PANTONE PANTONE PANTONE3015C PANTONE3015C 1 0.3 0 0.2 PANTONE Date of
publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2023.0322000

GSP acknowledges the project Business Finland QuTI (decision 41419/31/2020) as
well as support under a research grant agreement between Saab and Aalto
University. This work was performed as part of the Academy of Finland Centre
of Excellence program (project 352925).

Corresponding author: Hany Khalifa (e-mail: hany.khalifa@aalto.fi).

# Fault-tolerant one-way noiseless amplification for microwave bosonic quantum
information processing

HANY KHALIFA1  0000-0002-1276-5428 RIKU JÄNTTI1  GHEORGHE SORIN PARAOANU 2,3
Department of Information and Communications Engineering, Aalto University,
02150 Espoo, Finland  QTF Centre of Excellence, Department of Applied Physics,
Aalto University, 00076 Aalto, Finland  InstituteQ – the Finnish Quantum
Institute, Aalto University, 02150 Espoo, Finland

###### Abstract

Microwave quantum information networks require reliable transmission of single
photon propagating modes over lossy channels. In this article we propose a
microwave noise-less linear amplifier (NLA) suitable to circumvent the losses
incurred by a flying photon undergoing an amplitude damping channel (ADC). The
proposed model is constructed by engineering a simple one-dimensional four
node cluster state. Contrary to conventional NLAs based on quantum scissors
(QS), single photon amplification is realized without the need for photon
number resolving detectors (PNRDs). Entanglement between nodes comprising the
device’s cluster is achieved by means of a controlled phase gate (CPHASE).
Furthermore, photon measurements are implemented by quantum non demolition
detectors (QNDs), which are currently available as a part of circuit quantum
electrodynamics (cQED) toolbox. We analyze the performance of our device
practically by considering detection inefficiency and dark count probability.
We further examine the potential usage of our device in low power quantum
sensing applications and remote secret key generation (SKG). Specifically, we
demonstrate the device’s ability to prepare loss-free resources offline, and
its capacity to overcome the repeater-less bound of SKG. We compare the
performance of our device against a QS-NLA for the aforementioned
applications, and highlight explicitly the operating conditions under which
our device can outperform a QS-NLA. The proposed device is also suitable for
applications in the optical domain.

###### Index Terms:

Noiseless linear amplification (NLA), Cluster state quantum computing,
entanglement, Quantum non-demolition detection (QND), Remote entanglement,
Qubit protection, Secret key generation.

=-21pt

##  I Introduction

For a multitude of bosonic microwave quantum information processing (QIP)
tasks, the reliable transmission of a single flying photon is critical to
their success [1, 2]. However, due to the non-ideal transmissivity of
transmission media, the accumulation of propagation losses would eventually
result in a protocol failure. Seemingly, the most practical solution would be
to deploy phase insensitive amplifiers in order to combat channel attenuation.
Nonetheless, the process of amplification is always accompanied by at least
half a quantum of noise photons [3]. Unfortunately, a deterministic noise-free
amplifier is fundamentally prohibited by virtue of the no cloning theorem [4].
However, a nondeterministic noiseless amplifier (NLA) is physically possible
if the operation is restricted to a subset of states in the device Hilbert
space [5, 6].

Ever since the inception of NLAs, numerous experiments had been conducted in
order to demonstrate the concept [7]. On the application level, NLAs have
found great utility in fields such as entanglement distillation [8, 9],
device-independent quantum key distribution (DI-QKD) [10, 11], quantum
repeaters [12], and most recently quantum teleportation [13].

A crucial requirement for implementing NLAs as depicted in Fig. (1) is that
the detection process that heralds a successful probabilistic amplification is
accomplished by photon number resolving detectors (PNRDs), also know as photon
counters. This presents a major obstacle in front of physical implementations
in the microwave domain, as the low power of single microwave photons renders
the task of finding a suitable observable for photon numbers a challenging
one.

Recently the necessity of using PNRDs for the implementation of NLAs had been
relaxed by proposing a new model that employs a cluster state as a resource in
order to achieve noiseless amplification [14]. The concept followed the new
architecture of third generation quantum repeaters [15, 16], which is
characterized by its fault-tolerance capabilities. In the field of quantum
computation, the cluster state model is an alternative to the circuit-based
framework that utilizes a resource of highly entangled qubits [17]. The
cluster nodes (qubits) are initially prepared in the diagonal basis
$\\{\lvert+\rangle,\lvert-\rangle\\}$. Links (network edges) are established
between nodes by means of a CPHASE gate. Every quantum computing algorithm can
be realized on a cluster state resource by performing single qubit
measurements on the nodes comprising the cluster in a pre-defined Pauli basis
[17]. The same concept of cluster state computation or synonymously, the one-
way model, was later developed for the more versatile continuous variable (CV)
systems [18].

Our main objective in this article is to propose a NLA model suitable for
microwave bosonic QIP. Since PNRDs are currently not available in the
microwave domain, we propose utilizing microwave QNDs, which are efficient and
have low dark count rates [19, 20]. Furthermore, owing to their deterministic
generation process, we adopt bosonic temporal modes as an encoding scheme for
all the qubits involved in our protocol [21, 22, 23]. It is worth mentioning
at this point that the aforementioned characteristics are essentially
different from our previous proposal [14], that relied on coincidence counters
and polarization qubits.

Our proposed model is composed of a simple 4-node one dimensional cluster
state engineered in a manner that exactly simulates the dynamics of a heralded
NLA as depicted in Fig. (1). However, as we demonstrate in the upcoming
sections, this model is extremely powerful since it inherits the fault
tolerance capacity of the one-way quantum computing model.

In order to practically evaluate the performance of our proposed device, we
consider two important microwave QIP tasks that could be greatly enhanced by
the usage of our device. Firstly, we study the problem of storing a single
bosonic mode in a leaky memory element. Then we turn our attention towards the
task of entanglement generation between remote parties. We further consider
the remote generation of secure keys over the established entanglement
channel.

$\alpha\lvert 0\rangle_{I}+\beta\lvert 1\rangle_{I}$$\gamma$BS1
$E$$50/50$$t$BS2$\rho^{(\text{out})}$$\lvert
1\rangle_{F}$$A$$D_{-}$$D_{+}$$\tilde{E}$$\tilde{I}$$\tilde{F}$$\tilde{A}$NLA
Figure 1: Qubit noiseless amplification. The depicted setup is composed of
three beamsplitters and two PNRDs denoted by $D_{+}$ and $D_{-}$. The first
beamsplitter, BS1, models a lossy channel with transmissivity $\gamma$. The
second one, BS2, has a variable transmissivity $t$, whereas the third is
balanced. The output of a lossy channel is a mixture of single photon state
and vacuum. The amplifier rescales the probabilities of the channel’s output
density operator in favor of the single photon component. Noiseless
amplification succeeds probabilistically by registering one and only one click
in either $D_{+}$ or $D_{-}$.

This article is organized as follows: In Section (2) we start with a formal
description of the theory of NLA followed by a mathematical representation of
damping and losses. Then we focus on the inner workings of a heralded NLA
based on quantum scissors. Section (3) is dedicated to our one-way NLA.
Firstly we briefly introduce the mathematical formalism of bosonic temporal
modes. After that we describe in details the working principle of our device.
Then we study the effect of dark counts and non ideal efficiency on the
device’s success probability. Finally the last part of this section focuses on
the relation between the success probability and the tuning parameter of our
device. In section (4) we study two important applications of our device.
Firstly we tackle the problem of storing an idler mode in an entanglement-
enhanced sensing protocol. We show the cost at which our device is capable of
fully restoring the attenuated mode. Then we turn our attention towards
another interesting problem, that is, remote entanglement generation. We first
demonstrate the ability of our device to establish remote entanglement
deterministically, due to its fault tolerant capacity. After that we examine
the secret key rate that can be generated accordingly. We mainly focus in this
final section on showing the ability of our cluster state NLA to beat the
repeater-less bound of remote SKG, and we further highlight the operating
conditions under which it can outperform a NLA based on QS. Finally section
(5) is our conclusion.

##  II THEORY OF NLA

We begin this section with a brief definition of NLA, then formally define
channel losses. In the last part of this section we describe how amplification
is achieved with a QS-NLA.

###  II-A Formal definition of NLA

Ideally a NLA can be described by defining the following operator [5]

$\displaystyle\hat{T}$ $\displaystyle=g^{\hat{n}},$ (1)

where $\lvert g\lvert\geq 1$ is a real number characterizing the amplifier’s
gain, and $\hat{n}$ is the number operator.

Consider now a coherent state $\lvert\alpha\rangle$ undergoing the
aforementioned transformation

$\displaystyle\hat{T}\lvert\alpha\rangle$
$\displaystyle=g^{\hat{n}}\Big{[}e^{\frac{-\lvert\alpha\lvert^{2}}{2}}\underset{n=0}{\overset{\infty}{\sum}}\frac{\alpha^{n}}{\sqrt{n!}}\lvert
n\rangle\Big{]}$
$\displaystyle=e^{\frac{-\lvert\alpha\lvert^{2}}{2}}\underset{n=0}{\overset{\infty}{\sum}}\frac{\alpha^{n}}{\sqrt{n!}}g^{\hat{n}}\lvert
n\rangle$
$\displaystyle=e^{\frac{-\lvert\alpha\lvert^{2}}{2}}\underset{n=0}{\overset{\infty}{\sum}}\frac{(g\alpha)^{n}}{\sqrt{n!}}\lvert
n\rangle$ $\displaystyle=e^{\frac{\lvert\alpha\lvert^{2}}{2}(g^{2}-1)}\lvert
g\alpha\rangle,$ (2)

where $\lvert g\alpha\rangle=e^{\frac{-\lvert
g\alpha\lvert^{2}}{2}}\underset{n=0}{\overset{\infty}{\sum}}\frac{(g\alpha^{n})}{\sqrt{n!}}\lvert
n\rangle$,
$g^{\hat{n}}=e^{\ln{g^{\hat{n}}}}=e^{(\ln{g})\hat{n}}=\underset{n=0}{\overset{\infty}{\sum}}[(\ln{g})\hat{n}]^{n}/n!$,
$\hat{n}\lvert n\rangle=n\lvert n\rangle$, and
$P_{d}=e^{\lvert\alpha\lvert^{2}(g^{2}-1)}$ is the success probability of this
transformation which equals to the norm of the output state [5].

A linear operator in a Hilbert space, $\hat{O}\in\mathcal{L}(\mathcal{H})$,
$\hat{O}:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$, is bounded if there
exists a constant $\mathcal{K}\leq 0$ such that [24]

$\displaystyle\lvert\lvert\hat{O}v\lvert\lvert_{\mathcal{H}_{2}}\leq\mathcal{K}\lvert\lvert
v\lvert\lvert_{\mathcal{H}_{1}}\hskip 5.0pt\forall v\in\mathcal{H}_{1}$ (3)

where $\mathcal{H}_{1}$, $\mathcal{H}_{2}$ are the domain and target Hilbert
spaces respectively, $v$ is a ket vector, and the smallest possible value of
$\mathcal{K}$ is called the operator norm, which is denoted by
$\lvert\lvert\hat{O}\lvert\lvert$.

The transformation described in Eq. (2) effectively multiplies the coherent
field input amplitude $\alpha$ by a gain factor $g$ without the addition of
any extra noise, $\hat{T}\lvert\alpha\rangle\rightarrow c\lvert
g\alpha\rangle$, where $c$ is a complex number satisfying $0<\lvert
c\lvert<1$. However, in order for this transformation to be valid for all gain
values, the operator $\hat{T}$ has to be unbounded, and hence the entire
process becomes nonphysical. Overcoming this problem was achieved in the
original proposal [5] by splitting the input coherent state evenly on an
$N$-port beamsplitter, such that each individual photon comprising the
original input enters a quantum scissor (QS) device [25] similar to the one
depicted in Fig. (1). The overall success of the device is the independent
successes of each QS, which is heralded by registering one and only photon in
either $D_{+}$ or $D_{-}$. Finally the outputs of the $N$ quantum scissors are
interferometrically recombined on a second N-port beamsplitter in order to
produce the desired amplified output. When $N$ is large, the success
probability of this device becomes
$P_{d}\approx(1-t)^{\frac{N}{2}}e^{\lvert\alpha\lvert^{2}(g^{2}-1)}$ which
decreases when the variable transmissivity (see Fig. (1)) $t>1/2$, and $N$ is
large. Consequently this bounds the operator’s norm which deems the
transformation physical.

The vanishing success probability as the number of ports increases implies
that NLA is physically tangible only when considering a truncation of the
original input state. Alternatively by limiting the device’s operation to
small input states, $\lvert\alpha\lvert\ll 1$, the operator $\hat{T}$ can be
made physical by using only one QS.

###  II-B Channel model

In bosonic QIP information loss is dominantly due to energy dissipative
processes. Mathematically a lossy transmission medium is described as an
amplitude damping channel (ADC) [26]. In the language of quantum operations,
ADC transforms an input positive semidefinite (PSD) density operator
$\rho_{\text{in}}\geq 0$, into an output PSD, thus rendering it a completely
positive trace preserving map (CPTP) [27]. The quantum operation corresponding
to a qubit ADC can be written in terms of its Kraus representation [28],
$\mathcal{E}_{\text{AD}}(\rho_{\text{in}})=\underset{k=0}{\overset{1}{\sum}}K_{k}\rho_{\text{in}}K^{\dagger}_{k}=K_{0}\rho_{\text{in}}K^{\dagger}_{0}+K_{1}\rho_{\text{in}}K^{\dagger}_{1}$,
where $K_{0}=\left(\begin{smallmatrix}1&0\\\
0&\sqrt{\gamma}\end{smallmatrix}\right)$,
$K_{1}=\left(\begin{smallmatrix}0&\sqrt{1-\gamma}\\\
0&0\end{smallmatrix}\right)$ satisfy
$\underset{k=0}{\overset{1}{\sum}}K^{\dagger}_{k}K_{k}=I$, such that $I$ is
the two dimensional identity matrix, and $\gamma$ is the survaivability of one
photon. Practically, a qubit ADC can be modelled by a unitary passive
beamsplitter transformation, where the beamsplitter’s unused port injects a
vacuum mode, $U_{\theta}=e^{-i\theta H/\hbar}$, where
$H=i(a^{\dagger}b+b^{\dagger}a)$ is the beamsplitter’s Hamiltonian, $\theta$
is its angle, and $a$ and $b$ are its first and second modes respectively. An
arbitrary single photon state state occupying the first mode evolves under the
previous Hamiltonian as $U_{\theta}(\alpha\lvert 0_{a}\rangle+\beta\lvert
1_{a}\rangle)\otimes\lvert 0_{b}\rangle=(\alpha\lvert
0_{a}\rangle+\beta\cos{\theta}\lvert 1_{a}\rangle)\otimes\lvert
0_{b}\rangle+\beta\sin{\theta}\lvert 0_{a}\rangle\otimes\lvert 1_{b}\rangle$,
where $U_{\theta}a^{\dagger}U^{\dagger}_{\theta}U_{\theta}\lvert
0_{a}0_{b}\rangle=U_{\theta}a^{\dagger}U^{\dagger}_{\theta}\lvert
0_{a}0_{b}\rangle=a^{\dagger}\cos{\theta}+b^{\dagger}\sin{\theta}\lvert
0_{a}0_{b}\rangle$. Then, tracing out the environment mode $b$ leaves the
input state in a mixture,
$\rho_{\text{out}}=(1/(1+\beta^{2}\sin^{2}{\theta}))\lvert\Psi\rangle\langle\Psi\lvert+\big{(}(\beta^{2}\sin^{2}{\theta})/(1+\beta^{2}\sin^{2}{\theta})\big{)}\lvert
0\rangle\langle 0\lvert$, where $\lvert\Psi\rangle=(\alpha\lvert
0\rangle+\beta\cos{\theta}\lvert
1\rangle)/(\sqrt{\alpha^{2}+\beta^{2}\cos^{2}{\theta}})$. The Krauss
representation defined earlier can yield the same output mixed state by
defining $\gamma=\cos^{2}{\theta}$, and thus $1-\gamma=\sin^{2}{\theta}$ would
be the probability that a photon is lost.

It is useful at this point to introduce the gate-based model of amplitude
damping. This will be recalled when we develop our one-way NLA. Consider an
arbitrary single photon state, $\alpha\lvert 0\rangle_{c}+\beta\lvert
1\rangle_{c}$, as the control of a two-qubit controlled rotation gate, $\lvert
0\rangle_{c}\langle 0\lvert_{c}\otimes I_{t}+\lvert 1\rangle_{c}\langle
1\lvert_{c}\otimes R^{\theta}_{t}$, where
$R^{\theta}=\left(\begin{smallmatrix}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\end{smallmatrix}\right)$, and the gate’s target is
prepared in a vacuum state $\lvert 0\rangle_{t}$. The gate’s output can be
written as, $\big{[}\alpha\lvert 0\rangle_{c}\otimes\lvert
0\rangle_{t}\big{]}+\big{[}\beta\lvert
1\rangle_{c}\otimes\big{(}\cos{\theta}\lvert 0\rangle_{t}+\sin{\theta}\lvert
1\rangle_{t}\big{)}\big{]}$. Then using the same output as an input to a
controlled not (CNOT) entangling gate, $\text{CNOT}=\lvert 0\rangle_{c}\langle
0\lvert_{c}\otimes I_{t}+\lvert 1\rangle_{c}\langle 1\lvert_{c}\otimes X_{t}$,
where $X$ is the Pauli bit flip operation, and exchanging the roles of control
and target qubits, yields the following output state, $\alpha\lvert
0_{t}\rangle\otimes\lvert 0_{c}\rangle+\beta\cos{\theta}\lvert
1_{t}\rangle\otimes\lvert 0_{c}\rangle+\beta\sin{\theta}\lvert
0_{t}\rangle\lvert 1_{c}\rangle$. Finally after tracing out the control qubit,
it can be straightforwardly verified that the output is similar to that of the
aforementioned ADC.

An equivalent model for a qubit ADC can be obtained by using a controlled
phase operation (CPHASE), two Hadamard gates and employing the relation
$X=HZH$, where $H$ is a Hadamard gate, and $Z$ is a Pauli phase flip operator
[26]. This will be more clear in the next section when developing our device
model.

###  II-C QS-NLA

Fig. (1) depicts an experimental setup of a NLA based on a single QS. At a
transmitter an arbitrary single photon state $\alpha\lvert
0\rangle_{I}+\beta\lvert 1\rangle_{I}$, where
$\lvert\alpha\lvert^{2}+\lvert\beta\lvert^{2}=1$, is prepared. The photon
undergoes an ADC characterized by a unitary beamsplitter transformation as
described in the previous section. By choosing an appropriate beamsplitter
angle $\theta$, an ADC can be simply written as a $2\times 2$ matrix, ${\rm
BS1}=\left(\begin{smallmatrix}\sqrt{\gamma}&-\sqrt{1-\gamma}\\\
\sqrt{1-\gamma}&\sqrt{\gamma}\end{smallmatrix}\right)$. In the Heisenberg
picture the output modes are transformed by the beamsplitter matrix as

$\displaystyle\tilde{I}$ $\displaystyle=\sqrt{\gamma}I+\sqrt{1-\gamma}E$
$\displaystyle\tilde{E}$ $\displaystyle=\sqrt{\gamma}E-\sqrt{1-\gamma}I,$ (4)

whereas, in the Schrödinger picture an input state vector evolves as

$\displaystyle\lvert\Phi\rangle_{IE}$ $\displaystyle=(\alpha\lvert
0\rangle_{I}+\beta\lvert 1\rangle_{I})\otimes\lvert 0\rangle_{E}$
$\displaystyle=\alpha\lvert 00\rangle_{{I}{E}}+\beta I^{\dagger}\lvert
00\rangle_{{I}{E}}$ $\displaystyle\lvert\Phi\rangle_{\tilde{I}\tilde{E}}$
$\displaystyle=(\alpha\lvert 0\rangle_{\tilde{I}}+\sqrt{\gamma}\beta\lvert
1\rangle_{\tilde{I}})\otimes\lvert
0\rangle_{\tilde{E}}-\sqrt{1-\gamma}\beta\lvert
0\rangle_{\tilde{I}}\otimes\lvert 1\rangle_{\tilde{E}}.$ (5)

Simultaneously an auxiliary single photon is mixed with a vacuum input on a
second beamsplitter defined by ${\rm
BS2}=\left(\begin{smallmatrix}\sqrt{t}&-\sqrt{1-t}\\\
\sqrt{1-t}&\sqrt{t}\end{smallmatrix}\right)$. We assume that the
beamsplitter’s transmissivity $t$ is controllable and can be adjusted as
desired. Accordingly the beamsplitter’s modes transform as

$\displaystyle\tilde{F}$ $\displaystyle=\sqrt{t}F-\sqrt{1-t}A$
$\displaystyle\tilde{A}$ $\displaystyle=\sqrt{t}A+\sqrt{1-t}F,$ (6)

and hence the auxiliary photon state vector evolves as

$\displaystyle F^{\dagger}\lvert 0\rangle_{A}\lvert 0\rangle_{F}$
$\displaystyle=(\sqrt{t}\tilde{F}^{\dagger}+\sqrt{1-t}\tilde{A}^{\dagger})\lvert
00\rangle_{\tilde{A}\tilde{F}}$
$\displaystyle\lvert\Phi\rangle_{\tilde{A}\tilde{F}}$
$\displaystyle=\sqrt{1-t}\lvert 10\rangle_{\tilde{A}\tilde{F}}+\sqrt{t}\lvert
01\rangle_{\tilde{A}\tilde{F}}.$ (7)

The two modes, $\tilde{I}$, $\tilde{A}$ are then redirected towards a balanced
beamsplitter for a Bell state measurement (BSM). They evolve according to

$\displaystyle\tilde{I}$ $\displaystyle=\frac{1}{\sqrt{2}}(D_{+}-D_{-})$
$\displaystyle\tilde{A}$ $\displaystyle=\frac{1}{\sqrt{2}}(D_{+}+D_{-}),$ (8)

where $D_{+}$ and $D_{-}$ are detector operators.

Hence the overall state becomes

$\displaystyle\lvert\Phi^{(\mathrm{out})}\rangle$
$\displaystyle=\lvert\Phi\rangle_{\tilde{I}\tilde{E}}\otimes\lvert\Phi\rangle_{\tilde{A}\tilde{F}}$
$\displaystyle=(\alpha\lvert
00\rangle_{\tilde{I}\tilde{E}}+\beta\sqrt{\gamma}\lvert
10\rangle_{\tilde{I}\tilde{E}}-\beta\sqrt{1-\gamma}\lvert
01\rangle_{\tilde{I}\tilde{F}})$ $\displaystyle\otimes(\sqrt{1-t}\lvert
10\rangle_{\tilde{A}\tilde{F}}+\sqrt{t}\lvert 01\rangle_{\tilde{A}\tilde{F}})$
$\displaystyle=\alpha\sqrt{1-t}\lvert
0010\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}+\alpha\sqrt{t}\lvert
0001\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{\gamma(1-t)}\lvert
1010\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}+\beta\sqrt{\gamma t}\lvert
1001\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle-\beta\sqrt{(1-\gamma)(1-t)}\lvert
0110\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle-\beta\sqrt{(1-\gamma)t}\lvert
0101\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle=(\alpha\sqrt{1-t}\tilde{A}^{\dagger}+\alpha\sqrt{t}\tilde{F}^{\dagger}+\beta\sqrt{\gamma(1-t)}\tilde{I}^{\dagger}\tilde{A}^{\dagger}$
$\displaystyle+\beta\sqrt{\gamma
t}\tilde{I}^{\dagger}\tilde{F}^{\dagger}-\beta\sqrt{(1-\gamma)(1-t)}\tilde{E}^{\dagger}\tilde{A}^{\dagger}$
$\displaystyle-\beta\sqrt{(1-\gamma)t}\tilde{E}^{\dagger}\tilde{F}^{\dagger})\lvert
0000\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}.$ (9)

The required event for a successful amplification occurs when one and only one
detector clicks. Thus, when $D^{+}$ clicks, a BSM projects the overall state
to

$\displaystyle\lvert\Phi^{(\rm{out})}_{+}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(\alpha\sqrt{1-t}\lvert
0\rangle_{\tilde{F}}+\beta\sqrt{\gamma t}\lvert
1\rangle_{\tilde{F}})\otimes\lvert 0\rangle_{\tilde{E}}$
$\displaystyle-\frac{1}{\sqrt{2}}(\beta\sqrt{(1-\gamma)(1-t)})\lvert
0\rangle_{\tilde{F}}\otimes\lvert 1\rangle_{\tilde{E}},$ (10)

whereas when $D^{-}$ clicks the overall state becomes

$\displaystyle\lvert\Phi^{(\rm{out})}_{-}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(\alpha\sqrt{1-t}\lvert
0\rangle_{\tilde{F}}-\beta\sqrt{\gamma t}\lvert
1\rangle_{\tilde{F}})\otimes\lvert 0\rangle_{\tilde{E}}$
$\displaystyle-\frac{1}{\sqrt{2}}(\beta\sqrt{(1-\gamma)(1-t)})\lvert
0\rangle_{\tilde{F}}\otimes\lvert 1\rangle_{\tilde{E}},$ (11)

where the single photon sign flip in the previous equation can be fixed by an
application of a Pauli phase flip operator $Z$ on mode $\tilde{F}$.

Tracing out the environment mode $\tilde{E}$ yields the density operator of
the amplifier’s output

$\displaystyle\rho^{\text{(out)}}$
$\displaystyle=\frac{1}{N}\big{[}\lvert\Psi^{\text{(out)}}\rangle\langle\Psi^{\text{(out)}}\lvert+\beta^{2}(1-\gamma)(1-t)\lvert
0\rangle\langle 0\lvert\big{]},$ (12)

where $\lvert\Psi^{\text{(out)}}\rangle=\alpha\sqrt{(1-t)}\lvert
0\rangle_{\tilde{F}}+\beta\sqrt{\gamma t}\lvert 1\rangle_{\tilde{F}}$, and
$N=\alpha^{2}(1-t)+\beta^{2}[\gamma t+(1-\gamma)(1-t)]$ is a normalization
constant.

The transformation applied by a NLA on the input state lead to a change in the
weightings of the vacuum and single photon components. As can be seen, an
increase in the value of the tunable parameter $t$ rescales the input photon
probability amplitudes in favor of the single photon component. The purely
vaccuum output corresponds to an error event or more precisely a failed
amplification attempt which embodies the probabilistic nature of such device.

The success probability of the amplifier can be calculated as

$\displaystyle P_{\mathrm{succ}}$
$\displaystyle=2\langle\Phi^{(\rm{out})}_{+}\lvert\Phi^{(\rm{out})}_{+}\rangle$
$\displaystyle=\alpha^{2}(1-t)+\beta^{2}[(\gamma t+(1-\gamma)(1-t)],$

where the factor of $2$ comes from the fact that a click in either $D_{+}$ or
$D_{-}$ signifies a successful amplification.

The implementation of the previous amplification protocol in the microwave
domain is extremely challenging, since microwave single photon counters are
still at the conceptual level. Furthermore, when a QS-NLA fails, that is, none
of the detectors register any photon or they capture both photons, the entire
process has to be repeated again, since the amplifier’s output in this case is
two uncorrelated error modes. By exploiting a cluster state of entangled
photons, we demonstrate in the next section that we can achieve microwave NLA
by using the available microwave technology. Furthermore, we show that our
device is fault-tolerant and can operate on the protocol’s failed instances to
correct an error state, which is a feature that cannot be attained by QS-NLA.

##  III MICROWAVE ONE-WAY NLA

For microwave bosonic QIP, the most efficient state of the art encoding scheme
is photon temporal modes (TMs) [23, 22]. This is due to the unprecedented
controllability of light-matter couplings in cQED systems that facilitated the
generation and absorption of single photon wavepackets. Thus we spend the
first part of this section introducing the mathematical formalism of bosonic
temporal modes [23]. After that we proceed with explaining the inner workings
of our device.

###  III-A Bosonic temporal modes

A single bosonic mode occupying a particular TM can be represented as a
coherent superposition of a continuum of single bosonic monochromatic modes

$\displaystyle\ket{1_{\alpha}}=\underset{-\infty}{\overset{\infty}{\int}}d\omega\hskip
1.99997pt\alpha_{j}(\omega)a^{\dagger}(\omega)\ket{0},$ (13)

where $\alpha_{j}(\omega)$ is a complex-valued spectral function of the
corresponding wave-packet,
$\underset{-\infty}{\overset{\infty}{\int}}d\omega\lvert\alpha(\omega)\rvert^{2}=2\pi$,
$a^{\dagger}(\omega)$ is a monochromatic creation operator and the vacuum
state is assumed to be multimode, i.e.,
$\underset{m=1}{\overset{\infty}{\bigotimes}}\ket{0}$.

In the time domain the previous representation can be interpreted as a
coherent superposition of a continuum of creation times

$\displaystyle\ket{1_{\alpha}}$
$\displaystyle=\underset{-\infty}{\overset{\infty}{\int}}dt\hskip
1.99997pt\alpha_{j}(t)a^{\dagger}(t)\ket{0}$
$\displaystyle=\tilde{a}_{j}^{\dagger}\ket{0},$ (14)

where $\tilde{a}_{j}^{\dagger}$ is denoted as broadband operator, and the
following Fourier relations were assumed

$\displaystyle a(t)$
$\displaystyle=\frac{1}{2\pi}\underset{-\infty}{\overset{\infty}{\int}}d\omega
e^{-i\omega t}a(\omega)$ $\displaystyle a(\omega)$
$\displaystyle=\underset{-\infty}{\overset{\infty}{\int}}dte^{i\omega t}a(t)$
$\displaystyle\alpha(t)$
$\displaystyle=\underset{-\infty}{\overset{\infty}{\int}}d\omega e^{-i\omega
t}\alpha(\omega)$ $\displaystyle\alpha(\omega)$
$\displaystyle=\frac{1}{2\pi}\underset{-\infty}{\overset{\infty}{\int}}d\omega
e^{i\omega t}\alpha(t).$ (15)

Together with the convention that the adjoint operator is defined as
$a^{\dagger}(\omega)=[a(-\omega)]^{\dagger}=\underset{-\infty}{\overset{\infty}{\int}}d\omega
e^{i\omega t}a(t)$, and the wave-packet bosonic commutation relations
$[a(\omega),a^{\dagger}(\omega^{\prime})]=2\pi\delta(\omega-\omega^{\prime})$,
whereas in the time-domain
$[a(t),a^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$, we have a complete
characterization of the creation and annihilation operators in each of the
frequency and time domains.
TMs comprise a basis for a single boson wavepacket. Thus any arbitrary single
bosonic mode can be expanded as a linear superposition of TMs

$\displaystyle\ket{\Psi}=\underset{j=0}{\overset{\infty}{\sum}}c_{j}\tilde{a}_{j}^{\dagger}\ket{0},$
(16)

where $c_{j}$ is a complex amplitude.

Accordingly, a 2-dimensional bosonic space can be defined, such that any
arbitrary qubit is written as

$\displaystyle\ket{\Psi}=\underset{j=0}{\overset{1}{\sum}}c_{j}\tilde{a}_{j}^{\dagger}\ket{0},$
(17)

$\mathrm{X}$$\mathrm{X}$$E$ $I$ $A$ $F$ $\mathrm{CZ}$ $\mathrm{CZ}$
$\mathrm{CZ}$ Bell measurement$\tilde{E}$ $\tilde{F}$ Figure 2: A schematic of
the one-way NLA. The amplification process is composed of three steps. First
state preparation, this is done by entangling each of the $I-E$ and $A-F$
pairs by means of a CZ operation. After that, a BSM is performed on the middle
nodes, $I$ & $A$. This is achieved by first entangling the two nodes via a CZ
gate followed by two projective X-basis measurements. As a result, the two
nodes are removed from the cluster and nodes $E$ & $F$ are directly entangled.
Finally, according to the measurement outcomes we either perform error
correction or not. The protocol is a reminiscent of an entanglement swapping
based quantum repeater

where the logical zero basis state is defined as,
$\ket{\tilde{0}}=\tilde{a}^{\dagger}_{0}\lvert
0\rangle=\underset{-\infty}{\overset{\infty}{\int}}d\omega\hskip
1.99997pt\alpha_{j}(\omega)a^{\dagger}(\omega)\ket{0}$, whereas the logical
one is, $\ket{\tilde{1}}=\tilde{a}^{\dagger}_{1}\lvert
0\rangle=\underset{-\infty}{\overset{\infty}{\int}}d\omega\hskip
1.99997pt\beta_{j}(\omega)a^{\dagger}(\omega)\ket{0}$, such that
$\langle\alpha_{j}\lvert\beta_{j}\rangle=\underset{-\infty}{\overset{\infty}{\int}}d\omega\bar{\alpha_{j}}\beta_{j}=0$
are orthonormal complex temporal modes.

Experimentally an arbitrary TM qubit is prepared by splitting the photon’s
wavepacket on a beamsplitter, more precisely its microwave counterpart, the
quadrature hybrid [29, 21, 30], $\tilde{a}^{\dagger}_{b}\lvert
00\rangle=(\sqrt{T}\tilde{a}^{\dagger}_{1}+\sqrt{1-T}\tilde{a}^{\dagger}_{v})\lvert
00\rangle=c_{0}\lvert\tilde{0}\rangle+c_{1}\lvert\tilde{1}\rangle$, where
$\tilde{a}_{b}$ is the beamsplitter’s output, $T$ is its transmsissivity,
$c_{0}$, $c_{1}$ are equal to $\sqrt{1-T}$, $\sqrt{T}$ respectively and we
have assumed a dual-rail encoding defined as $\lvert
0_{1}1_{v}\rangle\rightarrow\lvert\tilde{0}\rangle$, $\lvert
1_{1}0_{v}\rangle\rightarrow\lvert\tilde{1}\rangle$. It is also worth
mentioning that a Hadamard transformation can be realized in a similar way
[31].

The qubit space defined in Eq. (17) can be regarded as analogous to the two
dimensional polarization space, where the basis vectors $\\{\lvert
H\rangle,\lvert V\rangle\\}$ correspond to the bosonic polarization state. For
ease of readability we drop all tildes, subscripts from all qubit TMs and
their corresponding operators, since it is now clear that a qubit’s TM space
is isomorphic [26] to a conventional two-dimensional Hilbert space.

###  III-B Device description

The core idea of our device is that a cluster state model is capable of
simulating the dynamics of any circuit-based quantum algorithm. Furthermore,
the model’s inherent redundancy guarantees fault tolerance. Our task in this
section is to show that the setup described in Fig. (1) can be simulated by a
cluster model (see Fig. (2)) with the capacity of being immune to failed
attempts, thus promising an uninterrupted performance.

Consider an arbitrary single photon undergoing a passive loss channel, an
amplitude damping channel (ADC), as previously described. By adopting the
gate-based ADC model explained earlier, the channel’s dynamics can be
described as follows

$\displaystyle\lvert\Phi\rangle_{EI}$ $\displaystyle=(\alpha\lvert
0\rangle_{E}+\beta\lvert 1\rangle_{E})\otimes\lvert 0\rangle_{I}$
$\displaystyle{\rm{CR}}^{\theta}\lvert\Phi\rangle_{EI}$
$\displaystyle=\alpha\lvert 00\rangle_{EI}+\beta\sqrt{1-\gamma}\lvert
10\rangle_{EI}$ $\displaystyle+\beta\sqrt{\gamma}\lvert 11\rangle_{EI},$
$\displaystyle({\rm{H}}\otimes{\rm{I}}){\rm{CR}}^{\theta}\lvert\Phi\rangle_{EI}$
$\displaystyle=\alpha\lvert+0\rangle_{EI}+\beta\sqrt{1-\gamma}\lvert-0\rangle_{EI}$
$\displaystyle+\beta\sqrt{\gamma}\lvert-1\rangle_{EI}$
$\displaystyle{\rm{CZ}}({\rm{H}}\otimes{\rm{I}}){\rm{CR}}^{\theta}\lvert\Phi\rangle_{EI}$
$\displaystyle=\alpha\lvert+0\rangle_{EI}+\beta\sqrt{1-\gamma}\lvert-0\rangle_{EI}$
$\displaystyle+\beta\sqrt{\gamma}\lvert+1\rangle_{EI},$

finally,

$\displaystyle({\rm{H}}\otimes{\rm{I}}){\rm{CZ}}({\rm{H}}\otimes{\rm{I}}){\rm{CR}}^{\theta}\lvert\Phi\rangle_{EI}=\lvert\Phi\rangle_{\tilde{E}\tilde{I}},$
$\displaystyle\lvert\Phi\rangle_{\tilde{I}\tilde{E}}=(\alpha\lvert
0\rangle_{\tilde{I}}+\beta\sqrt{\gamma}\lvert
1\rangle_{\tilde{I}})\otimes\lvert
0\rangle_{\tilde{E}}+\beta\sqrt{1-\gamma}\lvert
0\rangle_{\tilde{I}}\otimes\lvert 1\rangle_{\tilde{E}},$ (18)

where $\mathrm{CZ}=\lvert 0\rangle_{c}\langle
0\lvert_{c}\otimes\mathrm{I}_{t}+\lvert 1\rangle_{c}\langle
1\lvert_{c}\otimes\mathrm{Z}_{t}$, $\mathrm{I}=\lvert 0\rangle\langle
0\lvert+\lvert 1\rangle\langle 1\lvert$, $\mathrm{Z}=\lvert 0\rangle\langle
0\lvert-\lvert 1\rangle\langle 1\lvert$, $\mathrm{H}=\frac{1}{\sqrt{2}}(\lvert
0\rangle\langle 0\lvert+\lvert 0\rangle\langle 1\lvert+\lvert 1\rangle\langle
0\lvert-\lvert 1\rangle\langle 1\lvert)$, and $\lvert 0\rangle_{c}\langle
0\lvert_{c}\otimes{\rm{I}}_{t}+\lvert 1\rangle_{c}\langle
1\lvert_{c}\otimes{\rm{R}}^{\theta}_{t}$, where
${\rm{R}}^{\theta}=\cos{\theta}\lvert 0\rangle\langle 0\lvert$
$-\sin{\theta}\lvert 0\rangle\langle 1\lvert+\sin{\theta}\lvert
1\rangle\langle 0\lvert+\cos{\theta}\lvert 1\rangle\langle 1\lvert$,
$\lvert\alpha\lvert^{2}+\lvert\beta\lvert^{2}=1$, and
$\gamma=\cos^{2}{\theta}$.

Simultaneously two resource modes are prepared, such that, one is prepared as
a $\lvert+\rangle_{F}=(1/\sqrt{2})(\lvert 0\rangle_{F}+\lvert 1\rangle_{F})$,
while the other is a weighted superposition
$\lvert\Phi\rangle_{A}=\sqrt{1-t}\lvert 0\rangle_{A}+\sqrt{t}\lvert
1\rangle_{A}$, where $t$ serves as a tunable parameter as before. Then a CZ
entangles the two modes in a simple 2-node cluster [32] as follows

$\displaystyle\lvert\Phi\rangle_{AF}=\lvert\Phi\rangle_{A}\otimes\lvert+\rangle_{F}$
$\displaystyle\lvert\Phi\rangle_{AF}=\sqrt{\frac{1-t}{2}}\lvert
00\rangle_{AF}+\sqrt{\frac{1-t}{2}}\lvert
01\rangle_{AF}+\sqrt{\frac{t}{2}}\lvert 10\rangle_{AF}$
$\displaystyle+\sqrt{\frac{t}{2}}\lvert 11\rangle_{AF}$
$\displaystyle\mathrm{CZ}\lvert\Phi\rangle_{AF}=\lvert\Phi\rangle_{\tilde{A}\tilde{F}}$
$\displaystyle\lvert\Phi\rangle_{\tilde{A}\tilde{F}}=\sqrt{\frac{1-t}{2}}\lvert
00\rangle_{\tilde{A}\tilde{F}}+\sqrt{\frac{1-t}{2}}\lvert
01\rangle_{\tilde{A}\tilde{F}}+\sqrt{\frac{t}{2}}\lvert
10\rangle_{\tilde{A}\tilde{F}}$ $\displaystyle-\sqrt{\frac{t}{2}}\lvert
11\rangle_{\tilde{A}\tilde{F}},$

then,

$\displaystyle\lvert\Phi\rangle_{\tilde{A}\tilde{F}}=\sqrt{1-t}\lvert
0+\rangle_{\tilde{A}\tilde{F}}+\sqrt{t}\lvert 1-\rangle_{\tilde{A}\tilde{F}}$
$\displaystyle(\mathrm{I}\otimes\mathrm{H})\lvert\Phi\rangle_{\tilde{A}\tilde{F}}=\sqrt{1-t}\lvert
00\rangle_{\tilde{A}\tilde{F}}+\sqrt{t}\lvert 11\rangle_{\tilde{A}\tilde{F}}$
(19)

After that modes $\tilde{I}$, and $\tilde{A}$ enter another CZ gate, such that
mode $\tilde{I}$ is the gate’s control, whereas mode $\tilde{A}$ is its target

$\displaystyle\lvert\Phi^{({\rm{out}})}\rangle$
$\displaystyle=\lvert\Phi\rangle_{\tilde{I}\tilde{E}}\otimes\lvert\Phi\rangle_{\tilde{A}\tilde{F}}$
$\displaystyle=\alpha\sqrt{1-t}\lvert
0000\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}+\alpha\sqrt{t}\lvert
0011\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{\gamma(1-t)}\lvert
1000\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}+\beta\sqrt{\gamma t}\lvert
1011\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert
0100\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}+\beta\sqrt{t(1-\gamma)}\lvert
0111\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}},$
$\displaystyle\mathrm{CZ}\lvert\Phi^{({\rm{out}})}\rangle$
$\displaystyle=\alpha\sqrt{1-t}\lvert
0000\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}+\alpha\sqrt{t}\lvert
0011\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{\gamma(1-t)}\lvert
1000\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}-\beta\sqrt{\gamma t}\lvert
1011\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert
0100\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}+\beta\sqrt{t(1-\gamma)}\lvert
0111\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}},$
$\displaystyle(\mathrm{H}_{\tilde{I}}\otimes\mathrm{H}_{\tilde{A}})\lvert\Phi^{{\rm{(out)}}}\rangle$
$\displaystyle=\alpha\sqrt{1-t}\lvert+0+0\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}+\alpha\sqrt{t}\lvert+0-1\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{\gamma(1-t)}\lvert-0+0\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}-\beta\sqrt{\gamma
t}\lvert-0-1\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert+1+0\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{t(1-\gamma)}\lvert+1-1\rangle_{\tilde{I}\tilde{E}\tilde{A}\tilde{F}},$
(20)

where in the last step a change of basis was applied to modes $\tilde{I}$ and
$\tilde{A}$ by performing two Hadamard operations. It can be seen from the
previous expression that $\lvert\Phi^{{\rm{(out)}}}\rangle$ is a four mode
entangled state. Performing two X-basis projective measurements on $\tilde{I}$
and $\tilde{A}$, disentangles them and directly entangles $\tilde{E}$ and
$\tilde{F}$. The state’s form is dependent on the outcomes of the projective
measurements.

When the measurement outcomes are similar, we denote this as operating point
one ($\mathrm{OP}1$), the output state is

$\displaystyle\lvert\Phi^{({\rm{out}})}_{+}\rangle$
$\displaystyle=\frac{1}{\sqrt{N_{+}}}\Big{[}(\alpha\sqrt{1-t}\lvert
0\rangle_{\tilde{F}}-\beta\sqrt{\gamma t}\lvert
1\rangle_{\tilde{F}})\otimes\lvert 0\rangle_{\tilde{E}}$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert
0\rangle_{\tilde{F}}\otimes\lvert 1\rangle_{\tilde{E}}\Big{]},$ (21)

where $N_{+}=\alpha^{2}(1-t)+\beta^{2}[\gamma t+(1-\gamma)(1-t)]$ is a
normalisation constant.

The transformation of the 4-node cluster state into a 2-node one re-scaled the
weights of the vacuum and single photon components. In order to quantify the
amount of re-scaling, a gain expression is defined. By calculating the ratio
between the single photon component in the final mode, $\tilde{F}$, given a
successful BSM, and the single photon component entering the amplifier,
$\tilde{I}$, we arrive at the desired gain expression [33, 34]

$\displaystyle G_{\rm{OP1}}=\frac{t}{\alpha^{2}(1-t)+\beta^{2}[\gamma
t+(1-\gamma)(1-t)]}.$ (22)

It can be seen from the previous expression that $G_{\rm{OP1}}>1$ when
$t>1/2$. When $t=1/2$, $G_{\rm{OP1}}=1$ and the device operates as a
teleporter. Furthermore, ideal gain, $G_{\rm{OP1}}\approx t/(1-t)$ [5], is
achieved given a successful BSM and high input losses, i.e., $\gamma=1$.

On the other hand, in the event of different measurement outcomes, denoted as
operating point 2 ($\mathrm{OP}2$), the output state becomes

$\displaystyle\lvert\Phi^{({\rm{out}})}_{-}\rangle$
$\displaystyle=\frac{1}{\sqrt{N_{-}}}\Big{[}(\alpha\sqrt{t}\lvert
1\rangle_{\tilde{F}}-\beta\sqrt{\gamma(1-t)}\lvert
0\rangle_{\tilde{F}})\otimes\lvert 0\rangle_{\tilde{E}}$
$\displaystyle+\beta\sqrt{t(1-\gamma)}\lvert 1\rangle_{\tilde{F}}\otimes\lvert
1\rangle_{\tilde{E}}\Big{]},$ (23)

where $N_{-}=\alpha^{2}t+\beta^{2}[\gamma(1-t)+t(1-\gamma)]$.

As can be seen the weightings of the single photon and vacuum components are
switched. This can be redeemed by performing an error correction step. First
observe how the state transforms after applying a Pauli-X correction to the
final mode

$\displaystyle\lvert\Phi^{{\rm{(out)}}}_{-}\rangle$
$\displaystyle=\frac{1}{\sqrt{N_{-}}}\Big{[}(\alpha\sqrt{t}\lvert
0\rangle_{\tilde{F}}-\beta\sqrt{\gamma(1-t)}\lvert
1\rangle_{\tilde{F}})\otimes\lvert 0\rangle_{\tilde{E}}$
$\displaystyle+\beta\sqrt{t(1-\gamma)}\lvert 0\rangle_{\tilde{F}}\otimes\lvert
1\rangle_{\tilde{E}}\Big{]}.$ (24)

(a)

(b)

Figure 3: A plot of the amplifier’s gain against its tunable parameter $t$ for
OP1 and OP2. In OP1, increasing the value of $t$, higher gain values can be
achieved in order to restore an attenuated qubit initially prepared in a
balanced superposition state, i.e., $\alpha=\beta=1/\sqrt{2}$. On the other
hand, by lowering the value of $t$, OP2 can achieve higher gain values.
Effectively lowering the value $t$ in OP2 means higher reflectivity, while
increasing $t$ in OP1 means high transmissivity and lower reflectivity. The
gain is state dependent as depicted, it depends on how much attenuation is
introduced by the transmitting channel and the qubit’s initial weightings. We
have considered different channel transmissivities and their corresponding
gain curves. As can be seen gain is plotted for $\gamma=0.2,0.5,0.6$, and
$0.8$.

The corresponding gain expression in this case is

$\displaystyle
G_{\rm{OP2}}=\frac{1-t}{\alpha^{2}t+\beta^{2}[\gamma(1-t)+t(1-\gamma)]}.$ (25)

Contrary to the previous case, here a $G_{\rm{OP2}}>1$ operation is achieved
when $t<1/2$, whereas when $t=1/2$ the device operates as a teleporter as
before, and the ideal gain in this case would be $G_{\rm{OP2}}\approx(1-t)/t$.

The key feature of our device is that the BSM required for a successful
amplification event was achieved by a CZ operation followed by two qubit
X-basis measurements. This approach, as demonstrated in details, doesn’t
necessarily require photon counters in order to achieve amplification event.
This is due to the fact that the device’s final measurement is performed
individually on each qubit, unlike a QS-NLA which detects a photon number
interference event between two modes and thus requires photon counters to
post-select the desired amplified output. Furthermore, the extra redundancy in
our device, opened the way for an alternative operating point when a BSM
fails, which can not be replicated by a QS-NLA.

We end this section by analyzing the gain expressions of our device for both
OP1 and OP2. In Fig. (3) we have plotted the amplifier’s gain against its
tunable parameter $t$. We firstly observe that both gain expressions are state
dependent, i.e., they rely on $\alpha$, $\beta$, and channel loss $\gamma$.
Thus, without loss of generality, we have assumed for both OPs in Figs. (3a)
and (3b) respectively, that $\alpha=\beta=1/\sqrt{2}$. In OP1, by increasing
the transmissivity $t$, higher gain values can be achieved in order to restore
an attenuated qubit. On the other hand, OP2 has higher tunable gain when the
beamsplitter’s reflectivity increases, in other words, when its transmissivity
decreases.

We now turn our attention towards assessing the two models, QS-NLA and one-way
NLA, against eachother by considering detection imperfections. We show in the
next section the regime of operation where our device has a higher success
probability and hence outperform a QS-NLA.

###  III-C Effect of detector’s noise and inefficiency

In the previous sections it was pointed out that the state of the art QS-NLAs
rely on (PNRDs) for successful amplification. As of today, microwave single
photon counters remain elusive. Nonetheless, microwave photo-detectors have
enjoyed great success, specifically, the quantum non-demolition  (QND) type,
owing to the versatility of circuit quantum electrodynamics (cQED) platforms.
Practically, detection errors such as, non-ideal efficiency and dark count
probability, would affect the performance of any QIP protocol. Hence, we focus
in this section on studying these effects. Firstly, we begin with a brief
description of the working principle of QND detectors, followed by a
mathematical definition of the detection operators and then finally an
evaluation of the success probabilities corresponding to the two possible
device operating points. The last part of this section is dedicated to the
performance analysis of the one-way NLA, specifically, we study the relation
between the device’s success probability and gain for each operating point.
Furthermore, we compare our device to a practical QS-NLA and show its capacity
to achieve a higher success probability.

Suppose that our device performs the X-basis projective measurements required
for a successful amplification via QND detectors. Physically a QND detector is
a qubit-cavity system engineered such that the interaction between the qubit
and a single photon populating the cavity is described as,
$H_{\mathrm{int}}=i\Omega(a\sigma^{+}+a^{\dagger}\sigma^{-})$, where $\Omega$
is the qubit’s Rabi frequency, and $\sigma^{+}(\sigma^{-})$, $a^{\dagger}(a)$
are the qubit and photon raising (lowering) operators respectively [35].

When the cavity captures an arbitrary single photon wave-packet $\alpha\lvert
0\rangle+\beta\lvert 1\rangle$, a series of controlled qubit pulses evolve the
compound system into an entangled state that can be written as $\alpha\lvert
0g\rangle+\beta\lvert 1e\rangle$. As described earlier, applying a Hadamard
transformation to the photonic degree of freedom transforms the state into,
$\alpha\lvert+g\rangle+\beta\lvert-e\rangle$. When the single photon state is
$\lvert+\rangle$, the qubit assumes its ground state, whereas when the photon
state is $\lvert-\rangle$ the qubit is in the excited state. Thus, the
presence of a photon in any of the 2-dimensional diagonal basis states,
$\\{\lvert+\rangle,\lvert-\rangle\\}$, is detectable by measuring the qubit
state.

The aforementioned QND detector provides information on the presence or
absence of a specific single photon state. This process is denoted as on-off
detection [36, 37, 38]. The measurement operators describing an ideal on-off
detector are defined as

$\displaystyle\mathrm{M}_{\mathrm{off}}$ $\displaystyle=\lvert 0\rangle\langle
0\lvert$ $\displaystyle\mathrm{M}_{\mathrm{on}}$
$\displaystyle=\mathrm{I}-\mathrm{M}_{\mathrm{off}}=\underset{n=1}{\overset{\infty}{\sum}}\lvert
n\rangle\langle n\lvert.$ (26)

For the case at hand, the QND detector ”on” operator,
$\mathrm{M}_{\mathrm{on}}$, is truncated to the single photon level. Detector
imperfections manifest as dark count probability, which is incorporated to the
off operator, $M_{\rm{off}}$, and inefficiency, which is modeled by a
beamsplitter of transmissivity $\eta$ followed by a perfect detector. More
precisely, the mode to be detected impinges first on an attenuating
beamsplitter then followed by an on-off detector,
$a_{\rm{out}}=\sqrt{\eta}a_{\rm{in}}+\sqrt{1-\eta}a_{v}$, where $a_{\rm{in}}$
is the input to be detected, $a_{\rm{out}}$ is the beamsplitter’s output, and
$a_{v}$ is a vacuum mode entering from the beamsplitter’s unused port. In
addition to that, we recast the photon projector operators in the Hadamard
basis. Thus the on-off operators become

$\displaystyle\mathrm{M}_{\mathrm{off}}$
$\displaystyle=(1-\mu)\lvert+\rangle\langle+\lvert$
$\displaystyle\mathrm{M}_{\mathrm{on}}$
$\displaystyle=\mu\lvert+\rangle\langle+\lvert\hskip 5.0pt+\hskip
5.0pt\lvert-\rangle\langle-\lvert,$ (27)

where $\mu$ is the dark count probability.

Let us now focus on the output state of our device, Eq. (20). In our scheme,
two QND on-off detectors are utilised to perform a projective measurement on
the two middle cluster nodes. A successful amplification event is decided
based on measuring the following observables

$\displaystyle\mathrm{M}_{0}$
$\displaystyle=\mathrm{M}_{\mathrm{off}}\otimes\mathrm{M}_{\mathrm{off}}$
$\displaystyle\mathrm{M}_{1}$
$\displaystyle=\mathrm{M}_{\mathrm{off}}\otimes\mathrm{M}_{\mathrm{on}}$
$\displaystyle\mathrm{M}_{2}$
$\displaystyle=\mathrm{M}_{\mathrm{on}}\otimes\mathrm{M}_{\mathrm{off}}$
$\displaystyle\mathrm{M}_{3}$
$\displaystyle=\mathrm{M}_{\mathrm{on}}\otimes\mathrm{M}_{\mathrm{on}},$ (28)

where an ”off” event corresponds to measuring a photon in the $\lvert+\rangle$
state, whereas an ”on” event corresponds to measuring a $\lvert-\rangle$
state. After recasting Eq. (20) as follows

$\displaystyle\lvert\Phi^{{\rm{(out)}}}\rangle$
$\displaystyle=\alpha\sqrt{1-t}\lvert
00\rangle_{\tilde{E}\tilde{F}}\otimes\lvert\Psi_{0}\rangle+\alpha\sqrt{t}\lvert
01\rangle_{\tilde{E}\tilde{F}}\otimes\lvert\Psi_{1}\rangle$
$\displaystyle+\beta\sqrt{\gamma(1-t)}\lvert
00\rangle_{\tilde{E}\tilde{F}}\otimes\lvert\Psi_{2}\rangle-\beta\sqrt{\gamma
t}\lvert 01\rangle_{\tilde{E}\tilde{F}}\otimes\lvert\Psi_{3}\rangle$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert
10\rangle_{\tilde{E}\tilde{F}}\otimes\lvert\Psi_{0}\rangle$
$\displaystyle+\beta\sqrt{t(1-\gamma)}\lvert
11\rangle_{\tilde{E}\tilde{F}}\otimes\lvert\Psi_{1}\rangle,$ (29)

where $\lvert\Psi_{0}\rangle=\lvert++\rangle$,
$\lvert\Psi_{1}\rangle=\lvert+-\rangle$,
$\lvert\Psi_{2}\rangle=\lvert-+\rangle$, and
$\lvert\Psi_{3}\rangle=\lvert--\rangle$.

The probability of successfully obtaining a specific combination of the
aforementioned on-off observables becomes

$\displaystyle P(l)$
$\displaystyle=\langle\Phi^{{\rm{(out)}}}\lvert\mathrm{I}\otimes\mathrm{M}_{l}\otimes\mathrm{I}\lvert\Phi^{{\rm{(out)}}}\rangle,$
(30)

where $l=0,1,2,3$.

Consequently the success probabilities can be calculated as

$\displaystyle\langle\Phi^{{\rm{(out)}}}\lvert\mathrm{I}\otimes\mathrm{M}_{0}\otimes\mathrm{I}\lvert\Phi^{{\rm{(out)}}}\rangle$
$\displaystyle=(1-\mu)^{2}[(1-t)[\alpha^{2}+\beta^{2}(1-\gamma)]]$
$\displaystyle\langle\Phi^{{\rm{(out)}}}\lvert\mathrm{I}\otimes\mathrm{M}_{1}\otimes\mathrm{I}\lvert\Phi^{{\rm{(out)}}}\rangle$
$\displaystyle=(1-\mu)[\mu(1-\eta)+\eta][\alpha^{2}t+\beta^{2}t(1-\gamma)]$
$\displaystyle\langle\Phi^{{\rm{(out)}}}\lvert\mathrm{I}\otimes\mathrm{M}_{2}\otimes\mathrm{I}\lvert\Phi^{{\rm{(out)}}}\rangle$
$\displaystyle=(1-\mu)[\mu(1-\eta)+\eta][\beta^{2}\gamma(1-t)]$
$\displaystyle\langle\Phi^{{\rm{(out)}}}\lvert\mathrm{I}\otimes\mathrm{M}_{3}\otimes\mathrm{I}\lvert\Phi^{{\rm{(out)}}}\rangle$
$\displaystyle=\beta^{2}\gamma t[\mu(1-\eta)+\eta]^{2}.$ (31)

where detection inefficiency was manifested as the following transformation,
$\lvert-\rangle=\sqrt{\eta}\lvert-\rangle+\sqrt{1-\eta}\lvert+\rangle$.

According to the previous equation the device has two success probabilities
corresponding to its two operating points. When $\mathrm{OP1}$ is true, the
probability of success $P^{\mathrm{succ}}_{\rm{OP1}}$ corresponds to the sum
of on-on and off-off events. On the other hand, when $\mathrm{OP}2$ is true,
the success probability $P^{\mathrm{succ}}_{\rm{OP2}}$ corresponds to the on-
off and off-on events after error correction.

In practice the efficiency and dark count probability of a QND detector are
$\eta\approx 0.75-0.85$, and $\mu\approx 0.015$ respectively [19, 39, 20].
However, due to TM mismatch errors, the efficiency may drop to $50\%$ [40]. We
consider both cases in the next section and show the effect of reducing the
detector’s efficiency on the success probability of our device. We also
compare the performance of our device to a QS-NLA similar to the one depicted
in Fig. (1) utilizing PNRDs, and show that our device has an enhanced success
probability besides being fault-tolerant.

###  III-D Performance analysis

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Success probability curves of one-way NLAs and QS-NLA versus their
tuning parameter $t$ respectively. The channel’s input was assumed to be a
single photon, i.e., $\alpha=0$, $\beta=1$. In the above panels detector’s
efficiency is $\eta\approx 0.85$, whereas in the below ones is $\eta\approx
0.5$. A dark count probability, $\mu\approx 0.015$, was assumed for both one-
way NLAs, whereas $\mu\approx 0.5$ for QS-NLA. An efficieny of $\eta\approx
0.5$ was considered for all amplifiers in the below panels, while dark count
probability is the same as before respectively. Figs. (4a) and (4c) show that
both one-way NLAs outperform a QS-NLA for different channel transmissivities
respectively. In Fig. (4a) OP1’s success probability is controlled by the
transmissivity of the auxiliary photon, whereas OP2’s success probability is
controlled by its reflectivity. OP1 and OP2 outperform QS-NLA for all the
considered channel transmissivites except for the case when $\gamma=0.8$ where
QS-NLA outperforms OP1.

Fig. (4) is a plot of the success probability of a one-way NLA against its
tunable parameter for different detector efficiencies, channel transmissivites
and operating points. The plot also includes a QS-NLA in order to compare the
two models against eachother. Without loss of generality, we have assumed a
single photon as the channel’s input, i.e., $\alpha=0$, $\beta=1$. In the
upper panel Figs. (4a), (4b), and (4c) consider one-way NLA OP1, QS-NLA, and
one-way NLA OP2 respectively. Both one-way NLAs utilize QND detectors whereas
QS-NLA utilize PNRDs. The detection efficiency was assumed $\eta=0.85$ for all
detectors. Dark count probability for QND detectors was assumed $\mu=0.015$,
while that for PNRDs is $\mu=0.5$. This is due to the fact that ideal
microwave photon counters are currently unavailable [41]. As can be seen, both
one-way NLAs outperform QS-NLA in terms of the probability of success. This is
due to the fact that a BSM in one-way NLA heralds a successful amplification
event by performing two separate photodetections. Unlike QS-NLA where a BSM
rely on detecting a two mode photon number interference pattern with PNRDs.
For one-way OP1 and QS-NLA, when the channel’s transmissivity is $\gamma<1/2$,
the success probability decreases by increasing $t$. This can be justified by
observing that when the input’s weightings are already in favor of the vaccum
component, a large value of $t$ means that the auxilary photon is mostly
transmitted and a BSM fails since with high probability no photons enter the
bell state analyzer. On the other hand when $\gamma>1/2$ the success
probability increases by increasing $t$, since in this case the input
weightings are already in favor of the single photon component and by
increasing $t$ less fraction of the auxiliary photon interferes with the input
and a BSM succeeds since with high probability only one photon enters the bell
state analyzer. For one-way NLA OP2, the amplifier’s gain is controlled by
varying the auxiliary beamsplitter reflectivity $1-t$ (see Fig. (3)). As can
be seen from Fig. (4c) when $\gamma<1/2$ the success probability increases by
increasing the reflectivity, since the input weightings are already in favor
of the vacuum mode and large reflectivity means a larger portion of the
auxiliary mode enters the amplifier and hence enhances the success of a BSM.
On the other hand when $\gamma>1/2$, i.e., weightings are in favor of the
single photon component, the success probability decreases by increasing the
reflectivity, since this situation leads to a larger portion of the auxiliary
photon entering the amplifier and hence a BSM fails. In the lower panel the
same amplifiers were considered, however the detection efficiency was assumed
$\eta=0.5$ for all detectors. The dark count probability was assumed as before
respectively. Similarly it can be seen that both one-way OP1 and OP2
outperform QS-NLA for all channel transmissivities, except when $\gamma=0.8$,
it can be seen that QS-NLA can only beat OP1 while being outperformed by OP2.

##  IV APPLICATIONS OF ONE-WAY NLA

In this section we are mainly concerned with microwave quantum sensing
applications that thrive on low power signals. Quantum illumination [42, 43],
and entanglement-enhanced radar sensing [44] are the most studied protocols
fitting this category.

The aforementioned protocols utilize an entanglement resource in the form of a
TMSV in order to interrogate a target. A probe denoted as signal is sent
towards a target, whereas its twin (denoted as idler) is retained in a lossy
memory element for a correlation measurement with the target’s return.
Analogously, we can regard the dissipative dynamics of the stored mode as a
propagation through an ADC. Thus our objective here is to evaluate the cost at
which NLAs are capable of protecting an idler mode against storage losses
during an entanglement-based quantum sensing protocol.

###  IV-A Low power microwave quantum sensing

Suppose now that a TMSV is generated at a transmitter

$\displaystyle\lvert\Psi\rangle_{SI}=\underset{n=0}{\overset{\infty}{\sum}}\sqrt{\frac{N^{n}_{S}}{(N_{S}+1)^{n+1}}}\lvert
n\rangle_{S}\lvert n\rangle_{I},$ (32)

where $N_{S}$ is the mean photon number in each of the signal and idler modes,
i.e., $\langle a^{\dagger}_{S}a_{S}\rangle=\langle
a^{\dagger}_{I}a_{I}\rangle=N_{S}$. It is often more convenient to recast the
previous expression by defining the following parameter
$\lambda^{2}=\frac{N_{S}}{1+N_{S}}$, hence

$\displaystyle\lvert\Psi\rangle_{SI}=\sqrt{1-\lambda^{2}}\underset{n=0}{\overset{\infty}{\sum}}\lambda^{n}\lvert
n\rangle_{S}\lvert n\rangle_{I}.$ (33)

Assume now that the idler mode is undergoing an ADC representing a lossy
memory element with transmissivity $\gamma$. Similarly as before, this can be
modeled by a passive beamsplitter with the same transmissivity, where its
unused port injects vacuum. In the Heisenberg picture, the corresponding
beamsplitter unitary operator can be defined as,
$U_{\theta}=e^{\theta(a^{\dagger}b+b^{\dagger}a)}$, where the beamsplitter’s
transmissivity equals $\gamma=\cos^{2}{\theta}$, and $a$, $b$ are its modes.
Thus the input transforms as

$\displaystyle\lvert\Psi\rangle_{SI}$
$\displaystyle=\sqrt{1-\lambda^{2}}\underset{n=0}{\overset{\infty}{\sum}}\lambda^{n}\frac{{a^{\dagger}_{S}}^{n}}{\sqrt{n!}}\frac{{a^{\dagger}_{I}}^{n}}{\sqrt{n!}}\lvert
0\rangle^{\otimes 3}_{SIE}$ $\displaystyle U_{\theta}\lvert\Psi\rangle_{SI}$
$\displaystyle=\sqrt{1-\lambda^{2}}\underset{n=0}{\overset{\infty}{\sum}}\lambda^{n}\frac{{a^{\dagger}_{S}}^{n}}{\sqrt{n!}}\frac{[{U_{\theta}a^{\dagger}_{I}U_{\theta}^{\dagger}]}^{n}}{\sqrt{n!}}\lvert
0\rangle^{\otimes 3}_{SIE}$
$\displaystyle=\sqrt{1-\lambda^{2}}\underset{n=0}{\overset{\infty}{\sum}}\lambda^{n}\frac{{a^{\dagger}_{S}}^{n}}{\sqrt{n!}}\frac{[{\sqrt{\gamma}a^{\dagger}_{I}+\sqrt{1-\gamma}a^{\dagger}_{E}]}^{n}}{\sqrt{n!}}\lvert
0\rangle^{\otimes 3}_{SIE}.$

By utilizing the Binomial expansion theorem we get

$\displaystyle\lvert\Psi\rangle_{SIE}$
$\displaystyle=\sqrt{1-\lambda^{2}}\underset{n=0}{\overset{\infty}{\sum}}\lambda^{n}\underset{k=0}{\overset{n}{\sum}}{n\choose
k}\frac{{a^{\dagger}_{S}}^{n}}{\sqrt{n!}}$
$\displaystyle\frac{[\gamma^{\frac{n-k}{2}}{a^{\dagger}_{I}}^{n-k}][(1-\gamma)^{\frac{k}{2}}{a^{\dagger}_{E}}^{k}]}{\sqrt{n!}}\lvert
0\rangle^{\otimes 3}_{SIE},$

where $\lvert 0\rangle^{\otimes 3}_{SIE}=\lvert 0\rangle_{S}\otimes\lvert
0\rangle_{I}\otimes\lvert 0\rangle_{E}$, and $E$ is an environment mode.

Then finally by utilizing the properties of the factorial function we arrive
at

$\displaystyle\lvert\Psi\rangle_{SIE}$
$\displaystyle=\sqrt{1-\lambda^{2}}\underset{n=0}{\overset{\infty}{\sum}}\lambda^{n}\underset{k=0}{\overset{n}{\sum}}\sqrt{{n\choose
k}}\gamma^{\frac{n-k}{2}}(1-\gamma)^{\frac{k}{2}}$ $\displaystyle\lvert
n\rangle_{S}\lvert n-k\rangle_{I}\lvert k\rangle_{E}.$ (34)

For low powered quantum sensing applications, the number of photons in each of
the signal and idler modes is approximately $N_{S}\ll 1$. Thus we can truncate
the previous expansion to first order

$\displaystyle\lvert\Psi\rangle_{SIE}$
$\displaystyle\approx\sqrt{1-\lambda^{2}}\big{[}\lvert
000\rangle_{SIE}+\lambda\sqrt{\gamma}\lvert 110\rangle_{SIE}$
$\displaystyle+\lambda\sqrt{1-\gamma}\lvert 101\rangle_{SIE}\big{]}$
$\displaystyle\approx\alpha\lvert 000\rangle_{SIE}+\beta\sqrt{\gamma}\lvert
110\rangle_{SIE}+\beta\sqrt{1-\gamma}\lvert 101\rangle_{SIE},$ (35)

where $\alpha=\sqrt{1-\lambda^{2}}$, $\beta=\lambda\sqrt{1-\lambda^{2}}$.

In order to complete the amplification protocol, the receiver prepares an
auxiliary mode defined as

$\displaystyle\lvert\Psi\rangle_{AF}=\sqrt{1-t}\lvert
00\rangle_{AF}+\sqrt{t}\lvert 11\rangle_{AF}.$ (36)

Thus the whole state of the amplifier becomes

$\displaystyle\lvert\Psi\rangle_{SIEAF}$
$\displaystyle=\lvert\Psi\rangle_{SIE}\otimes\lvert\Psi\rangle_{AF}.$ (37)

After that modes $I$ and $A$ are redirected towards a CZ gate, such that, mode
$I$ is the control, whereas mode $A$ is the target. We note that exchanging
the roles of the control and target modes won’t affect the protocol’s outcome
due to the symmetry of the CZ gate

$\displaystyle\lvert\Psi\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle=\alpha\sqrt{1-t}\lvert
0+0+0\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\alpha\sqrt{t}\lvert
0+0-1\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{\gamma(1-t)}\lvert
1-0+0\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle-\beta\sqrt{\gamma t}\lvert
1-0-1\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert
1+1+0\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}}$
$\displaystyle+\beta\sqrt{t(1-\gamma)}\lvert
1+1-1\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}},$ (38)

where $\tilde{I}$ and $\tilde{A}$ are written in the diagonal basis.

Then two X-basis measurements are performed on modes $\tilde{I}$ and
$\tilde{A}$, yielding the following output state when the measurement outcomes
are the same

$\displaystyle\lvert\Psi\rangle_{\tilde{S}\tilde{E}\tilde{F}}$
$\displaystyle=\frac{1}{\sqrt{N_{+}}}\Big{[}(\alpha\sqrt{(1-t)}\lvert
00\rangle_{\tilde{S}\tilde{F}}-\beta\sqrt{\gamma t}\lvert
11\rangle_{\tilde{S}\tilde{F}})\otimes\lvert 0\rangle_{\tilde{E}}$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert
10\rangle_{\tilde{S}\tilde{F}}\otimes\lvert 1\rangle_{\tilde{E}}\Big{]}.$ (39)

Conversely, when the outcomes are different we get

$\displaystyle\lvert\Psi\rangle_{\tilde{S}\tilde{E}\tilde{F}}$
$\displaystyle=\frac{1}{\sqrt{N_{-}}}\Big{[}(\alpha\sqrt{t}\lvert
01\rangle_{\tilde{S}\tilde{F}}-\beta\sqrt{\gamma(1-t)}\lvert
10\rangle_{\tilde{S}\tilde{F}})\otimes\lvert 0\rangle_{\tilde{E}}$
$\displaystyle+\beta\sqrt{(1-\gamma)t}\lvert
11\rangle_{\tilde{S}\tilde{F}}\otimes\lvert 1\rangle_{\tilde{E}}\Big{]},$ (40)

where $N_{+}$ and $N_{-}$ are normalization constants defined as before.

Then we can operate on the erroneous state in Eq. (40) by applying a Pauli X
correction on mode $\tilde{F}$ as described earlier in details (see section
3B). Thus, the amplifier’s output state for each case respectively becomes

$\displaystyle\rho^{(\rm{out})}_{+}$
$\displaystyle=\frac{1}{N_{+}}\Big{[}\lvert\Psi^{\rm{(out)}}_{+}\rangle\langle\Psi^{\rm{(out)}}_{+}\lvert+\beta^{2}(1-\gamma)(1-t)\lvert
10\rangle\langle 01\lvert\Big{]},$ (41)

where
$\lvert\Psi^{\rm{(out)}}_{+}\rangle=\frac{1}{\sqrt{N_{+}}}\big{[}\alpha\sqrt{1-t}\lvert
00\rangle_{\tilde{S}\tilde{F}}-\beta\sqrt{\gamma t}\lvert
11\rangle_{\tilde{S}\tilde{F}}\big{]}$

$\displaystyle\rho^{(\rm{out})}_{-}$
$\displaystyle=\frac{1}{N_{-}}\Big{[}\lvert\Psi^{\rm{(out)}}_{-}\rangle\langle\Psi^{\rm{(out)}}_{-}\lvert+\beta^{2}(1-\gamma)t\lvert
10\rangle\langle 01\lvert\Big{]},$ (42)

where
$\lvert\Psi^{\rm{(out)}}_{-}\rangle=\frac{1}{\sqrt{N_{-}}}\big{[}\alpha\sqrt{t}\lvert
00\rangle_{\tilde{S}\tilde{F}}-\beta\sqrt{\gamma(1-t)}\lvert
11\rangle_{\tilde{S}\tilde{F}}\big{]}$.

Considering the previous equations more carefully, it can be seen that when
the environment mode is in the vacuum state, the amplifier’s output is an
entangled state, whereas error occurs when the environment captures an input
photon. As previously described in great details, by increasing (decreasing)
$t$ for OP1(OP2), the probability amplitudes of the entangled output can be
rescaled in favor of the single photon component, and thus restoring the lost
photon.

We end this section by considering a practical entanglement-enhanced sensing
scenario where a one-way NLA is utilized to restore a non-ideally stored idler
mode $I$.

Let’s consider a microwave storage device of $50\%$ efficiency. This
corresponds to a lossy channel with transmissivity equal to $\gamma=0.5$. The
number of signal-idler probe pairs that can be generated in a typical
microwave quantum sensing protocol is approximately $M\approx 10^{5}-10^{6}$.
Due to the probabilistic nature of NLAs, this number is rescaled by the
device’s success probability. Thus the number of generated pairs has to be
increased in order to compensate for this reduction. Considering the worst
case scenario first, it can be seen from Fig. (4) that when $\mathrm{OP1}$ is
true, and the detector’s efficiency is $0.5$, the success probability
corresponding to a complete restoration of the single photon component is
$\approx 0.5$, whereas when $\mathrm{OP2}$ is true it becomes $0.24$. Since
one-way NLA is fault tolerant and has uninterruptible operation, its success
probability on average is $\approx 0.4$. On the other hand for the same case a
QS-NLA has a success probability of $\approx 0.2$. Consequently the minimum
number of required pairs would be $2.5\times 10^{5}$ for one-way NLA, and
$5\times 10^{5}$ for QS-NLA which is double that of one-way NLA. For
completeness we consider the case where the efficiency is $\eta\approx 0.85$,
the average success probability of one-way NLA is approximately $\approx 0.5$
and that of QS-NLA is $\approx 0.3$ thus the minimum number of required pairs
would be $\approx 2\times 10^{5}$ for the former and $\approx 3.5\times
10^{5}$ for the latter.

To conclude, we have shown that a one-way NLA is capable of complete
restoration of a lossy idler mode participating in a quantum sensing protocol.
The corresponding cost was an increase in the number of utilized probe pairs,
however compared to a QS-NLA our device requires a fewer number of pairs which
would result in a faster quantum sensing protocol.

###  IV-B Entangling remote superconducting qubits

We turn our attention towards the task of entangling two remote network nodes.
As initially proposed, successful entanglement is established by performing a
joint BSM in a mid-way station between the two communicating parties [45].
Recently the intriguing idea of NLA has been deployed in order to implement
this BSM [46, 47, 48].

When BS1 $\&$ BS2 in Fig. 1 are considered as two channels linking two remote
nodes to a midway balanced beamsplitter, the resemblance between entanglement
sharing and NLA protocols is clear. However, as pointed out earlier, heralded
NLA requires

$\rm CZ$$\rm CZ$$\rm CZ$$\mathrm{X}$$\mathrm{X}$$S$$I$$A$$F$$\rm{BSM}$ Figure
5: Remote entanglement between two superconducting qubits. Each qubit-photon
pair is composed of a transmon and a flying photon entangled system.
Entanglement between different bosons is achieved by a CZ gate. BSM is
performed with QNDs as described in the main text.

single photon counters, furthermore it is non-fault tolerant when
amplification fails.

Here we investigate utilizing our one-way NLA in order to perform the required
BSM for remote entanglement. In Fig. (5) we consider an instance of our
proposed protocol. Nodes $S$ & $F$ are two remote nodes to be entangled, while
$I$ & $A$ are intermediate ones. At each node a transmon-cavity system is
prepared in a qubit-photon entangled state, $\frac{1}{\sqrt{2}}(\lvert
0e\rangle+\lvert 1g\rangle)$ [49]. After that a photonic CZ gate entangles ;
$SI$, $AF$, and $IA$ pairs respectively. The whole state of the system can
then be written as

$\displaystyle\lvert\varphi\rangle$
$\displaystyle=\Big{[}\frac{1}{\sqrt{2}}(\lvert 0e\rangle_{S}+\lvert
1g\rangle_{S})\Big{]}\otimes\Big{[}\frac{1}{\sqrt{2}}(\lvert
0e\rangle_{I}+\lvert 1g\rangle_{I})\Big{]}$
$\displaystyle\otimes\Big{[}\frac{1}{\sqrt{2}}(\lvert 0e\rangle_{A}+\lvert
1g\rangle_{A})\Big{]}\otimes\Big{[}\frac{1}{\sqrt{2}}(\lvert
0e\rangle_{F}+\lvert 1g\rangle_{F})\Big{]}.$ (43)

Applying a CZ between $SI$ and $AF$, transforms the state into

$\displaystyle\lvert\varphi\rangle=$ $\displaystyle\Big{[}\frac{1}{2}(\lvert
00ee\rangle_{SI}+\lvert 01eg\rangle_{SI}+\lvert 10ge\rangle_{SI}-\lvert
11gg\rangle_{SI})\Big{]}\otimes$ $\displaystyle\Big{[}\frac{1}{2}(\lvert
00ee\rangle_{AF}+\lvert 01eg\rangle_{AF}+\lvert 10ge\rangle_{AF}-\lvert
11gg\rangle_{AF})\Big{]}.$ (44)

Then another CZ is applied between nodes $I$ & $A$

$\displaystyle\lvert\varphi\rangle=$ $\displaystyle\frac{1}{4}\Big{(}\lvert
0++0eeee\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}+\lvert
0++1eeeg\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}$ $\displaystyle+\lvert
0+-0eege\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}-\lvert
0+-1eegg\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}$ $\displaystyle+\lvert
0-+0egee\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}+\lvert
0-+1egeg\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}$ $\displaystyle-\lvert
0--0egge\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}+\lvert 0--
1eggg\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}$ $\displaystyle+\lvert
1++0geee\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}+\lvert
1++1geeg\rangle_{SIAF}$ $\displaystyle+\lvert 1+-0gege\rangle_{SIAF}-\lvert
1+-1gegg\rangle_{SIAF}$ $\displaystyle-\lvert
1-+0ggee\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}-\lvert
1-+1ggeg\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}$ $\displaystyle+\lvert
1--0ggge\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}-\lvert 1--
1gggg\rangle_{\tilde{S}\tilde{I}\tilde{A}\tilde{F}}\Big{)}.$ (45)

Finally performing two X-basis measurements on the middle nodes completes the
protocol. When the measurement outcomes are the same, we get

$\displaystyle\lvert\varphi^{\rm{(out)}}_{+}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\lvert
01eg\rangle_{\tilde{S}\tilde{F}}+\lvert
10ge\rangle_{\tilde{S}\tilde{F}}\big{)}.$ (46)

On the other hand, when the outcomes are different, the state becomes

$\displaystyle\lvert\varphi^{\rm{(out)}}_{-}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\lvert
00ee\rangle_{\tilde{S}\tilde{F}}+\lvert
11gg\rangle_{\tilde{S}\tilde{F}}\big{)}.$ (47)

By applying a Pauli-X correction to the second mode on both the photonic and
atomic degrees of freedom, we get the same output as in Eq. (46).

The previous protocol is a specific realization of the more general one-way
NLA described earlier. When $\gamma=1$, and $t=1/2$ a one-way NLA reduces to
the aforementioned remote entanglement swapping.

From Fig. (4) we can directly estimate the success probability of the
protocol. When detection efficiency is $\approx 0.85$, a one-way NLA based
remote entanglement has an average success probability of $\approx 0.7$,
whereas that of a QS based protocol is $\approx 0.45$. The success probability
drops significantly when detection efficiency is further reduced. For the case
of $\approx 0.5$ efficiency, a one-way NLA based protocol has on the average a
probability of success of $\approx 0.38$, such that, OP1 success probability
is $\approx 0.26$, while OP2’s is $\approx 0.5$. On the other hand the success
probability of a QS is $\approx 0.36$ which is clearly outperformed by OP2.

Remote entanglement is the basis of many QKD protocols [10]. Many practical
imperfections play a role in limiting the distance over which remote
entanglement can be generated. The primary causes for the case at hand are
qubit decoherence and propagation losses. For long distance entanglement
sharing the qubit state has to be preserved for milliseconds. Most recently,
significant progress has been made towards longer qubit coherence times.
Approximately the coherence time that can be achieved with the current state
of the art technology is in the range of $1-3\hskip 1.99997pt\text{ms}$ [50,
51], which suggests a maximum sharing distance of $\approx 30\hskip
1.99997pt\text{km}$. As for attenuation due to propagation losses, the
survivability of a microwave wavepacket in a transmission medium decreases
exponentially as a function of its length $L_{0}$. This is captured by the
famous Beer’s formula,[52] $P(L_{0})=\exp{(-L_{0}/2L_{\mathrm{att}})}$, where
$L_{\mathrm{att}}$ is the attenuation length.

In the next section we study in details the effect of transmission losses on
SKR. More specifically, we consider two remote parties establishing a shared
secure key by generating remote entanglement over a lossy channel. As before,
we compare our one-way NLA entanglement sharing protocol to a QS’s, and
demonstrate the conditions under which our protocol outperforms both direct
transmission and a QS.

###  IV-C Remote secret key generation

(a)

(b)

(c)

(d)

Figure 6: SKR of our one-way NLA based on weak TMSV and QND detection. The
plots depict SKR in (bits/use) against transmission distance in (km). Included
in all plots the performance of a QS-NLA for comparison. All detectors in the
first two plots are assumed ideal. It can be seen that OP2 and QS-NLA have the
best performance in Fig. (6a), however, they both outperform a point to point
link only for distances greater than $\approx 500$ $\mathrm{km}$ as shown in
Fig. (6b). In Fig. (6c) detector’s efficiency is assumed $\eta\approx 0.85$,
and a dark count probability of $\mu=0.015$ for both one-way NLAs and
$\mu=0.5$ for QS-NLA. In Fig. (6d) both detector’s efficiency and dark count
probability were assumed $0.5$ for all detectors. OP2 outperforms QS-NLA in
Fig. (6c) and achieves the best performance, whereas OP1 has the least SKR.
For a wireless transmission link, Fig. (6d) shows that OP2 & QS-NLA has the
highest SKR only for distances over 1200 $\mathrm{km}$. As before, OP1 has the
least SKR.

Consider now a slight variation of the previous protocol where the $S$ & $I$
nodes (see Fig. (5)) are prepared in a weak TMSV state identical to that in
Eq. (35), while $A$ and $F$ are prepared in the same resource state described
before, $\lvert\Phi\rangle_{AF}=\sqrt{1-t}\lvert 00\rangle_{AF}+\sqrt{t}\lvert
11\rangle_{AF}$. Suppose further, that modes $I$ and $A$ undergo two ADCs with
transmissivities $\gamma$, and $\tilde{\gamma}$ respectively.

Practically in such scenario each of the input and auxiliary photons are
subjected to damping quantified by the aforementioned tranmissivities. After
applying a CZ between nodes $I$ and $A$, we can write an expression for the
overall state by including the two environment lossy modes as follows

$\displaystyle\lvert\varphi\rangle$ $\displaystyle=\alpha\sqrt{1-t}\lvert
0+0+00\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}}$
$\displaystyle+\alpha\sqrt{t\tilde{\gamma}}\lvert
0+0-10\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}}$
$\displaystyle+\alpha\sqrt{t(1-\tilde{\gamma})}\lvert
0+0+11\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{\gamma(1-t)}\lvert
1-0+00\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}}$
$\displaystyle-\beta\sqrt{\gamma\tilde{\gamma}t}\lvert
1-0-10\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{\gamma(1-\tilde{\gamma})t}\lvert
1-0+11\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert
1+1+00\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{t(1-\gamma)\tilde{\gamma}}\lvert
1+1-10\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{t(1-\gamma)(1-\tilde{\gamma})}\lvert
1+1+11\rangle_{\tilde{S}\tilde{I}\tilde{E}\tilde{A}\tilde{F}\tilde{E^{\prime}}},$
(48)

where $\tilde{E}$, $\tilde{E^{\prime}}$ are two environment modes each in a
vacuum state, and nodes $\tilde{I}$ and $\tilde{A}$ are written in the
diagonal basis.

After performing two X-basis measurements on nodes $\tilde{I}$, and
$\tilde{A}$, the state becomes the following when the measurement outcomes are
the same

$\displaystyle\lvert\varphi\rangle$
$\displaystyle=\frac{1}{\sqrt{N_{+}}}\Big{[}(\alpha\sqrt{1-t}\lvert
00\rangle_{\tilde{S}\tilde{F}}+\beta\sqrt{\gamma\tilde{\gamma}t}\lvert
11\rangle_{\tilde{S}\tilde{F}})\otimes\lvert
00\rangle_{\tilde{E}\tilde{E^{\prime}}}$
$\displaystyle+\alpha\sqrt{t(1-\tilde{\gamma})}\lvert
01\rangle_{\tilde{S}\tilde{F}}\otimes\lvert
01\rangle_{\tilde{E}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{(1-\gamma)(1-t)}\lvert
10\rangle_{\tilde{S}\tilde{F}}\otimes\lvert
10\rangle_{\tilde{E}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{t(1-\gamma)(1-\tilde{\gamma})}\lvert
11\rangle_{\tilde{S}\tilde{F}}\otimes\lvert
11\rangle_{\tilde{E}\tilde{E^{\prime}}}\Big{]}.$ (49)

However, when the outcomes are different, it becomes

$\displaystyle\lvert\varphi\rangle\textasciidieresis$
$\displaystyle=\frac{1}{\sqrt{N_{-}}}\Big{[}(\alpha\sqrt{t\tilde{\gamma}}\lvert
01\rangle_{\tilde{S}\tilde{F}}+\beta\sqrt{\gamma(1-t)}\lvert
10\rangle_{\tilde{S}\tilde{F}})\otimes\lvert
00\rangle_{\tilde{E}\tilde{E^{\prime}}}$ $\displaystyle+\beta\sqrt{\gamma
t(1-\tilde{\gamma})}\lvert 11\rangle_{\tilde{S}\tilde{F}}\otimes\lvert
01\rangle_{\tilde{E}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{\tilde{\gamma}t(1-\gamma)}\lvert
11\rangle_{\tilde{S}\tilde{F}}\otimes\lvert
10\rangle_{\tilde{E}\tilde{E^{\prime}}}\Big{]}.$ (50)

Then after applying a bit-flip operation on the final mode, $\tilde{F}$, the
state becomes

$\displaystyle\lvert\varphi\rangle\textasciidieresis$
$\displaystyle=\frac{1}{\sqrt{N_{-}}}\Big{[}(\alpha\sqrt{t\tilde{\gamma}}\lvert
00\rangle_{\tilde{S}\tilde{F}}+\beta\sqrt{\gamma(1-t)}\lvert
11\rangle_{\tilde{S}\tilde{F}})\otimes\lvert
00\rangle_{\tilde{E}\tilde{E^{\prime}}}$ $\displaystyle+\beta\sqrt{\gamma
t(1-\tilde{\gamma})}\lvert 10\rangle_{\tilde{S}\tilde{F}}\otimes\lvert
01\rangle_{\tilde{E}\tilde{E^{\prime}}}$
$\displaystyle+\beta\sqrt{\tilde{\gamma}t(1-\gamma)}\lvert
10\rangle_{\tilde{S}\tilde{F}}\otimes\lvert
10\rangle_{\tilde{E}\tilde{E^{\prime}}}\Big{]},$ (51)

where
$N_{+}=\alpha^{2}(1-t\tilde{\gamma})+\beta^{2}[t\tilde{\gamma}(2\gamma-1)+1-\gamma]$,
and
$N_{-}=\alpha^{2}t\tilde{\gamma}+\beta^{2}[\gamma+\tilde{\gamma}t(1-2\gamma)]$
are normalization constants.

Let us now analyze the previous equations in more details. In Eq. (49) the
first term on the right side represents an entangled state established between
the remote parties when each environment mode is in a vacuum state. The rest
of the terms are error states corresponding to the following loss events
respectively: when the second environment mode $\tilde{E^{\prime}}$ captures a
photon, loss occurs to the auxiliary mode, whereas when $\tilde{E}$ captures a
photon, it occurs to the input. The final term means that both photons were
lost. Similarly, after error correction, the terms in Eq. (51) corresponds to
the following: the first term represents a successful entangled state as can
be verified by inspecting the environment vacuum states, whereas the rest of
the error terms correspond to a loss in either the input or the auxiliary
modes depending on which environment mode captured a photon. Unlike the state
in Eq. (49), the event where both the input and auxiliary modes are lost
doesn’t occur, since the physical meaning of different measurement outcomes
corresponds to detecting one and only one photon in either $\tilde{I}$ or
$\tilde{A}$.

Focusing now on SKG, the repeater-less bound [53] for two remotely
communicating parties defines a relation between the SKR that can be generated
and the tranmsissivity of the linking channel,
${\rm{SKR}}\approx-{\rm{log}}_{2}(1-\chi^{1/K})$, where $\chi$ is the
channel’s transmissivity, and $K$ is the number of repeater links. When $K=1$,
SKR corresponds to a point-to-point direct transmission link, whereas when
$K=2$ it corresponds to a single repeater case.

The inner workings of one-way NLA resembles that of a quantum repeater, this
can be shown by noting that the device’s BSM swaps the entanglement between
the middle nodes to the remote ones. Furthermore, a secret key rate (SKR) can
be estimated according to the success probability of the device [54], since it
establishes an entanglement channel between the involved remote parties.

When the outcomes of a BSM are identical, i.e., OP1, the success probability
is
$P^{\mathrm{succ}}_{S}\approx\alpha^{2}(1-t)+\beta^{2}\gamma\tilde{\gamma}t$,
whereas when they are different, OP2, it becomes
$P^{\mathrm{succ}}_{\mathrm{D}}\approx\alpha^{2}t\tilde{\gamma}+\beta^{2}\gamma(1-t)$.
In the limit of small squeezing $\lambda<1$, $\alpha^{2}\approx 1$,
$\beta^{2}\approx\lambda$, symmetric loss,
$\gamma=\tilde{\gamma}=\sqrt{\chi}$, $t=1/2$, it can be seen that the success
probabilities become $P_{S}^{\mathrm{succ}}\propto\chi/2$, and
$P^{\mathrm{succ}}_{\mathrm{D}}\propto\sqrt{\chi}/2$ respectively. The
previous approximations can be justified by observing that the first term in
both success probability expressions is a vacuum offset, since it corresponds
to the event were both the input and the auxiliary modes are in vacuum states.

In Fig. (6) we have studied the behaviour of SKR (in bits/use) against the
transmission distance (in km) for two different transmission media and both
ideal and non-ideal detection. We have included in this analysis SKR of a
point to point link as an indicative of the PLOB bound. The first two panels
study the case where the device’s detectors are operating in the ideal limit,
whereas the last two plots consider detection imperfections. For the ideal
case, the performance of a one-way NLA OP2 and a QS-NLA coincide.

We have taken in Fig. (6a) a cryogenic waveguide transmission medium with 0.6
$\rm{dB/km}$ attenuation factor, whereas in Fig. (6b) a wireless transmission
link with $0.0063{\rm{dB/km}}$ attenuation factor [55] was assumed. In the
former, from Beer’s law, the attenuation length was found to be
$L_{\rm{att}}=7.3\hskip 1.99997pt\rm{km}$, whereas in the latter it was
$L_{\rm{att}}\approx 700\hskip 1.99997pt\rm{km}$. The last two plots consider
the same transmission media, however, detection non idealities were
considered. Similar to Fig. (6a), Fig. (6c) considers the same channel
attenuation factor, while detector’s efficiency is $\eta\approx 0.85$.
Furthermore, we have assumed dark count probability of $\mu\approx 0.015$ for
one-way NLA and $\mu\approx 0.5$ for QS-NLA, since microwave PNRD are
currently unavailable as mentioned earlier. On the other hand, in Fig. (6d) a
wireless transmission medium was assumed as in Fig. (6b), however, both
detector’s efficiency and dark count probability were assumed $0.5$ for all
detectors.

In Fig. (6a), it can be seen that one-way NLA operating under OP2 and QS-NLA
have the best SKR. This can be justified by noticing that when the channel’s
transmissivity is small, splitting the overall distance by a NLA outperforms
direct transmission, due to its entanglement properties. Comparatively a
higher channel transmissivty as in Fig. (6b), namely $0.0063\hskip
1.99997pt\rm{dB/km}$, resulted in an improved performance over direct
transmission only for large distances, $L_{0}>500\rm{km}$. We note that in
both of the previous cases when one-way NLA operated under OP1 it had the
least SKR.

When detection imperfections were considered, SKR of all protocols dropped as
shown in Fig. (6c), however, a practical QS-NLA would be outperformed by a
one-way NLA operating under OP2, due to the unavailability of perfect
microwave PNRD. Similarly as before one-way NLA OP1 has the least performance.
For the last case where the detection limits of all detectors are the same,
Fig. (6d) showed that direct transmission had the best SKR among all channels
for distances less than $\approx 1200\rm{km}$, nonetheless, when the distance
exceeds this limit, it can be seen that one-way NLA OP2 & QS-NLA have the best
SKR, thus suggesting that they are more suitable for long distance microwave
QIP.

To summarize, in this section we have shown that due to the inherent
resemblance between one-way NLA and a quantum repeater, our device can be
utilised to establish a remote entanglement channel for SKG. The device’s
success probability translates to a SKR that can be generated. Furthermore, it
can beat the repeater-less bound when operating under OP2 and outperform a QS-
NLA for lossy cryogenic links, thus offering a more secure option for
cryogenic chip-to-chip communications [56].

##  V CONCLUSION

In this paper we have proposed a novel microwave bosonic NLA based on a simple
one dimensional linear cluster of four nodes. Unlike conventional QS-NLAs, our
proposed device doesn’t rely on PNRDs to signal the success of the BSM
required for amplification. Instead, our device utilizes QND detectors
operating in the photodetection limit, which significantly eases the
experimental requirements for a microwave NLA. Operationally, the proposed
device inherits the fault tolerant capabilities of a cluster state, which
renders it capable of correcting the erroneous output and achieve
amplification. This feature is not possible in a conventional QS-NLA, which
has to repeat the whole process when amplification fails.

Owing to the low dark count probability of a QND detector, our device showed
an enhanced probability of success over a QS-NLA operating with microwave
PNRDs. Furthermore, our device also demonstrated a great ability to enhance
microwave QIP tasks. For quantum sensing applications, we have shown that with
the aid of a one-way NLA, entanglement enhanced sensing protocols can fully
restore an attenuated idler beam despite detection imperfections. This was
achieved at the expense of a fewer increase in the number of probe pairs,
which is still significantly lower than that required by a QS-NLA performing
the same task. Furthermore, we have also shown that our device can generally
establish secret keys between remote parties at a rate surpassing that of a
direct communication link, both for a cryogenic ADC and a long distance
wireless channel. More specifically, it has showed higher SKR than that of a
QS-NLA for cryogenic ADC when detection inefficiencies were considered.

Finally we note that the proposed device model can be also employed in the
optical domain, thus rendering it a new addition to the QIP toolbox.

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