\left|-i\epsilon/\sqrt{2}\right\rangle}{2}-\left|-_{i}\right\rangle\frac{e^{i\pi/4}\left|-2\alpha+i\epsilon/\sqrt{2}\right\rangle-e^{-i\pi/4}\left|-2\alpha-i\epsilon/\sqrt{2}\right\rangle}{2i}$.
For an $n$-mode ECSs, we can similarly apply local schemes to all $n$ modes to
transform into more loss-robust states. These examples explain well the
logical steps to be applied to any specific superposition of finite number of
coherent states in different modes.

##  III Results

###  III.1 Single mode superposition states

As a measure of performance of the protocol, we can first look at fidelity
Jozsa (1994); Uhlmann (1976). Fidelity is defined as
$F(\rho_{\mathrm{in}},\rho_{\mathrm{out}})=\mathrm{tr}(\sqrt{\sqrt{\rho_{\mathrm{in}}}\rho_{\mathrm{out}}\sqrt{\rho_{\mathrm{in}}}})^{2}$
between input $\rho_{\mathrm{in}}$ and output $\rho_{\mathrm{out}}$. For large
$\alpha\gg 2$, the superpositions of coherent states represent a qubit space
that is well characterized by fidelity. The average performance of the schemes
on 2-SCSs $\mu\left|\alpha\right\rangle+\nu\left|-\alpha\right\rangle$ with a
fixed $\alpha$ but arbitrary $\mu$ and $\nu$ over the “Bloch sphere” of
coherent states can be measured by the channel fidelity Albert et al. (2018).
It is defined as the fidelity between an input virtual entangled state
$2^{-1/2}(\left|\mathbf{0}\right\rangle\left|\alpha\right\rangle+\left|\mathbf{1}\right\rangle\left|-\alpha\right\rangle)$
and the output state after the noise channels, where
$\\{\left|\mathbf{0}\right\rangle,\left|\mathbf{1}\right\rangle\\}$ are
fictitious qubit basis. We note that in contrast to a conventional qubit, the
statistical weight of states represented in the channel fidelity is not
homogeneous over all coefficients $\mu,\nu$. For example, the weight of
balanced superposition states
$\left|\alpha\right\rangle\pm\left|-\alpha\right\rangle$ has the extrema
values, but this inhomogeneity is negligible for $\alpha\gtrsim 2$.

Figure 2:  Performance of bypass protection protocol in Fig. 1 (a) for a
superposition of two coherent states in a bosonic mode compared to the
Gaussian strategy under various dephasings. (a) The logarithmic decay rate
$\log\gamma$ of channel fidelity modeled as
$F_{c}(\eta)=F_{c}(1)e^{-\gamma(1-\eta)}$ for various cases. The curves for
different $p$’s are overlapped. (b) Channel fidelity vs. loss parameter for
$\alpha=6$. This result shows the qualitative difference between bypass and
Gaussian strategies.

We compared the strategy of Gaussian squeezing strategy prior to transmission
through the lossy channel Filip (2013); Jeannic et al. (2018) as a benchmark.
The optimization of Gaussian strategy was with respect to the average photon
number in the CV modes at each $\alpha$. The optimal squeezing parameter
$r_{\mathrm{opt}}$ for the initial state
$\mu\left|\alpha\right\rangle+\nu\left|-\alpha\right\rangle$ is given as

$\displaystyle
r_{\mathrm{opt}}=\frac{1}{4}\left(\log\left[\frac{4e^{2\alpha^{2}}\alpha^{2}\mu^{*}}{e^{2\alpha^{2}}\mu^{*}+\nu\left(\mu^{*}\right)^{2}+\mu^{2}\nu}+1\right]-\log\left[1-\frac{4\alpha^{2}\nu\left(\left(\mu^{*}\right)^{2}+\mu^{2}\right)}{e^{2\alpha^{2}}\mu^{*}+\nu\left(\mu^{*}\right)^{2}+\mu^{2}\nu}\right]\right).$
(8)

In Fig. 2, we show the effectiveness of our strategy quantified by the channel
fidelity for general 2-SCSs of various amplitudes against bosonic loss
(fidelity for individual states are calculated in Appendix A). This is well
evidenced by the decay rate $\gamma$ of the fidelity model
$F_{c}(\eta)=F_{C}(1)e^{-\gamma(1-\eta)}$, where a larger $\gamma$ represents
a faster decay. Fig. 2 (a) clearly shows the qualitative difference of our
qubit bypass strategy showing decreased $\gamma$ approximately described as
$0.65-0.025\alpha^{1.2}$. In contrast, the $\gamma$s for no protection and
Gaussian squeezing protection are described by increasing trends respectively
as $0.17+\alpha^{1.92}$ and $0.29+0.96\alpha$. In Fig. 2 (b), we can see that
the channel fidelity of bypass scheme significantly surpasses the Gaussian
strategy for a mild level of dephasing for a large amplitude $\alpha\gg 2$,
and loss is not critically detrimental to our scheme. At complete lossy case
$\eta=0$ for $p=0$ corresponding to the complete blockage of the oscillator,
the channel fidelity of our scheme can be approximately given in the large
$\alpha$ limit as

$\displaystyle
F_{c}\approx\frac{e^{-\frac{3\pi^{2}}{64\alpha^{2}}}+e^{-\frac{\pi^{2}}{64\alpha^{2}}}}{2}.$
(9)

We can see in Fig. A.1 (a,b) in Appendix A that keeping the oscillator is
still beneficial even though it carries little information.

Figure 3:  Protection of negative peaks of Wigner functions of superposition
of two coherent states as in Fig. 1 (b). (a,b) Interference fringes in cross
section of the Wigner function for
$\left|\alpha\right\rangle+e^{i\phi}\left|-\alpha\right\rangle$ at $\alpha=6$
with $\phi=\pi/2,\pi$ under $1-\eta=0.01$ of loss and $p=0,0.05,0.1$ of
dephasing. The interference fringes for complete lossy channel $\eta=0$ and
under various $p$’s are overlapped largely with the initial curve. (c) The
depths of the negative peak located at phase space origin vs. loss parameter
$1-\eta$ at $\alpha=6$ for the initial state
$\left|\alpha\right\rangle-\left|-\alpha\right\rangle$ (curves with $p>0$ are
overlapped), (d) and the logarithm of slopes of the curves in (c) vs.
$\alpha$.

The negative part of the Wigner functions can serve as a quantitative measure
of the non-classical features, highly sensitive to both losses and noises
Zurek (2003); Schlosshauer (2005). The equal-weight 2-SCSs
$\left|\alpha\right\rangle+e^{i\phi}\left|-\alpha\right\rangle$ have
interference fringes around the origin of the phase space. In Fig. 3 (a),
these interference fringes along P-axis in phase space are shown for various
$\phi$ under various levels of loss and dephasing. We note that the negativity
of all interference fringes are preserved, not only in the largest one. The
bypass scheme protects the interference fringes even under a large dephasing,
especially for a large $\alpha\gg 2$. Although the input state
$\left|\alpha\right\rangle+i\left|-\alpha\right\rangle$ is not fully protected
in terms of fidelity under dephasing, the interference fringes are not
critically destroyed, although not immune. Mathematically, the value of the
Wigner function at the phase space origin is proportional to the average value
of the parity operator as
$W(0,0)=\frac{2}{\pi}\langle\rho{(-1)}^{\hat{n}}\rangle$. The bypass scheme
protects the interference fringes even under a large dephasing, especially for
a large $\alpha\gg 2$. The deepest negative peak of the initial state
$\left|\alpha\right\rangle-\left|-\alpha\right\rangle$ is located at the phase
space origin, and its first-order approximation for our protocol vs. the loss
parameter $1-\eta$ and the dephasing parameter $p$ scales approximately in the
large $\alpha$ limit as

$\displaystyle
W(0,0)\approx-\frac{2-4e^{-2\alpha^{2}}}{\pi}+(1-\eta)\frac{2e^{-2\alpha^{2}}\alpha^{2}}{\pi}+p\frac{1-e^{-\frac{\pi^{2}}{8\alpha^{2}}}}{2\pi}.$
(10)

Here the effect of loss and dephasing evidenced in the second and third terms
is _weakening_ for a large $\alpha$. This is because the state
$\left|\alpha\right\rangle-\left|-\alpha\right\rangle$ is mapped onto the
oscillator ground state (Eq. (3)), which is unaffected by both boson loss and
dephasing. Fig. 3 (b) shows that our scheme protects this negative peak better
than the Gaussian squeezing protection especially at large $\alpha$, whose
negative peak at the origin is described approximately by
$-\frac{2}{\pi}+0.61(1-\eta)\alpha^{1.60}$. In comparison, without any
protection the negative peak at the origin scales as
$\frac{2e^{2\alpha^{2}(1-\eta)}-2e^{2\alpha^{2}\eta}}{\pi\left(e^{2\alpha^{2}}-1\right)}$,
which is influenced more harshly than the others.

In Appendix B, we calculated the Wigner functions and channel fidelity of the
4 SCSs for a special encodings as in Leghtas et al. (2013); Ofek et al. (2016)
under various levels of loss and dephasing. Similarly as for 2-SCSs, we can
also protect the negative peaks of 4-SCSs with an increased fidelity in large
$\alpha$ limit against loss under a modest level of dephasing on the ancilla.

###  III.2 Two mode entangled states

Figure 4:  Protection of entangled coherent states
$\left|\mathrm{ECS}_{-}\right\rangle$ in Fig. 1 (c) quantified by numerically
calculated logarithmic negativities (a) vs. $\alpha$ for $1-\eta=0.01$ and (b)
vs. $1-\eta$ for $\alpha=3$.

Entangled coherent states Sanders (2012); Ourjoumtsev et al. (2009) were
engineered experimentally from trapped ions Sarlette et al. (2012); Arenz et
al. (2013) and superconducting circuits Wang et al. (2016); Liu et al. (2016).
We briefly note that in terms of fidelity, similar tendencies, e.g.
enhancement at large $\alpha$ beyond Gaussian strategies, can be obtained as
of single mode states. An important question is whether a non-local property
such as entanglement of these states can also be protected by the local bypass
schemes as in Fig. 1 (c), which cannot be simply predicted by the fidelity.
For our protocol, intuitively we expect this to be the case, as we transfer
the entanglement from the bosonic modes to the ancillas, thus bypassing
losses. As a measure of entanglement which can characterize the bipartite
states $\rho$, we choose logarithmic negativity
$N_{L}[\rho]=\log||\rho^{\mathrm{PT}}||$ with partial transposition
$\mathrm{PT}$ and the trace norm $||\cdot||$, which operationally is connected
to the upper bound on the distillable entanglement Plenio (2005). The even
superposition state $\left|\mathrm{ECS}^{+}\right\rangle$ has $0$ logarithmic
negativity at small $\alpha$ as it is close to the vacuum and thus has no
entanglement. On the other hand, the odd superposition state
$\left|\mathrm{ECS}^{-}\right\rangle$ possesses a value $\ln 2$ at all
$\alpha$.

At each $\alpha$, the squeezing parameters for Gaussian strategies were
optimized to minimize the average photon number before the lossy channel. The
Gaussian squeezing protection can be applied on both modes, with the optimal
squeezing parameter with respect to average photon number is given as

$\displaystyle
r_{\mathrm{opt}}=\frac{1}{4}\left(\log\left[2\alpha^{2}+2\alpha^{2}\coth\left(2\alpha^{2}\right)+1\right]-\log\left[-2\alpha^{2}+2\alpha^{2}\coth\left(2\alpha^{2}\right)+1\right]\right).$
(11)

In addition, we optimized the Gaussian protection numerically at each $\alpha$
as well.

The scaling of the logarithmic negativity vs. $\alpha$ and $\eta$ can be
described by a fitted function $\log 2-(1-\eta)f[\alpha]$ when the dephasing
is absent. The loss effect functions are approximately given as
$f[\alpha]=1.59\alpha^{2}+0.12\alpha+0.59$ for no protection,
$f[\alpha]=-0.26+1.34\alpha-0.70\log\alpha$ for Gaussian squeezing protection,
and $f[\alpha]=0.44-0.22\log\alpha$ for bypass protocol. In Fig. 4, the curves
for the logarithmic negativities of ECS $\left|\mathrm{ECS}^{-}\right\rangle$
after various noise channels were drawn numerically. We notice again that the
entanglement of the bipartite states suffers weaker effect by qubit bypass
protocol than the Gaussian strategies, with enhanced performance in large
$\alpha$ limit under a moderate qubit dephasing.

Figure 5:  Protection in Fig. 1 (c) of the Wigner function negative peak
depths at the phase space origin of $\left|\mathrm{ECS}^{-}\right\rangle$
after a projective vacuum measurement on one mode. (a) vs. loss parameter
$1-\eta$ at $\alpha=6$ (curves for $p>0$ are overlapped), (b) the logarithm of
the slope of the curves in (a). The curves for the protection by squeezing
were calculated numerically. The dephasing noise does not affect the depth of
the negative peaks, as is evidenced by the overlap of the curves.

Additionally, the negative Wigner function of the projected state by a local
detection shows a sensitive witness of the entanglement. If the bipartite
state becomes factorizable in coherent basis due to decoherence as
$\left|\alpha\right\rangle_{1}\left\langle\alpha\right|\otimes\left|\alpha\right\rangle_{2}\left\langle\alpha\right|+\left|-\alpha\right\rangle_{1}\left\langle-\alpha\right|\otimes\left|-\alpha\right\rangle_{2}\left\langle-\alpha\right|$,
no measurement in mode $1$ can create a negative peak in the phase space of
mode $2$. In contrast, ECSs allow measurement-induced preparation of coherent
superposition states which possess negative Wigner function. As the simplest
examples, we can consider homodyne measurements
$\left|P\right\rangle_{1}\left\langle P\right|$ or
$\left|X\right\rangle_{1}\left\langle X\right|$ with $X=0$ or $P=0$, or a
projection on vacuum state $\left|0\right\rangle_{1}\left\langle 0\right|$.
For an ideal entangled state $\left|\mathrm{ECS}^{\pm}\right\rangle$, all
projections will prepare superposition states
$\left|\alpha\right\rangle_{2}\pm\left|-\alpha\right\rangle_{2}$.

In Fig. 5, we compared how the deepest negative peak of Wigner function
behaves for different $\alpha$ and $\eta$. Without protection, the formula for
the largest negative peak depth is given as
$w_{0}=\frac{2\left(e^{-4\alpha^{2}(\eta-1)}-e^{2\alpha^{2}\eta}\right)}{\pi\left(e^{-2\alpha^{2}(\eta-2)}-1\right)}$,
and by squeezing protection approximately as $(w_{0})^{e^{-0.18}}$. The
approximate formula for qubit bypass scheme is given as
$-\frac{2\exp[{\left(1.46\alpha^{2}-7.14\alpha+11.0\right)(\eta-1)}}{\pi}]$.
We can see that our protocol is clearly superior to the Gaussian squeezing
strategies in protecting the locally induced negativity of of Wigner functions
in all parameter regions of $\alpha$s and $\eta$s. Interestingly, the
squeezing is not highly effective in protecting the negative interference
fringes here.

Notably for a large $\alpha$, this Wigner function negative peak of the local
projected state is impacted by loss even though the entanglement itself is
preserved by our protection. This discrepancy is caused by the effect of loss
on local projection in the control mode. It might limit some applicability of
the protected states to non-local tasks in multi-modes, and further evaluation
of the loss tolerance of conditional states may be needed. We speculate that
certain non-local protocols may enhance the protection of non-local aspects.

##  IV Conclusions

In this work, we proposed a deterministic bypass protocol utiliaing Rabi
couplings to lossless qubit ancillas under a low level of dephasing, to
mitigate decoherence by stable loss on the unknown single-mode and two-mode
superposition of coherent states with various number of components. Our method
depends on experimental realization of the Rabi couplings: high strengths of
the coupling, and the existence of counter-rotating term in the Hamiltonians.
The generation of high strength gate can be in principle assisted by inline
squeezings (see Appendix E). This protocol can be implemented in various
cavity QED systems which support the precise control of Rabi couplings Forn-
Diaz et al. (2019); Kockum et al. (2019); Mueller et al. (2020); Flühmann et
al. (2018); Lv et al. (2018); Langford et al. (2017); Markovic et al. (2018);
van Loock et al. (2008). We analyzed the performance by various measures in
comparison to optimal Gaussian strategies, and confirmed the superiority of
our protocol.

Our protocol possesses following unique characteristics: (i) it utilizes a
minimal number of resources in contrast to ultimate but challenging quantum
error correction schemes, (ii) the performance improves for larger $\alpha$ in
contrast to existing protocols optimal in the opposite regime, (iii) CV
channel plays non-negligible role except for a very large $\alpha\gtrsim 10$,
and (iv) it can be combined with conventional error correction schemes on
qubits Ofek et al. (2016); Noh et al. (2020); Fabre et al. (2019); Heeres et
al. (2017); Hastrup et al. (2021); Michael et al. (2016); Noh and Chamberland
(2020); Fluhmann et al. (2019); Campagne-Ibarcq et al. (2020); Grimm et al.
(2020); Hu et al. (2019); Puri et al. (2020); Tzitrin et al. (2020); Terhal et
al. (2020) or a Gaussian protection scheme on CV states Jeannic et al. (2018);
Filip (2001); Serafini et al. (2004); Filip (2013). Our work implies that
robust qubit channels can be exploited for future extension of the method to
various classes of CV quantum non-Gaussian or non-local resources.

## Acknowledgment

This project was supported by the Danish National Research Foundation through
the Center of Excellence for- Macroscopic Quantum States (bigQ, DNRF0142).
R.F. acknowledges project LTAUSA19099 and 8C20002 from the Ministry of
Education, Youth and Sports of Czech Republic. K.P. acknowledges project
19-19722J of the Grant Agency of Czech Republic (GAČR). This project has
received funding from the European Union’s 2020 research and innovation
programme (CSA - Coordination and support action, H2020-WIDESPREAD-2020-5)
under grant agreement No. 951737 (NONGAUSS).

## Data availability

The numerical data presented in this study is available from the authors upon
request.

## Author contributions

All authors designed the protocol. K.P. and J.H. performed the calculations.
R.F. and U.L.A. supervised the work. All authors discussed and interpreted the
results, and contributed to the writing of the manuscript.

## Competing interest

The authors declare that there are no competing interests.

##  Appendix A Channel fidelity vs. $\alpha$ and fidelity for individual 2-SCS

Figure A.1: Channel fidelity vs. coherent amplitude $\alpha$ under losses (a)
$1-\eta=0.01$, (b) $0.02$, and dephasings $p=0.00,0.05,0.10$.

In Fig. A.1, the channel fidelity vs. $\alpha$ are shown at small losses. We
can observe a clear enhancement over Gaussian strategy. We can also see that
keeping the oscillator is still beneficial even though it carries little
information.

Figure A.2: Fidelity vs. $\alpha$ under the different levels of loss for
various 2-SCSs. Our scheme protects all states better than the optimal
Gaussian protection for a large $\alpha$. We note that the effect of dephasing
is negligible for the real coefficients of the superpositions. The loss
(a,b,c) for $1-\eta=0.01$ and (d,e,f) for $1-\eta=0.04$, on input states (a,d)
$\left|\alpha\right\rangle+\left|-\alpha\right\rangle$, (b,e)
$\left|\alpha\right\rangle+i\left|-\alpha\right\rangle$, and (e,f)
$\left|\alpha\right\rangle-\left|-\alpha\right\rangle$.

In Fig. A.2, it is shown how individual 2-SCS states evolve by the noise
channels in terms of fidelity. Our bypass protocol is surpassing the Gaussian
protocol in terms of the fidelity. Here, we can see that in some cases the
dephasing errors are not affecting the fidelities in the limit of large
$\alpha$. These cases correspond to the case where the qubit enters the
channel in the eigenstates of $\hat{\sigma}_{z}$.

##  Appendix B Performance on 4-SCSs

Figure B.1:  Various measures for the 4-SCS $\left|\alpha
e^{i\pi/4}\right\rangle-\left|\alpha e^{3i\pi/4}\right\rangle-\left|\alpha
e^{5i\pi/4}\right\rangle+\left|\alpha e^{7i\pi/4}\right\rangle$ for
$\alpha=8$. (a) The cross section of Wigner function along the X-axis for
$1-\eta=0.01$, and identical curves is obtained along the other axis due to
the symmetry. The curves for different dephasing factor $p$ are nearly
overlapped. (b) The scaling of the negative peak at the phase space origin vs.
$1-\eta$. (c) The fidelity vs. $\alpha$ at $1-\eta=0.01$. (d) The fidelity vs.
$1-\eta$ for $\alpha=8$.

In Fig. B.1, we calculated the Wigner functions and channel fidelity for the 4
SCSs under various levels of loss and dephasing. We analyzed the cases
corresponding to $\mu(\left|\alpha e^{i\pi/4}\right\rangle-\left|\alpha
e^{i5\pi/4}\right\rangle)+\nu(\left|\alpha
e^{i3\pi/4}\right\rangle-\left|\alpha e^{i7\pi/4}\right\rangle)$ for a special
encodings as in Leghtas et al. (2013); Ofek et al. (2016), especially for
$\mu=-\nu$. The Wigner functions of 4-SCSs are comprised of Gaussian peaks of
coherent components, the interference fringes of superposition of 2
components, and the interference fringes of superposition of 4 components
located near the phase space origin. The negative peaks of 4-components
interference near the phase space origin are well preserved even under the
effect of dephasing. Similarly as for 2-SCSs, we can also protect 4-SCSs with
an increased fidelity in large $\alpha$ limit against various level of losses
even under a modest level of qubit dephasing.

##  Appendix C  Conditional enhancement of protocol by detections

We might consider strategies compromising the deterministic nature in order to
enhance the robustness of the protocol. There are two ways to introduce the
projective detections in the protocols, i.e. in the oscillator and the
ancillary mode. If we measure the oscillator after the pre-processing, we can
partially filter out the erroneous components with a high success probability.
In (3) as in Fig. 1 (a), if we detect $\left|0\right\rangle\left\langle
0\right|$ on APD after the pre-processing $\hat{U}_{T}$, we are left with
nearly target state
$i\mu\left|-_{i}\right\rangle-i\nu\left|+_{i}\right\rangle$. If we detect non-
zero photon, we are left nearly with an erroneous mixed state of
$i\mu\left|-_{i}\right\rangle-i\nu\left|-_{i}\right\rangle$ and
$i\mu\left|-_{i}\right\rangle+i\nu\left|-_{i}\right\rangle$. If we have access
further to a parity detection or corresponding qubit detection through
dispersive interaction, we get
$i\mu\left|-_{i}\right\rangle-i\nu\left|+_{i}\right\rangle$ for even photon
number outcomes and
$i\mu\left|-_{i}\right\rangle+i\nu\left|-_{i}\right\rangle$ for odd outcomes.
After a conditional operation $\hat{\sigma}_{x}$ or $\hat{\sigma}_{z}$
depending on the outcomes, we get the target state from both states.

The projective qubit detection of the input qubit state after the entire
protocol can additionally filter out the error by loss and dephasing, and the
protective power is enhanced. The channel fidelity can be improved, e.g. from
approximately $0.98$ to $1-5\times 10^{-5}$ for $\alpha=2$ at loss
$1-\eta=0.04$ in the absense of dephasing. The success probability can be
close to 1 if the loss is low, e.g. $99.5\%$ in this example. This filtering
effect can appear for the other cases similaly, e.g. 4 SCS or ECSs.

##  Appendix D  Hybrid error correction of dephasing using Rabi coupling

We can actively correct the dephasing error in the qubits using Rabi coupling
with ancillary CV modes as follows. Suppose the qubit is in a state
$c_{1}\left|+\right\rangle+c_{2}\left|-\right\rangle$. A Rabi gate
$\exp[-i\sqrt{2}\beta\hat{\sigma}_{x}\hat{P}_{1}]$ with a small $\beta$ with
ancillary oscillator mode in vacuum $\left|0\right\rangle_{1}$ creates an
entangled form
$c_{1}\left|+\right\rangle\left|\beta\right\rangle+c_{2}\left|-\right\rangle\left|-\beta\right\rangle$.
We can ignore the loss in the oscillator if $\beta$ is small. Now, if qubit
state suffers a dephasing $\hat{\sigma}_{z}$, we have a state
$c_{1}\left|-\right\rangle\left|\beta\right\rangle+c_{2}\left|+\right\rangle\left|-\beta\right\rangle$
which corresponds to a bit-flip error of qubit in $\left|\pm\right\rangle$
basis. Now we apply another Rabi gate
$\exp[-i\sqrt{2}\beta\hat{\sigma}_{x}\hat{P}_{2}]$ on the qubit and a second
ancillary oscillator in vacuum $\left|0\right\rangle_{2}$ after the channel.
Then the state without error becomes
$c_{1}\left|+\right\rangle\left|\beta\right\rangle_{1}\left|\beta\right\rangle_{2}+c_{2}\left|-\right\rangle\left|-\beta\right\rangle_{1}\left|-\beta\right\rangle_{2}$,
and the erroneous state becomes
$c_{1}\left|-\right\rangle\left|\beta\right\rangle_{1}\left|-\beta\right\rangle_{2}+c_{2}\left|+\right\rangle\left|-\beta\right\rangle_{1}\left|\beta\right\rangle_{2}$
with dephasing. Now a phononic 50:50 beam splitter interaction is applied
between the ancillary modes which transforms the states as follows:

$\displaystyle\hat{U}_{BS}(c_{1}\left|+\right\rangle\left|\beta\right\rangle_{1}\left|\beta\right\rangle_{2}+c_{2}\left|+\right\rangle\left|-\beta\right\rangle_{1}\left|-\beta\right\rangle_{2})=c_{1}\left|+\right\rangle\left|\sqrt{2}\beta\right\rangle_{1}\left|0\right\rangle_{2}+c_{2}\left|+\right\rangle\left|-\sqrt{2}\beta\right\rangle_{1}\left|0\right\rangle_{2}.$
$\displaystyle\hat{U}_{BS}(c_{1}\left|-\right\rangle\left|\beta\right\rangle_{1}\left|-\beta\right\rangle_{2}+c_{2}\left|+\right\rangle\left|-\beta\right\rangle_{1}\left|\beta\right\rangle_{2})=c_{1}\left|-\right\rangle\left|0\right\rangle_{1}\left|-\sqrt{2}\beta\right\rangle_{2}+c_{2}\left|+\right\rangle\left|0\right\rangle_{1}\left|\sqrt{2}\beta\right\rangle_{2}.$
(12)

Now we notice that these two states can be distinguished: when the dephasing
occured, the phonons are found only in the mode 2, while without dephasing
only in the mode 1. Therefore, we can detect in which mode are phonons, and
depending on the outcome apply a correction operation on the qubit. Note that
the number of redundant oscillators affects the error rate and the success
probability of this protocol. Other types of error such as bit-flip error
$\hat{\sigma}_{x}$ can be similarly achieved by using the same formalism with
a different Rabi coupling with the substition of
$\hat{\sigma}_{x}\rightarrow\hat{\sigma}_{z}$.

An equivalent strategy also exist: the intermediate entangled state in
$c_{1}\left|+\right\rangle\left|\beta\right\rangle+c_{2}\left|-\right\rangle\left|-\beta\right\rangle$
becomes after another Rabi gate $\exp[-i\sqrt{2}\beta\hat{\sigma}_{x}\hat{P}]$
as
$c_{1}\left|+\right\rangle\left|2\beta\right\rangle+c_{2}\left|-\right\rangle\left|-2\beta\right\rangle$.
On the contrary, for the erroneous intermediate state in
$c_{1}\left|-\right\rangle\left|\beta\right\rangle+c_{2}\left|+\right\rangle\left|-\beta\right\rangle$,
after the same Rabi gate, we have
$c_{1}\left|-\right\rangle\left|0\right\rangle+c_{2}\left|+\right\rangle\left|0\right\rangle$
due to the opposite shift. If some phonons are detected, we will know that no
dephasing occured, and vise versa. These protocol requires phononic detectors
that are currently challenging, but can be achieved with complex Rabi circuits
and qubit detections.

##  Appendix E JC interactions and squeezings

JC interactions are atom-light interaction similar to Rabi gates, but more
available in higher frequency experiments. A JC interaction alone cannot
replace Rabi gates for our purposes as it lacks the counter-rotating term, but
there exist methods which can be derived using JC interactions. A digital
simulation of Rabi gate has been realized by combining JC interactions with a
synthetic anti-JC interactions by a Suzuki-Trotter formula Pedernales et al.
(2015). Alternatively, the JC interaction can be expressed as the combination
of two non-commuting Rabi gates as

$\displaystyle\exp[i\kappa(\hat{\sigma}_{+}a+\hat{\sigma}_{-}a^{\dagger})]=\exp[i\kappa\\{\frac{\hat{\sigma}_{x}+i\hat{\sigma}_{y}}{2}a+\frac{\hat{\sigma}_{x}-i\hat{\sigma}_{y}}{2}a^{\dagger}\\}]=\exp[i2^{-1/2}\kappa(\hat{\sigma}_{x}\hat{X}-\hat{\sigma}_{y}\hat{P})].$
(13)

Therefore, a Rabi gate can be achieved by removing one of the Rabi terms in JC
interaction.

In-line squeezing helps our scheme in many ways. First, squeezing can be
combined with our scheme to reduce the effect of loss by reducing the number
of excitations in the oscillator as in Filip (2013); Jeannic et al. (2018).
Second, squeezing increases the strength of Rabi gates from $T$ to $Te^{2r}$
as
$S[r]\exp[iT\hat{\sigma}_{x}\hat{X}]S[-r]=\exp[iTe^{2r}\hat{\sigma}_{x}\hat{X}]$,
so the accessible strength of Rabi gate can be increased. Third, the digital
simulation of Rabi gates by JC interactions can be improved by a squeezing
without accessing rapid qubit rotations: sandwiching a JC interaction with
squeezing and anti-squeezing, we suppress one of the Rabi terms, and obtain
$S[r]\exp[i2^{-1/2}\kappa(\hat{\sigma}_{x}\hat{X}-\hat{\sigma}_{y}\hat{P})]S[-r]=\exp[i2^{-1/2}\kappa(\hat{\sigma}_{x}e^{r}\hat{X}-\hat{\sigma}_{y}e^{-r}\hat{P})]$.
This single JC operation together with squeezing and anti-squeezing therefore
can play the composite Rabi gate $\hat{U}_{T}$ alone. Furthermore, by the
rotations in the qubit, we get a counter-rotating term
$\exp[i\pi\hat{\sigma}_{x}/2]\exp[i2^{-1/2}\kappa(\hat{\sigma}_{x}e^{r}\hat{X}-\hat{\sigma}_{y}e^{-r}\hat{P})]\exp[-i\pi\hat{\sigma}_{x}/2]=\exp[i2^{-1/2}\kappa(\hat{\sigma}_{x}e^{r}\hat{X}+\hat{\sigma}_{y}e^{-r}\hat{P})]$.
When $r$ is large, these two terms can be combined to achieve a single Rabi
gate $\exp[i2^{1/2}\kappa\hat{\sigma}_{x}e^{r}\hat{X}]$ efficiently.

For the oscillators where in-line squeezing is unavailable, squeezing can be
synthesized by alternating non-commuting Rabi gates
$\exp[i\tau\hat{\sigma}_{x}\hat{X}]$ and $\exp[i\tau\hat{\sigma}_{y}\hat{P}]$
as

$\displaystyle\exp[i\tau\hat{\sigma}_{x}\hat{X}]\exp[i\tau\hat{\sigma}_{y}\hat{P}]\exp[-i\tau\hat{\sigma}_{x}\hat{X}]\exp[-i\tau\hat{\sigma}_{y}\hat{P}]\approx\exp[-2i\tau^{2}\hat{\sigma}_{z}(\hat{X}\hat{P}+\hat{P}\hat{X})/2]\exp[i\tau^{2}\hat{\sigma}_{z}].$
(14)

Now when the qubit ancilla is in a ground state, this operation is reduced to
a single mode squeezing operation. These synthetic squeezings can be used to
enhance the strength of JC interactions in return. Alternatively, we can use
the squeezed state Hastrup et al. (2020b) for the ancilla to induce squeezing
on the target mode.

##  Appendix F  Enhancement by multiple Rabi gates

Figure F.1:  Channel fidelity vs. $\alpha$ of general 2-SCSs under losses (a)
$1-\eta=0.01$ and (b) $0.02$, and dephasings $p=0.00,0.05,0.10$. Using
multiple gates, the fidelity is improved, especially under a heavy loss.

The overall performance of the protocol can be improved further if a
constraint in the number of available Rabi gates is relaxed, by engineering a
nonlinear Rabi gate $\hat{U}_{D}^{V}=\exp[i\hat{\sigma}_{x}V(\hat{X})]$ with a
nonlinear potential $V(\hat{X})$. An ideal controlled qubit rotation is
written as
$\hat{U}_{D}^{\mathrm{sign}}=\exp[i\pi/4\hat{\sigma}_{x}\mathrm{sgn}[\hat{X}]]$,
where $\mathrm{sgn}[\hat{X}]$ is the sign function. By this interaction, the
qubit ancilla in $\left|g\right\rangle$ is rotated perfectly to
$\left|-\right\rangle_{i}$ for the oscillator states lying in right half-plane
of phase space, and to $\left|+\right\rangle_{i}$ for the oscillator states in
the left half-plane. A variant of Rabi gate
$\hat{U}_{D}^{\mathrm{tanh}}=\exp[i\pi/4\hat{\sigma}_{x}\tanh[\hat{X}]]$ acts
nearly as $\hat{U}_{D}^{\mathrm{sign}}$ for the states with a large
excitation. For large values of quadrature variable $X$, the functional
$\tanh[\hat{X}]$ takes a value of $1$ regardless of the value of $X$. However,
$\hat{U}_{D}^{\mathrm{tanh}}$ is a highly nonlinear operation which is
difficult to be implemented. A polynomial Rabi gate
$U^{\prime}_{1}=\exp[i\epsilon^{\prime}\hat{\sigma}_{x}\hat{X}^{k}]$ with
$\epsilon^{\prime}(\sqrt{2}\alpha)^{j}=\pi/4$ and odd $k$ generates qubit-
oscillator entanglement in a similar way as for $k=1$, but the state in the
oscillator is heavily disturbed due to the high nonlinearity of the
interaction degrading the performance. In contrast, another advanced nonlinear
Rabi gate has an optimal feasibility and performance:
$\hat{U}_{D}^{\mathrm{sin}}=\exp[-i\frac{\pi}{4}\hat{\sigma}_{x}\sin[\frac{\pi}{2\sqrt{2}\alpha}\hat{X}]]$.
This sine-Rabi gate can be approximately achieved by a sequence of three Rabi
gates and qubit rotations as in Park et al. (2018):

$\displaystyle\exp[-it_{1}\hat{X}\hat{\sigma}_{y}]\exp[it_{2}\hat{\sigma}_{z}]\exp[2it_{1}\hat{X}\hat{\sigma}_{y}]\exp[-it_{2}\hat{\sigma}_{z}]\exp[-it_{1}\hat{X}\hat{\sigma}_{y}]\approx\exp[i2t_{2}\hat{\sigma}_{x}\sin[2t_{1}\hat{X}]]$
(15)

at $t_{1}=\frac{\pi}{4\sqrt{2}\alpha}$ and $t_{2}=-\pi/8$, where the
approximation is good for small $t_{1,2}$. This gate acts very precisely like
a controlled qubit rotation $\hat{U}_{D}^{\mathrm{sign}}$ on the state whose
peaks in wavefunction are centered around $X=\pm\sqrt{2}\alpha$ as in Fig.
F.1, showing a larger enhancement in the protocol especially when the loss is
heavy. This result implies further that individual optimal protective schemes
might exist for different types of oscillator state.

##  Appendix G  protocols for non-conventional superposition of coherent
states

We now briefly describe how other superposition of coherent states can be
protected by a bypass strategy. We start from an illustrative example of 3-SCS
states $\left|\alpha\right\rangle+\left|\alpha
e^{i2\pi/3}\right\rangle+\left|\alpha e^{i4\pi/3}\right\rangle$, where the
three coherent peaks are forming a triangle in phase space. These peaks can be
shifted to be located at the vertices of a rectangle. First, we displace this
state by $\hat{D}[-\alpha/4]$ to make the peaks located symmetrically about
the P-axis of phase space. After a Rabi gate
$\exp[i\epsilon\hat{\sigma}_{x}\hat{X}]$ with $\epsilon=\pi/3\sqrt{2}\alpha$,
an entangled state is formed approximately as
$c_{1}\left|+_{i}\right\rangle\left|\alpha-\alpha/4\right\rangle+c_{2}\left|-_{i}\right\rangle\left|\alpha
e^{i2\pi/3}-\alpha/4\right\rangle+c_{3}\left|-_{i}\right\rangle\left|\alpha
e^{i4\pi/3}-\alpha/4\right\rangle$ with some coefficients $c_{j}$. Applying
another Rabi gate $\exp[-i\frac{\sqrt{6}}{4}\hat{\sigma}_{y}\hat{X}]$, the
total state becomes
$c_{1}\left|+_{i}\right\rangle\left|3\alpha/4-i\sqrt{3}\alpha/4\right\rangle+\left|-_{i}\right\rangle
c_{2}\left|-3\alpha/4-i\sqrt{3}\alpha/4\right\rangle+\left|-_{i}\right\rangle
c_{3}\left|-3\alpha/4+i3\sqrt{3}\alpha/4\right\rangle$ with coherent peaks
located symmetricaly along the X-axis. Now applying
$\hat{D}[-\sqrt{3}\alpha/4]$, this state becomes
$c_{1}\left|+_{i}\right\rangle\left|3\alpha/4-i\sqrt{3}\alpha/2\right\rangle+\left|-_{i}\right\rangle
c_{2}\left|-3\alpha/4-i\sqrt{3}\alpha/2\right\rangle+\left|-_{i}\right\rangle
c_{3}\left|-3\alpha/4+i\sqrt{3}\alpha/2\right\rangle$ where the peaks are now
located at the vertices of a rectangle, and thus the protective scheme of
4-SCS can finally be applied with use of an additional qubit ancilla.

Another possible extension is the superposition of coherent states located on
a line Shukla et al. (2019), such as
$\left|-3\alpha\right\rangle+\left|-\alpha\right\rangle+\left|\alpha\right\rangle+\left|3\alpha\right\rangle$.
Applying $\exp[i\frac{\pi}{4\sqrt{2}\alpha}\hat{\sigma}_{y}\hat{X}]$ with the
qubit in the ground state, an entangled state generated is
$c_{-3\alpha}\left|-\right\rangle\left|-3\alpha\right\rangle+c_{-\alpha}\left|+\right\rangle\left|-\alpha\right\rangle+c_{\alpha}\left|-\right\rangle\left|\alpha\right\rangle+c_{3\alpha}\left|+\right\rangle\left|3\alpha\right\rangle$.
The information encoded into qubit can again be used to bring the coherent
peaks closer to the phase space origin. Applying
$\exp[i\sqrt{2}\alpha\hat{\sigma}_{x}\hat{P}]$, the state will evolve into
$c^{\prime}_{-3\alpha}\left|-\right\rangle\left|-2\alpha\right\rangle+c^{\prime}_{-\alpha}\left|+\right\rangle\left|-2\alpha\right\rangle+c^{\prime}_{\alpha}\left|-\right\rangle\left|2\alpha\right\rangle+c^{\prime}_{3\alpha}\left|+\right\rangle\left|2\alpha\right\rangle$.
We can now use the formalism of 2-SCS bypass protocol to transform the
oscillator into vacuum.

A non-trivial question is if we can improve the protocol using more resources.
We note that a 2-SCS
$\mu\left|\alpha\right\rangle+\nu\left|-\alpha\right\rangle$ can be understood
as a special case of superposition of 4-SCS with zero weight in the 2 other
vertices of a rectangle $\left|-\alpha\right\rangle$ and $\alpha$. This
connection can be made clear by a Gaussian transformation
$R[-\pi/4]\hat{D}[i\alpha]$ to
$\mu\left|\sqrt{2}\alpha\right\rangle+\nu\left|\sqrt{2}i\alpha\right\rangle$,
which is a special case of
$\mu_{1}\left|\sqrt{2}\alpha\right\rangle+\mu_{2}\left|\sqrt{2}i\alpha\right\rangle+\mu_{3}\left|-\sqrt{2}\alpha\right\rangle+\mu_{4}\left|-\sqrt{2}i\alpha\right\rangle$.
Here we note that the absolute amplitude of the states has been increased from
$\alpha$ to $\sqrt{2}\alpha$. Therefore, using the 4-SCS bypass formalism with
2 ancillas, we can expect an enhanced protection.

The trick of using more ancillas for enhanced protection can be applied to
other n-SCSs using proper controlled rotations and displacements as follows.
For a general n-SCS, we can think of the following example. Let us assume the
n-SCS as the coherent states aligned at the grid-like locations. As this grids
can be viewed as the alignment in 2-dimension, each axis can be treated
separately as in the case of superposition of coherent states on a line. This
idea can also work to an arbitrary input unknown states Hastrup et al. .

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