Federated Quantum Neural Network With Quantum Teleportation For Resource Optimization in Future Wireless Communication

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 72, NO.
11, NOVEMBER 2023 14717
Federated Quantum Neural Network With Quantum

Teleportation for Resource Optimization in
Future Wireless Communication
Bhaskara Narottama , Member, IEEE, and Soo Young Shin , Senior Member, IEEE
Abstract—The following study introduces FT-QNN, a federated TABLE I

and quantum teleportation –based quantum neural network, uti- RELATED WORKS
lized to optimize resource allocation for future wireless communi-
cations. The proposed FT-QNN consists of edge quantum neural
networks (QNNs) and a cloud QNN, while quantum teleportation
allows the cloud QNN to obtain the outputs of edge QNNs without
requiring prior measurements on the output states, allowing the
cloud to process the outputs directly as quantum states. As a
particular case to demonstrate its applicability for wireless resource
allocation, FT-QNN is then employed to optimize transmit power
allocation coefficients in a power domain non-orthogonal multiple
access (NOMA)-based system, aiming to maximize the achievable
sum-rate. FT-QNN yields lower complexity compared to a dis-
tributed QNN scheme without quantum teleportation, while the
numerical results also demonstrated that the FT-QNN is capable
to achieve a similar sum-rate compared to the scheme without
quantum teleportation. scenario requires user devices to send their information to the
cloud, which could impose computational burdens on the cloud
Index Terms—6G, quantum neural networks, quantum
teleportation, wireless communications. and cause communication overhead, and also raise concerns over
data privacy [6].
To address this issue, federated learning frameworks [7], [8],
I. INTRODUCTION each of which allows local data sets to be processed in a dis-
ESOURCE allocation is regarded as one of the vital as- tributed manner, have recently been considered [9]. Accordingly,
R pects to fulfill the demanding requirements of future wire-
less networks [2], [3]. Owing to the growing scale of wireless
the following studies have utilized federated learning for various
resource allocation problems: In [10], federated learning was
systems, obtaining optimization solutions via analytical-based used to optimize the bandwidth while maintaining energy effi-
resource allocation methodologies has become increasingly ciency. In [11], a federated learning methodology with coopera-
challenging as it typically involves NP-hard problems [2], [3], tion and augmentation schemes was utilized to optimize power
[4]. To mitigate this issue, neural network –based methodolo- allocation. In [12], the interplay between federated learning and
gies, which generally employ prediction models with adjustable extreme value theory –based approach was employed to jointly
parameters, can be used to estimate the solutions to the resource optimize the power management and resource allocation of
allocation problems [3], [4]. To date, centralized neural network an ultra-reliable low-latency communication (URLLC) system.
architectures, generally processed by cloud processing units, Nonetheless, as indicated in Table I, the afore-mentioned works
have typically been considered [5]. However, in most cases, this utilize classical neural networks (cNNs), employing classical
computations.1 However, the complexity of cNNs generally
Manuscript received 31 August 2022; revised 20 December 2022 and 28 increases as the number of layers and neurons grows [13].
March 2023; accepted 6 May 2023. Date of publication 29 May 2023; date of On the other hand, quantum computing, widely studied as
current version 14 November 2023. This work was supported by the Institute of an alternative to classical computing, is not performed upon
Information and communications Technology Planning and Evaluation (IITP)
funded by the Korea Government (MSIT) under Grant 2021-0-02120, Research classical bits; rather, it computes quantum bits, or “qubits”,
on Integration of Federated and Transfer learning between 6G Base Stations each of which is capable of representing multiple basis states.
exploiting Quantum Neural Networks. The review of this article was coordinated These states, in particular, can be represented |0 and |1, and
by Prof. Zhu Han. (Corresponding author: Soo Young Shin.)
Bhaskara Narottama is with the Institut National de la Recherche Scien- thereby their superposition can be expressed as: α|0 + β|1,
tifique (INRS), University of Quebec, Montreal, QC H5A 1K6, Canada (e-mail: and |α|2 + |β|2 = 1. These terms, α and β, correspond to |0
bhaskara.narottama@inrs.ca).
Soo Young Shin is with the WENS Laboratory, Department of IT Convergence
Engineering, Kumoh National Institute of Technology, Gumi 39177, South
Korea (e-mail: wdragon@kumoh.ac.kr). 1 Hereon, “classical computation” refers to an computation process involving
Digital Object Identifier 10.1109/TVT.2023.3280459 classical bits, each of which represents its information as “0” or “1”.
0018-9545 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information.
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14718 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 72, NO. 11, NOVEMBER 2023
To date, initial experiments have been made on quantum

teleportation [32], [33], [34]. The primary advantage of quantum
teleportation is arguably the ability to transmit an unknown
quantum state without performing measurement on the state.
Consequently, the transmitted quantum state can remain un-
known to both the transmitter and receiver during the quantum
teleportation process, thus maintaining its privacy and secu-
rity. In addition, quantum teleportation offers the advantage of
direct transmission of quantum state over a long distance, as
demonstrated in [35]. This mitigates the need for intermediary
re-transmissions and reduces the number of nodes required in
the transmission process. Thanks to its aforementioned benefits,
Fig. 1. The proposed FT-QNN framework, which incorporates federated QNN quantum teleportation has been investigated to serve as a key
models and quantum teleportation, compared with a typical classical federated enabler for realizing quantum-based communication and com-
learning methodology.
putation networks [33], envisioned as the quantum internet [32].
Acknowledging its potential for model-based optimization,
this study aims to explore the possibility of using quantum
and |1, respectively [18]. This aspect is called quantum super- teleportation in a distributed QNN setting. In addition, to further
position, and uniquely attributed to quantum computations. With demonstrate its application in wireless resource allocation, the
other attributes such as quantum entanglement [20], it responsi- proposed FT-QNN is utilized for power allocation in power-
ble for the processing benefits of the quantum computing [19], domain non-orthogonal multiple access (NOMA) [36].2
[20].
Furthermore, built upon quantum-based operations, quantum A. The Contributions of This Work
neural networks (QNNs) have recently garnered research at- In the following, the main contributions of this study are can
tention as one of the emerging quantum-based solutions [14], be outlined.
[15], [16]. In practice, their potential advantages includes im- 1) To the best of the authors’ knowledge, as of the time of
proved training progression and reduced computational com- writing, the proposed FT-QNN is an initial attempt to
plexity [17], [21]. The utilizations of QNNs have already been utilize quantum teleportation within a distributed QNN
demonstrated in the following works: In [14], a training algo- framework for the purpose of optimizing wireless com-
rithm for multi-layer QNNs was presented. In [15], the learning munications. The architecture of FT-QNN consists of
ability of QNNs was examined. In [16], a quantum generative two-tier QNNs, consisting of edge QNNs and a cloud
adversarial network was presented. In [17], QNN schemes were QNN. Quantum teleportation [22], [23], [24] is utilized to
utilized for resource allocation in an wireless system. However, deliver information from the edge QNN and cloud QNN,
quantum-based distributed/federated learning has not been em- as presented in Fig. 1. In this way, edge QNNs can send
ployed in the afore-mentioned studies. their states to the cloud QNN without collapsing their
Motivated by recent advancements of QNN–and their promis- superposition states (in particular, from α|0 + β|1 to |0
ing roles in distributed/federated learning–this work proposes or |1) due to measurement.
a federated and quantum teleportation –based neural network 2) In order to facilitate practical implementation, the quan-
framework called FT-QNN, consisting of edge QNNs and a tum circuits of the FT-QNN are described in detail.
cloud QNN, as illustrated in Fig. 1. In this scheme, the cloud The computational complexity of the particular quantum-
QNN receive information from each edge QNN via quantum based operations composing the FT-QNN is also thor-
teleportation [22], [23], [24]. Quantum teleportation leverages oughly analyzed.
quantum entanglement, allowing the transmission of informa- 3) To demonstrate its feasibility, the introduced FT-QNN is
tion between different quantum system [22], [23], [24]. It is utilized for a regression task that is optimizing power
worth noting that quantum-based federated learning schemes allocation in NOMA systems, in order to examine the prac-
have been addressed in prior studies [25], [26], [27]: In [25], tical implementation of FT-QNN in a particular resource
federated learning scheme that employs quantum parameterized allocation case of B5G/6 G communications. Although
model was studied. In [26], federated learning to collaboratively they can provide general approximations of the wireless
process the data was addressed. The authors of [27] considered problems at hand, classical learning models can be suscep-
quantum data for the federated learning scheme. Nevertheless, tible to performance limitations related to the increasing
the utilization of quantum teleportation has not been examined scale of wireless networks, e.g., a growing number of
in these studies. In the context of a distributed and connected
quantum-based machine intelligence framework, quantum tele- 2 Hereafter, the term “NOMA” refers to power-domain non-orthogonal multi-
portation can be leveraged to establish direct and secure con- ple access. In addition, acknowledging the initial state of quantum teleportation,
nections between different learning models, as it enables the an alternative scheme without quantum teleportation is also presented later in
transmission of information between an edge computation plat- Section V. In this alternative scheme, quantum measurements are performed
to obtain the output of an edge QNN. The resulting classical bits are then
form and the cloud via previously entangled qubits, bypassing transmitted to the cloud, which processes the classical information from the
the need for intermediate nodes. edge.
NAROTTAMA AND SHIN: FEDERATED QNN WITH QUANTUM TELEPORTATION FOR RESOURCE OPTIMIZATION 14719
TABLE II
NOTATIONS
Fig. 2. The utilization of FT-QNN for resource allocation. The global model
[m]
Qcloud is processed by the cloud QNN. The m–th local model Qedge is processed
by the m–th edge QNN. The dataset of users served by the m–th access point/
is sent to the cloud via quantum teleportation protocol, which
base station is stored locally by the m–th edge Em . The resource allocation will be addressed in detail in Section III. It is worth not-
solution from FT-QNN is then used by Em . ing that each edge store and process its respective user data
locally.
Each edge platform is assumed as an integrated unit that
transmit antennas, relays, and receiving terminals, which includes a transmit access point, a quantum-based process-
directly affects the training time and accuracy [28]. More ing unit, and the hardware necessary for supporting quan-
often than not, these kinds of setbacks can be mitigated tum teleportation.3 Each m–th edge, denoted by Em , m ∈
[m]
through feature preprocessing techniques, in particular, {1, . . . , Nedge }, ios employed to serve Ngroup user groups.
reduction of the input dimensions [29], [30]. Nonetheless, The n–th group among the groups that belong to Em is
these practices did not solve the core issue, which is in the [m] [m,n]
denoted as Gm,n , n ∈ {1, . . . , Ngroup }. It consists of Nuser
learning model itself. To this end, quantum-based models users. Each user among this group is denoted by Um,n,k ,
can be capitalized, which in this work they even coupled where k specifies the index of the user. Accordingly, Em
with quantum teleportation to reduce inter-model bottle- is responsible for the downlink (DL) transmission to serve
necks due to quantum measurements; in prior studies, Um,n,k . Moreover, Em , m ∈ {1, . . . , Nedge }, receives the chan-
quantum-based learning models have shown comparable nel information about its corresponding users via prior chan-
(or even better) performances [21], [31] while having nel estimation, which involves pilot signaling. In addition,
lower computational complexity and faster convergence hm,n,k denotes the channel between the transmitter and
rate [17], [21]. Um,n,k .
Prior to data transmission, the cloud and Em allocate the
B. The Organization of This Paper resources for Gm,n by leveraging the FT-QNN for resource
The rest of this work is structured as follows. The NOMA allocation optimization. Accordingly, let us define the set of
system is considered in Section II. Section III describes the resource allocation solutions for the groups served by Em as
proposed FT-QNN framework, providing a detailed discussion sm {sm,1 , . . . , sm,N [m] }.
group
on the integration of quantum teleportation and QNNs. In Sec- Notations: The absolute value and Euclidean norm are de-
tion IV, the utilization of FT-QNN for maximizing the sum-rate noted by | · | and · , respectively. The identity matrix is
of the NOMA system is described. Section V investigates the denoted as 1. The notation of “⊗” refers to the Kronecker
performance of FT-QNN. Finally, this study is summarized in product operator. The real and complex numbers are indi-
Section VI. cated by the symbols R and C, respectively. Time com-
plexity is examined using the notation O(·). Other nota-
II. THE SYSTEM MODEL tions used in this article are listed in Table II. Different
quantum operations used in this article are presented in
The assumed system model is presented in Fig. 2. Given Table III.4
consideration to the rapid advancements in quantum comput-
ing [33], this work assumes that the cloud and edges have
access to quantum computation, which can be achieved through 3 Hereafter, for simplicity, the term “m–th edge” will refer to both the m–th
either classical simulation or by directly performing quantum access point and the corresponding edge quantum computation platform.
operations on noisy intermediate-scale quantum (NISQ) pro- 4 Here, quantum gates can be understood as the fundamental operators in
cessors [18]. quantum computing systems, which can be composed into QNNs (will be
described in Section III). In an effort to provide a clearer description, the
The cloud computation, which processes the cloud QNN authors took the liberty of specifying the index of the designated qubit within
model, in which the operation is denoted by Qcloud , cooperates the notation of the corresponding quantum gate. For instance, this work uses
with the Nedges edge computations. Meanwhile, each m–th CX(|qtarget ||qcontrol ) to denote the controlled-X operation applied to the target
qubit |qtarget , with |qcontrol serving as the control qubit. In addition, the operation
edge processes an edge QNN model, in which the operation of RZ (θ; |qtarget ) is applied to |qtarget , where θ denotes the parameter of the
[m] [m]
is denoted by Qedge . The information about the output of Qedge gate.
Fig. 3. The proposed FT-QNN. In this scenario, the local channel information serves as the input for the local QNN model, represented by the quantum operation
[m] [m]
Uedge . Subsequently, each m–th edge platform sends the output as the quantum state |ϕent,1 via the quantum teleportation protocol. Subsequently, the said state
[m]
|ϕent,1 is employed as the input for the cloud quantum model, denoted by the quantum operation Ucloud .
TABLE III [m]

QUANTUM-BASED OPERATIONS [19] Qedge processes the dataset locally, taking it as input, while
Qcloud estimates the resource allocation solution as its output.
Moreover, the proposed scheme uses quantum teleportation to
convey information directly from the m–th edge QNN to the
cloud QNN [1]. The information is transferred as a quantum state
[m]
denoted by |ϕout . Moreover, each m–th edge Em serves several
m }. To conclude, the
user groups, denoted by {Gm,1 , . . . , Gm,Ngroup
proposed FT-QNN can be observed as a suite of distributed QNN
framework that leverages quantum teleportation.
To paint a picture how the FT-QNN can be applied in practical
wireless scenarios, Fig. 4 presents its two deployment phases:
training and prediction. These phases serves as an integrated
protocol, and their details will be described in the following
sub-sections.
1) The Training Phase: The objective of this phase is to train
the edge and cloud QNNs using the previously obtained network
III. THE PROPOSED SCHEME
data, which, in this case, is the acquired channel information (cf.
A. An Overview of the Proposed FT-QNN Section IV). The outcome of the training phase yields the opti-
mized set of weights for the edge and cloud QNNs, which, in this
An overview of the proposed scheme is discussed in this sec- [m]
tion. This study explores the possibility of integrating different context, are their respective gate parameters denoted as Θedge ,
QNN models using a quantum-based communication methodol- ∀m ∈ {1, . . . , Nedge }, and Θcloud , respectively. The proposed
ogy, specifically the quantum teleportation protocol [22], [23], protocol for the i–th training phase is shown in Fig. 4.5 Following
[m]
[24]). This way, the output of the edge QNN, which is the the computation of Qedge , its output, which is the resultant state
[m]
first learning model, can be transmitted and processed as a |φout , is transmitted to the cloud via the quantum teleportation
quantum state by the the cloud QNN, the second model. In protocol. After Qcloud is computed, its output is used to calculate
the future, this approach may become useful for task offloading [m]
the loss value L(Θcloud ). Then, weight parameters Θedge and
or decentralization cases in the quantum-based multi-hardware Θcloud are optimized based on the loss value.
processing systems, such as those found in the emerging quan- 2) The Prediction Phase: The objective of this phase is to
tum internet [37]). Among many, potential benefits of the decen- take the current information about the wireless network, e.g.,
tralized QNN framework are the distribution of learning tasks, the channel data, as the input for conducting feed-forward
the minimization of the amount of information uploaded to the inference using the trained QNN models. This results in the
central computation unit (as training datasets are being trained acquired resource allocation solution that is subsequently sent
by local edge computation units), and the enhancement of user [m]
to the edge. During this phase, both Qedge and Qcloud employ
privacy (as user data are being processed on the premises) [38],
[39].
5 As illustrated in Fig. 4, E
The specific application of the above concept can be specified m sends the ideal outputs included in the training
as follows. The proposed scheme, illustrated in Fig. 3, incorpo- data, denoted by odata,i
m,n , to the cloud, to facilitate the calculation of the loss value
L (cf. Section IV). However, to maintain user privacy, the data belonging to the
rates two QNN models: one processed by the edge, denoted as users served by Em , denoted as xdata,i
[m] m,n , ∀m ∈ {1, . . . , Nedge }) are processed
Qedge , and the other computed by the cloud, denoted as Qcloud . locally by Em .
Fig. 4. Proposed deployment protocol of the introduced FT-QNN. The left figure exhibits the training phase, whereas the right figure shows the prediction phase.
[m]
In each step of the training phase, the output of the edge QNN Qedge is transferred via quantum teleportation method to the cloud QNN Qcloud . Subsequently, the loss
calculation and weight optimization are conducted by the cloud. Once the training phase is completed, FT-QNN leverages the instantaneous channel information
to estimate the current wireless solutions throughout the prediction phase. (a) Training phase. (b) Prediction phase.
[m] [m] cloud,[m]

the previously selected sets of weights, denoted as Θ̂edge and described. Initially, the entanglement of |ϕent,1 and |ϕent,1
Θ̂cloud , respectively. These weights are obtained from the training is made possible via a bell pair entanglement [22], [23], [24],
phase. The instantaneous channel information, in which each which can be expressed as follows.
M
entry ĥm,n,k correspond to the k–th user served by the edge Definition 1 (The entanglement operation of Uent ): The first
[m]→cloud
Em , is pre-processed to obtain the input values x̂m,n,k for entanglement operation in Utele is defined as [22], [23],
[m] [m] [24] 7
Qedge , n ∈ {1, . . . , Ngroup }. After processing Qcloud , the cloud
proceeds to transmits the estimated solution back to Em . The M cloud,[m] [m] [m]
Uent CX(|ϕent,1 ||ϕent,1 )H(|ϕent,1 ). (1)
solution is denoted by om = {om,1 , . . . , om,N [m] }. Em then use
group N
the solution to optimize its wireless system. Afterwards, Em applies Uent , which is defined as follows.
N
Definition 2 (Entanglement operation of Uent ): The second
[m]→cloud
B. The Considered Quantum Teleportation Protocol entanglement operation in Utele is defined as [22], [23],
[24]
The mentioned quantum teleportation, which used to deliver
[m] [m]
H |ϕinf,1 CX |ϕent,1 |ϕinf,1 .
N [m]
information from edge QNN to cloud QNN, is described as Uent (2)
follows.6
1) Initialization: As demonstrated in Fig. 5, let us consider a Next, quantum measurement is performed; as the result,
[m] [m] [m]→cloud
quantum teleportation scheme, where edge Em and the cloud act |ϕinf,1 and |ϕent,1 are collapsed as classical bits cX and
as the transmitter and receiver, respectively, m ∈ {1 . . . Nedge }. [m]→cloud
cZ , respectively, then the measurement results are used
Herein, the edge Em employs two qubits for quantum telepor- [m]→cloud
in Uchannel .
[m] [m]
tation: |ϕinf,1 is used to encode information while |ϕent,1 is 2) The Operation of Quantum Teleportation: A single op-
cloud,[m] [m]
utilized to entangle with |ϕent,1 (a qubit controlled by the eration of quantum teleportation to transfer |ψout from
cloud). In what follows, the quantum teleportation scheme is [m] [m]→cloud
Qedge to Qcloud is denoted by Utele . Collectively, the
6 This study envisages general use of the quantum protocol, although currently, [m] [m] [m]
7 Consider the state of |ϕent,1 after Hadamard gate as |ϕent,1 = αent,1 |0 +
quantum communication is still in its infancy. [32]. Moreover, the authors [m] M is used to prepare the state of the pre-shared entangled qubits
acknowledge that if the distance between a particular edge and the cloud is βent,1 |1. Uent
sufficiently large, quantum repeater may be required [40]. Hence, to mitigate [m] [m] [m] [m]
as αent,1 |00 + βent,1 |11. The notations αent,1 and βent,1 are related to the
the need for repeaters, this study assumes single-hop quantum communication
[m] 2 [m] 2
as a simplifying assumption. probability of obtaining |0 and |1, where |αent,1 | + |βent,1 | = 1.
Fig. 5. The quantum operation representing the considered quantum teleportation protocol [22], [23], [24]. In the figure, the dashed-dotted line indicates the
[m] cloud,[m]
separation of particles containing the quantum states |ϕent,1 and |ϕent,1 [32]. These states are distributed priori to the m–th edge unit and the cloud,
[m]
respectively, m ∈ {1 . . . Nedge }. Within this context, the state |ϕout is regarded as the output of the edge quantum model, presented by the quantum operator
[m] cloud,[m]
Qedge . Afterwards, the state of |ϕent,1 is employed as the input of the cloud quantum model, denoted by Qcloud .
[1] [N ] In our ideal scenario, the edges perform quantum teleportation

quantum teleportation to send {|ψout , . . . , |ψoutedge } from
[1] [N ] simultaneously in the same time frame, resulting in Utele ∈
{Qedge , . . . , Qedgeedge }, respectively, is defined as Utele .
O(1).
Definition 3 (The quantum teleportation operation
[m]→cloud In the following, the QNN models for edge and cloud com-
Utele ): The quantum teleportation used to transfer putation units are described in Sections III-C and III-D, re-
[m] [m]
|ψout from Qedge to Qcloud can be expressed as [22], [23], [24]8 spectively, whereas the algebraic operation of each gate has
[m]→cloud [m]→cloud m m
N M been described in Table III, accordingly. Both edge and cloud
Utele Uchannel ◦ M : |ψinf,1 , |ψent,1 Uent ◦ Uent , QNN models employ various quantum gates for different pur-
(3) poses [43], [44], [45], [46], [47]:
with r Hadamard gates, each presented as H, are used to introduce
B A
CZ |ϕBent,1 cA
[m]→cloud
Z CX |ϕent,2 cX ,
Uchannel (4) quantum superpositions. In addition, the target qubit may
m m
also be specified; for instance, H(qi ) indicates Hadamard
and M : |ψinf,1 , |ψent,1 indicates that quantum measurement is operation on qubit qi [43], [44].
m
performed on the system, resulting in qubits |ψinf,1 m
and |ψent,1 r Each Rotation gate, e.g., RZ , is used to encode classical
being collapsed to classical bits. Furthermore, to teleport the operation [45] or to apply trainable parameter (related to
outputs of Nedge edge QNNs, quantum teleportation is defined weight in a conventional neural network) [43], [44].
as9 r Each controlled gate, e.g., CX(qi |qj ), which applies Pauli
Nedge
[m]→cloud
operation on particular qubit qi according to the state of
Utele Utele . (5) qubit qj , is invoked for connecting qubits. This connection
m=1 is analogous to inter-neuron or inter-layer connection in
Lemma 1: The complexity of quantum teleportation protocol classical neural networks [43], [44].
Utele is presented as Utele ∈ O(1).10 Furthermore, at the end of the inference process, quantum
[m]→cloud measurement is used to observe the final state of each qubit,
Proof: Let us examine Utele first. The operation of
[m]→cloud collapsing it into classical bit, with a value of 0 and 1 [43], [44].
Uchannel yields O(1). Additionally,
m
(M : |ψinf,1 m
, |ψent,1 N
)Uent yields O(1) + O(1) ≈ O(1). Sub- Formally, a single quantum measurement on the output related
M [m]→cloud to the solution can be expressed as
sequently, Uent yields O(1). The complexity of Utele can be
†
[m]→cloud
presented as Utele ∈ O(1) + O(1) + O(1) ≈ O(1). Sub- o[m]
n = xinit Qforward Tz Qforward xinit , (6)
sequently, the complexity of Utele can be analyzed as follows. where Tz is the observable related to Pauli Z operator [43], [44]
and xinit is the prepared initial quantum state (e.g., xinit = |0).
[m]→cloud In this instance, the FT-QNN output related to the solution of
8 Initially, the qubits are set to |0. Hence, we assume that U
tele |0⊗2 for
implementation. Moreover, the notation “◦” is used in (3) to indicate sequential the n–th user group of the m–th edge can be defined as
[m]→cloud m , |ψ m )U N ), and U M .
channel composition of Uchannel , ((M : |ψinf,1
= E[o[m]
ent ent
n ],
ent,1
9 Herein, this study assumes Kronecker product “⊗” in U
tele due to the
oFT-QNN
m,n (7)
[m]→cloud
assumption that each Utele , m ∈ {1, . . . , Nedge }, is performed as parallel where E[·] indicates an expected value. Later on, (19) and (27)
communication link (between m–th edge and the cloud) to each other.
10 Hereafter, the time complexity for a fraction of quantum operation, i.e., a specify the methods for obtaining outputs and decoding them
single quantum gate, is considered as O(1). into the particular NOMA-related solutions, respectively.
⎛ ⎞⎞
C. Edge Quantum Neural Network [m]
Nlayer Nneuron
[m,l]
⎜
⎜ [m] [m] ⎟⎟
The considered m–th edge QNN is described as follows.
⎝ RZ θl,r ; q1,r ⎟ ⎟
⎠⎠ ,
Consider the qubits of m–th edge QNN as l=2 r=1
weight encoding
⎧
⎧ ⎫ ⎪ [m] [m] [m]
⎪ ⎪ ⎪
⎨ CX q l+1,1 q [m,l] for l < Nlayer ,
⎪
⎪ ⎪
⎪ [m,l]
l,Nneuron

⎪
⎪ ⎪
⎪ UCX
⎨ ⎬ ⎪ inter-layer connection
[m] [m] [m] [m] [m] [m] ⎪
⎩
qedge q1,1 , . . . , q [m,1] , q2,1 , . . . , q , q . 1
[m]
for l = Nlayer .
⎪
⎪
1,Nneuron

[m,N
[m]
]
out
⎪
⎪
⎪
⎪ N
[m]
,N layer
⎪
⎪
⎪ ⎪ (10)
neuron

layer
⎩ l=1 ⎭
l>1
(8) where CX(·), CZ(·), and Rz (·) operations are used as inter-
[m]
Initially, the input value is encoded via Uencode . layer connections, inter-neuron connections, and weight encod-
Definition 4 (Encoding of m–th edge): Based on [43], [44], ing operations, respectively [43], [44], [46], [47]. The weights
[45], the encoding operation of m–th edge is defined as of the m–th edge QNN could be concatenated as
⎧ ⎫
⎛ ⎞ ⎪
⎪ ⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎨ ⎪
⎬
⎜ ⎟
[m] [m,n]
Ngroup
⎜N [m] [m] [m] [m] [m]
[m] ⎟ Θedge = θ1,1 , . . . , θ [m] , θ2,1 , . . . , θ .
user
[m]
(ℵm ) ⎜ RZ ‫ג‬n,k ; q1,n H q1,n ⎟
[m] [m] ⎪
⎪
[m]
]⎪
Nlayer ,Nneuron layer ⎪
1,Ngroup [m,N
Uencode ⎜ ⎟, ⎪ [m]
⎪
⎝ ⎠ ⎪
⎪ ⎪
⎪
n=1 k=1
⎩ l=1 ⎭
l>1
forGm,n (11)
(9) [m,1] [m,N ] [m]
[m]
[m]
where ‫ג‬n,k ∈ ℵm is the pre-processed input of the k–th user in Lemma 3: In the case of Nneuron = . . . = Nneuronlayer = Nlayer ,
[m] [m] [m]
the n–th group. Considering the channel gain as the input, the Ngroup = Nlayer , m ∈ {1, . . . , Nedge }, the complexity of Uedge
2
[m] e2hm,n,k −1 [m] [m] 2
pre-processed input is given as ‫ג‬n,k = 2hm,n,k 2
. can be asserted as Uedge ∈ O((Nlayer ) ).
e +1
Lemma 2: Considering
[m]
n ∈ {1, . . . , Ngroup }, m∈ Proof: First, the inter-layer and inter-neuron connections uti-
[m] lize CX(·) ∈ O(1) and CZ(·) ∈ O(1), respectively. Hence,
{1, . . . , Nedge }, the complexity of Uencode can be expressed [m] [m,l] [m] 2
[m] [m,n] this part yields O(Nlayer Nneuron ) ≈ O((Nlayer ) ). Second, the
as Uencode ∈ O(Nuser ). [m,l]
Proof: The RZ (·) operations are performed consecutively weight encoding operation utilizes RZ (·) ∈ O(1) for Nneuron
[m,n] [m]
for {Um,n,1 , . . . , Um,n,Nuser
m,n }, resulting in O(N
user ). Parallel neurons in Nlayer layers. Hence, the weight encoding part
[m] [m,n] [m] [m] 2
operations for Ngroup groups yields O(Nuser ). yields O(1). In conclusion, Uedge yields O((Nlayer ) ) + O(1) ≈
The main operations of the m–th edge QNN that is presented [m] 2
[m] O((Nlayer ) ).
in this work, i.e., Uedge , is adapted from the QNN models in [43], Definition 6 (The m–th edge QNN): The quantum circuit of
[44], [46], [47]. the m–th edge QNN is defined as11
Definition 5 (The main operation of the m–th edge QNN):

[m] [m]
The main operation of the m–th edge QNN is defined as
Qedge CZ qout q [m]
[m] [m] [m]
[m,l] Uedge Θedge
Nlayer ,Nneuron
⎛ ⎛ [m] ⎞ ⎞
[m] Nlayer Nneuron
[m,l]
Uedge Θedge CX q [m] q [m]
[m] [m] [m]
⎜ [m] ⎜ [m] ⎟ [m] ⎟
1,Ngroup 1,Ngroup −1
⎝Uencode (ℵm) ⊗ ⎝ H ql,r ⎠ ⊗H qout ⎠,
inter-layer connection l=2 r=1
⎛ ⎞ (12)
[m]
⎜
Ngroup −1 ⎟
⎜ [m] [m] ⎟
⎝ CZ q q
1,n 1,n−1 ⎠ [m]
where Θedge is the set of weight values used for the m–th edge
n=1 [m]
inter-neuron connection QNN. Let θl,n be the weight value for the n–th neuron in the
⎛ ⎛ ⎞⎞ [m]
Nlayer
[m] [m,l] lth layer. Eventually, the output of Qedge can be conveyed as the
⎜ N ⎟⎟
[m,l] ⎜ [m] [m]
neuron
[m]
⎜ U ⎜ CZ q q ⎟⎟ quantum state |ϕout .
⎝ CX ⎝ l,r l,r−1 ⎠⎠
l=2 r=2
inter-neuron connection
⎛⎛ [m]
⎞
Ngroup

⎝⎝ [m] [m]
RZ θ1,n ; q1,n ⎠ ⊗ initialization, the qubits are set to |0. Therefore, we consider
11 For
[m] [m,l]
[m] ⊗N N
n=1 Qedge |0 layer neuron for implementation.
[m,n] [m,1] considering

Lemma 4: Assuming that Nuser = Nneuron = . . . =
[m]
[m,Nlayer ] [m] ⎧
Nneuron = Nlayer , m ∈ {1, . . . , Nedge }, the complexity ⎪ cloud cloud
⎨ CX ql+1,1 ql,Nneuron
cloud
cloud,l for l < Nlayer ,
[m] [m]
of Qedge can be expressed as Qedge ∈ O((Nlayer ) ).
[m] 2 cloud,l
UCX
[m]
⎪
⎩ inter-layer connection
Proof: First, taking into consideration that CZ(·) and Uedge 1 for l = Nlayercloud
,
[m] 2
yield O(1) and O((Nlayer ) ), respectively, the first part shall ⎧ [m,l] cloud cloud
⎪
⎪
Nneuron
CZ ql,r ql,r−1
[m] 2 [m] ⎪
⎪
r=2

yields O((Nlayer ) ). Second, in view of Lemma 2, Uencode yields ⎪
⎪
⎪
⎪ inter-neuron connection
[m,n] [m]
O(Nuser ) ≈ O(Nlayer ). The single operation of H(·) yields ⎪
⎨ cloud
for l < Nlayer ,
cloud,l
O(1) whereas the parallel operations of H(·) yield O(1), ac- UCZ [m] (15)
⎪
⎪ Nedge Ngroup
CZ cloud cloud
[m]
cordingly. Hence, the second part yields O(Nlayer ). In closing, ⎪
⎪ m=1 n=1 q q
l,a+n l,(a+n)−1
⎪
⎪
⎪
⎪
[m] [m] [m] 2
Qedge yields O(Nlayer ) + O((Nlayer ) ) ≈ O((Nlayer ) ).
[m] 2
⎪
⎩ inter-neuron connection
for l = Nlayer
cloud
,
D. Cloud Quantum Neural Network where CX(·), CZ(·), and Rz (·) operations are used as inter-
The considered cloud QNN is described as follows. Hereon, layer connections, inter-neuron connections, and weight encod-
the set of qubits used to process the cloud QNN are registered ing operations [43], [44], [47]. Moreover, the set of weight values
as used for the cloud QNN is expressed as
⎧ ⎧
⎪
⎪ ⎪
⎪
⎨ ⎨
qcloud q1,1cloud
, . . . , q cloud [Nedge ] , q2,1
cloud
,..., Θcloud θ1,1 cloud
, . . . , θcloud [Nedge ] , θ2,1
cloud
,...,
⎪
⎪ 1,Nedge Ngroup ⎪
⎪ N
⎩ ⎩
1,N

edge group

l=1 l=1
⎧ ⎫ ⎧ ⎫
⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪
⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪
⎨ ⎬ ⎨ ⎬
cloud cloud cloud cloud
qN cloud , . . . , q [Nedge ] . (13) θN cloud , . . . , θ . (16)
⎪ layer ,1 ⎪ layer ,1
[Nedge ]
⎪
cloud ,N
Nlayer edge Ngroup ⎪ ⎪
⎪ ⎪
cloud ,N
Nlayer edge Ngroup
⎪
⎪
⎩ ⎪
⎪
⎭ ⎪
⎪
⎩ ⎪ ⎪
⎪
⎭
l=Nlayer
cloud
l=Nlayer
cloud
The main operation of the cloud QNN, i.e., Ucloud , is based on [m]
the QNN operation in [43], [44], [46], [47]. Lemma 5: Under the assumption of Nedge = Ngroup = Nlayer
cloud
,
cloud,1 cloud,N cloud
Definition 7 (The main operation of the cloud QNN): The m ∈ {1, . . . , Nedge } and Nneuron = . . . = Nneuron layer =
main operation of the cloud QNN is defined as cloud
Nlayer , the complexity of Ucloud can be conveyed as
⎛ 3
Ucloud ∈ O((Nlayer
cloud
) ).
⎜ cloud Proof: The complexity analysis of the first and last layers is
Ucloud (Θcloud ) ⎜
⎝CX q1,Nneuron
cloud
cloud,1 q2,1 presented as follows. The CX(·) operation yields O(1). The se-
[m] cloud 2
inter-layer connection rial CZ(·) operations yield O(Nedge Ngroup ) = O((Nlayer ) ), as-
⎛ ⎞ [1]
suming Ngroup = . . . = Ngroupedge [N ]
. The parallel RZ (·) operations
[m]
⎜
Nedge Ngroup
⎟ yield O(1). Hence, the first and last layers yield O((Nlayer cloud 2
) ).
⎜ cloud cloud
CZ q1,a+n q1,(a+n)−1 ⎟
⎝ ⎠ The time complexity for 2 ≤ l ≤ (Nlayer − 1) is analyzed
cloud
m=1 n=2 cloud,l
inter-neuron connection as follows. The serial CZ(·) operations yield O(Nneuron )≈
⎛ ⎛ ⎞⎞⎞ O(Nlayer ). Parallel RZ (·) operations yield O(1). Hence,
cloud
Nlayer
cloud cloud,l
N
neuron
2 ≤ l ≤ (Nlayer
cloud
− 1) yields O(Nlayer cloud
). In conclusion, Ucloud
⎝ cloud,l ⎝
UCX cloud,l ⎠⎠⎠
UCZ
cloud 2
N cloud
−1 cloud 2
l=2 r=2 yields O((Nlayer ) ) + l=2 layer
O(Nlayer
cloud
) + O((Nlayer ) )≈
⎛⎛ ⎞ O((Nlayer
cloud 2
) ).
[m]
Nedge Ngroup
⎜⎜ cloud
cloud ⎟ Definition 8 (The circuit of cloud QNN): The operation of the
⎝⎝ RZ θ1,a+n ; q1,a+n ⎠⊗
cloud QNN is expressed as12
m=1 n=1
weight encoding ⎛ [m]
⎞
⎛ ⎞⎞ Nedge Ngroup

Nlayer
Qcloud ⎝ ⎠
cloud
M : qN
cloud
X(qN cloud ,r )H
cloud cloud
cloud,l
⎜ N neuron
cloud cloud ⎟⎟ cloud ,r qN cloud ,r
RZ θl,r ; ql,r ⎠⎠ ,
layer layer layer
⎝ m=1 n=1
l=2 r=1

weight encoding

0 if m = 1,
a(m) m−1 [b] (14) 12 As the qubits are initially set to |0, consider Qcloud |0
⊗(Nlayer
cloud −1)N cloud
neuron
b=1 Ngroup if m > 1, for implementation.

⎛ ⎞
Nlayer
cloud cloud,l IV. FT-QNN APPLICATION FOR NOMA SYSTEMS
N neuron
Ucloud (Θcloud ) ⎝ cloud ⎠

H(ql,r ) , (17) This section describes the utilization of FT-QNN to optimize
l=2 r=1 power allocation in NOMA [36], as a particular case of wireless
resource allocation. Calling NOMA into play, a different power
where M : qN
cloud
cloud ,r indicating that quantum measurement is per- ratio is assigned to a different user in a NOMA group, allowing
layer
formed over the system; which leads to |qN cloud being collapsed
cloud each receiving terminal to better encode its designated message.
layer ,r
to a classical bit. On that account, power allocation can be considered as an critical
cloud,1 cloud,N cloud part in NOMA deployment.
Lemma 6: Assuming that Nneuron = . . . = Nneuron layer =
Each edge Em , which has a base station (BS) in the middle
cloud
Nlayer , the complexity of Qcloud can be analyzed as Qcloud ∈
2 of the cell, employs NOMA to transmit superimposed messages
O((Nlayer
cloud
) ). to different user group (denoted as Gm,1 , . . . , Gm,Ngroup , where
Proof: First, parallel operations of MX(·)H(·) ∈ O(1) Ngroup is the number of groups served in Em ). The superim-
yield O(1). Next, as examined in Lemma 5, Ucloud yields posed message transmitted to Gm,n using NOMA is denoted as
cloud 2 [m,n] !
Nuser
O((Nlayer ) ). Finally, parallel operations of H(·) yield O(1).
2 m,n =
χNOMA k=1 λm,n,k P sm,n,k , where λm,n,k and sm,n,k
In conclusion, Qcloud yields O(1) + O((Nlayer
cloud
) ) + O(1) ≈ are the assigned power ratio and designated message for the user
2
O((Nlayer
cloud
) ). Um,n,k , respectively. [36] Let P be the total transmit power of
the BS.
Moreover, let us assume zero-mean Rayleigh flat-fading chan-
E. The Feed-Forward Process and Output Decoding
nels [50], [51], where −κ indicating the pathloss exponent.
Methodology
Considering d−κ m,n,k as the distribution variance, let hm,n,k ∼
In the following, a single feed-forward inference of the FT- CN (0, d−κ )
m,n,k be the channel coefficient indicating the channel
QNN is defined. between BS and Um,n,k , modelled as complex-normal distri-
Definition 9 (Feed-forward inference, Qforward ): Combining bution. Accordingly, let us denote hm,n,k 2 as the gain of the
[m] [m]→cloud
Qedge , Qtele , and Qcloud , ∀m ∈ {1, . . . , Nedge }, a single channel between BS and Um,n,k .
feed-forward inference of the FT-QNN can be represented as In NOMA, the power ratio is assigned based on the channel
the quantum operation gain of each user. Taking that into account, let us order the
⎛ ⎞ user based on the acquired channel gains; the channel gain for
[m]
Nedge
Ngroup
each user in group Gm,n can be sorted as hm,n,1 2 ≥ . . . ≥
⎝ [m]→cloud ⎠ [m]
Qforward Qcloud Qtele Qedge . (18) hm,n,N [m,n] 2 , where hm,n,k 2 indicating the channel gain
user
m=1 n=1
of Um,n,k , accordingly.
edge
The transmit signal-to-interference-plus-noise ratio (SINR) of
Lemma 7: Considering Nlayer cloud
= Nlayer = Nlayer , the com- [m]
NOMA user Um,n,k , m ∈ {1, . . . , Nedge }, n ∈ {1, . . . , Ngroup },
plexity of Qforward can be expressed as Qforward ∈ O((Nlayer )2 ). respectively, can be are expressed as [36]
cloud 2
Proof: First, Qcloud yields O((Nlayer ) ) ≈ O((Nlayer )2 ) (see ⎧
Lemma 6). ⎨ρhm,n,k 2 λm,n,k for k = 1,
Nedge Ngroup
[m]
[m]→cloud [m]
NOMA
γm,n,k = ρhm,n,k 2 λm,n,k (20)
Second, m=1 ( n=1 Qtele )Qedge yields ⎩ k−1 2 for k > 1.
j=1 ρh
m,n,k λm,n,j+1
[m] 2
O((Nlayer ) ) (see Lemmas 1 and 4).
Let ρ = P/σ2 be the transmit signal-to-noise ratio (SNR); the
After the measurements (17), Qforward resulted in an array
[1] [Nedge ] noise variance is indicated by σ 2 . Subsequently, the achievable
of classical bits O = {o1 , . . . , o [N edge ]
}. Subsequently, the de- rate of Um,n,k by using NOMA can be expressed as
Ngroup
coding operation of the FT-QNN can be described as follows.
Definition 10 (Decoding operation, Udecode ): Considering
NOMA
Rm,n,k = log2 1 + γm,n,k
NOMA
. (21)
[m]
on ∈ O as the measurement result of M(qN cloud ,mn ), m ∈
cloud
layer
NOMA
The sum rate of each NOMA group is given by Rm,n =
{1, . . . , Nedge }, n ∈
[m]
{1, . . . , Ngroup }, the operation to decode the Nuser
[m,n]
NOMA
k=1 Rm,n,k .
output of Qforward is defined as
Nshot A. The Objective Function
1
Udecode : Qforward ⇔ oFT-QNN
m,n = o[m]
n
, (19)
Nshot r In this study, the goal of the optimization is to maximize the
r=1 sum rate achieved by the user devices in the NOMA system.
Considering Λ as the set of user power allocation, the optimiza-
where oFT-QNN
m,n is the FT-QNN output for the n–th group of the
tion problem can be defined as [53]:
m–th edge. The number of quantum measurements is denoted
as Nshot . In the considered NOMA scenario (Section IV), the [m]
Nedge Ngroup
FT-QNN decoding output oFT-QNN
m,n is used as the coefficient of maximize NOMA
Rm,n (22a)
the strong user power allocation, λm,n,str ← oFT-QNN
m,n .
Λ
m=1 n=1
subject to NOMA
Rm,n,k ≥ Rm,n,k
OMA
, ∀k, ∀n, ∀m, (22b) coefficient for each strong user given by [53]
#
2
λm,n,k ≤ 1, ∀k, ∀n, ∀m, (22c) data,i
1 + hm,n,weak ρ − 1
k λdata,i
m,n,str = 2
. (25)
0 ≤ λm,n,k ≤ 1, ∀k, ∀n, ∀m. (22d) hdata,i
m,n,weak ρ
Further details regarding (25) are provided in Appendix A.

The constraint specified in (22b) ensures that each user
achieves a higher (or equal) rate by using NOMA compared to C. The Model Training Methodology
that of OMA. By using OMA, the achievable rates for each k-th
OMA
user can be calculated as: Rm,n,k = 21 log2 (1 + γm,n,k
OMA
); the re- The considered training protocol is presented in Algo-
[1] [N ]
OMA
ceive receive signal-to-noise ratio can be expressed as γm,n,k = rithm 1.13 Let Θ = {Θcloud , Θedge , . . . , Θedgeedge } be the set of
2 weights of the cloud and edges. The training process is aimed to
hm,n,k ρλm,n,k [36], [50], [51]; the transmit signal-to-noise
optimize Θ. Assessing the performance of the given quantum-
ratio is represented as ρ = Pt/σ2 , where Pt denotes the transmit
based prediction model, the training loss can be calculated as
power at BS and σ 2 denotes the noise variance. The constraint
follows:
described in (22c) is related to the NOMA power budget con-
Nedge Ngroup
straint. The constraint outlined in (22d) defines the range of 1 2
λm,n,k . L(Θ) = |oFT-QNN
m,n − odata,i
m,n | , (26)
Nedge Ngroup m=1 n=1
Hereafter, this study assumes a particular case of NOMA
systems where each group consists of two users. Within a partic- where odata,i
m,n serves as the data labelling corresponding to the ith
ular NOMA group Gm,n , we distinguish between users based data sample.
on their channel gains: The user with a higher channel gain is Accordingly, FT-QNN are used for estimating the suitable
identified as the “strong” user, denoted as Um,n,str . Meanwhile, power allocation coefficients as regression problems [54]. Ow-
the user with a lower channel gain is referred to as the “weak” ing to the decoding method in (19), the output range of the
user, denoted as Um,n,weak . The estimation of user channels can FT-QNN related to the solution of the m–th group (specifically,
be attained through pilot signaling, prior to the deployment of the measurement output of Ucloud as in (19), within the range of
NOMA. oFT-QNN
m,n ∈ {0, 1}) is mapped to the range of each variable of the
particular wireless optimization; for instance, in NOMA power
allocation we may have oFT-QNN
m,n → λFT-QNN FT-QNN
m,n,str , where λm,n,str is the
B. The Training Data
power allocation for user with a stronger channel gain. In order
The data used during training phase and testing phase is to satisfy the NOMA power assignment constraints in (22c) and
described as follows. (22d), the range of the power allocation coefficient of the strong
The power allocation coefficients of the strong users within user within each group is defined as λFT-QNN
m,n,str ∈ {0, 0.5} [53]. To
the NOMA system can be compiled as this end, a simple transformation is utilized to acquire NOMA
" power allocation from the output of FT-QNN:
Λstr = λ1,1,str , . . . , λ [N edge ] , (23) λFT-QNN
m,n,str = ( /2)om,n
1 FT-QNN
. (27)
Nedge ,Ngroup ,str
Without restriction of generality, the loss calculation in (26) can
where λm,n,weak = 1 − λm,n,str . then be redefined as
We now can assume the possession of the training (testing) Nedge Ngroup
train
dataset consisting of Ndata test
(Ndata ) data points. Each i–th data 1 2
L(Θ) = |λFT-QNN
m,n − λdata,i
m,n | . (28)
point can be expressed as Nedge Ngroup m=1 n=1
During the training phase, the maximization of sum rate in (22)

data,i
τm,n = {xdata,i
m,n ; om,n },
data,i
(24) is projected as the minimization of the training loss, which can
be expressed as
data,i 2
data,i 2 2
where m,n = {hm,n,str , hm,n,weak },
xdata,i hdata,i
m,n,str ≥ min L(Θ). (29)
2 Θ
hdata,i
m,n,weak ,
data,i
m,n = λm,n,str ,
odata,i ∀m ∈ {1, . . . , Nedge },
[m] Accordingly, the solution provided by FT-QNN can be perceived
∀n ∈ {1, . . . , Ngroup }. The channel of the strong user is denoted
as the collection of solutions reserved for different edges. In
by hdata,i data,i
m,n,str . Meanwhile, hm,n,weak indicates the channel of the addition, each of the reference outputs included in the dataset,
weak user.
e.g., λdata,i
m,n , is the predefined solution for the corresponding edge.
The optimization process can be simplified by focusing on ac-
quiring the power allocation coefficient of the strong user within
each group, denoted as λdata,i
m,n,str . Subsequently, the power allo-
13 The training process is not the focus of this work. The training protocol
cation of the weak user can simply be calculated as λdata,i

m,n,weak =
is presented solely for the purpose of implementing FT-QNN and analyzing
its performance. More advanced training methodologies, particularly those
data,i
1 − λm,n,str . As the reference in the dataset, the optimized power leveraging gradient descent methods, are the subjects of future work.
The simulation was conducted as follows. The quan-

Algorithm 1: The Iterative Training Protocol.
tum computation was processed via IBM Qiskit platform
train
Input: Ndata × τm,n
data,i
, ∀m ∈ {1, . . . , Nedge }, (“qiskit”) [57]; the backend of “qasm_simulator” was em-
[m]
∀n ∈ {1, . . . , Ngroup } (see (24)). ployed; the libraries of “QuantumCircuit” were considered
[m]
Output: optimized Θcloud and Θedge , ∀m ∈ {1, . . . , Nedge } to generate the quantum circuit. The number of quantum mea-
Initialization: surements was set as Nshot = 1024, while the model outputs
[m]
1: Set all qubits in qedge , ∀m ∈ {1, . . . , Nedge }, and qcloud were obtained via averaging calculation in (19). The FT-QNN
as |0. was trained using the dataset presented in Section IV-C. Kindly
[m] note that the training data follows the optimal power alloca-
2: Set all weights in Θcloud and Θedge ,
m,n,str = om,n ) in (22) (which is based on the opti-
tion (λdataset dataset
∀m ∈ {1, . . . , Nedge }, in a random uniform
distribution with a range of {−2π, . . . , 2π}. mal power allocation in [53] - Appendix A). Therefore, al-
3: Set Ω ← {ω1 , . . . , ωNω }, where ωl , ∀l ∈ {1, . . . , Nω }, though not explicitly expressed, the output of the FT-QNN
is calculated using (30). (i.e., oFT-QNN
m,n ) ideally satisfies the constrains of (22). For test-
Weight optimization: ing, the dataset was generated according to the description in
4: for all Ndatatrain
dataset do Section IV-C. However, the testing used different datasets (with
test
5: for all m–th edge, m ∈ {1, . . . , Nedge } do a size of Ndata ) from training. For evaluation, E1 utilizes the
[m] output of the FT-QNN as the power allocation coefficient for
6: for all weights in Θedge do
strong users in the m–th group, i.e., λm,n,str ← odata,i m,n , ∀n ∈
7: for all Nω weight options do
[m] {1, 2}. Afterward, the average sum rate of E1 can be obtained
8: Set wj ← ωl , then update W. as
9: Do forward propagation Qforward (see (18)).
1 NOMA
2
Then, perform decoding operation Udecode (see NOMA 1
Rsum,1 = R1,n,str + R1,n,weak
NOMA
, (31)
(19)). 2 n=1 2
10: Calculate L(Ω) (see (28)).
11: end for NOMA NOMA
where R1,n,str and R1,n,weak are obtained from (21). Addition-
[m]
12: Set wj ← ωj∗ , where ωj∗ = arg min(L(Ω)) ally, the average rates for the strong and weak users in the group
13: end for E1 are calculated as
14: end for
NOMA 1 NOMA
15: for all weights in Θcloud do Rstr,1 = R1,1,str + R1,2,str
NOMA
,
16: for all Nω weight options do 2
17: Set wjcloud ← ωl , then update W. NOMA 1 NOMA
Rweak,1 = R1,1,weak + R1,2,weak
NOMA
. (32)
18: Do forward propagation Qforward (see (18)). 2
Then, perform decoding operation Udecode (see The performance analysis of FT-QNN in NOMA can be
(19)). presented as follows. For the sake of comparison, different
19: Calculate L(Ω) (see (28)). approaches were investigated:
20: end for r Scheme A, which refers to the proposed FT-QNN in Sec-
21: Set wjcloud ← ωj∗ , where ωj∗ = arg min(L(Ω)) [m]
tion III. It employed Qtele to transmit quantum states |ϕout
22: end for [m]
from Q to Qcloud .
23: end for r Schemeedge B, which is presented in Appendix B. Although
employed QNN in the cloud and edge, the information
Furthermore, given Nω weight option values, the l–th entry was sent classically without quantum teleportation in this
of these weight options can be defined as scheme. For this purpose, the edge performed measurement
[m]
at Qedge ; the measurement result is then transmitted to the
2π
ωl = −2π + (l − 1) , ωl ≤ 2π, l ≥ 1. (30) cloud and used as the input of Qcloud .
Nω The key difference between Scheme A and Scheme B is
emphasized in Fig. 9. As presented in Fig. 9, Scheme B employs
V. PERFORMANCE ANALYSIS classical method to transmit information from edge to cloud. The
A. The Achieved Sum Rate information that is being transferred is the output of edge model
[m]
(denoted as Qedge ). To this end, the output qubits of the edge
This section analyses the performance of FT-QNN in NOMA model are measured and decoded, so that the edge output can
power allocation. Unless otherwise specified, the simulation be transmitted to the cloud as classical information (e.g., via
parameters are given in Table V.14 The considered circuit is control plane [59], [60]). Subsequently, the cloud encodes the
presented in Fig. 7. Taking into account the quantum circuit transmitted edge output information and uses it as the input of
details provided in Fig. 7 and Table V, we can deduce that cloud NOMA
the model. Fig. 6 presents Rsum,1 by utilizing FT-QNN with
requires 8 qubits while the edge requires 8 qubits, accordingly.
respect to SNR ρ. The “optim.” refers to the NOMA power
NOMA
14 Owing to the limitation on the number of qubits in the current NISQ system,
optimization in (25) (presented in [53]). Moreover, Rstr,1 and
NOMA
this study only considers an edge and a NOMA group in the simulation. Rweak,1 in accordance with the given transmitted SNR ρ are
TABLE IV
THE COMPLEXITY COMPARISON
TABLE V
THE SIMULATION PARAMETERS
NOMA
Fig. 6. The average sum rate Rsum,1 (cf. (31)) with respect to the transmit
SNR ρ, achieved using the proposed FT-QNN. Herein, Scheme A employs the
[m] [m]
quantum teleportation protocol to transmit the quantum state |ϕout from Qedge
to Qcloud . In contrast, Scheme B does not use quantum teleportation and instead
relies on quantum measurements to obtain the output of the edge QNN.
presented in Fig. 8. As presented in Fig. 6, Schemes A and B

achieve similar average sum rate.
In addition, the authors acknowledge the possibility of errors
and noise in quantum channels [67]. However, due to the lack A” and “Scheme B”, respectively. Moreover, the details of
[m] 2
of access to widely accepted noise and error models based Scheme B are described in Appendix B. Assume O((Nlayer ) ) =
on real-world measurements of quantum channels at the time 2
O((Nlayer
cloud
) ) = O((Nlayer )2 ), ∀m ∈ {1, . . . , Nedge }. For the
of writing, the study assumes a perfect quantum channel for sake of simplicity, consider Nneuron = Nlayer .
quantum teleportation in Scheme A. Consequently, to ensure The complexity comparison is presented in Table IV. For
a fair comparison, the classical output of the edge QNNs in [m]
Scheme A, the complexities of Qedge , Qcloud , and Qforward are
Scheme B is also transmitted under the assumption of an ideal presented in Lemmas 4, 6, and 7, respectively. For Scheme B,
classical communication channel without the influence of any [m]
the complexities of Qedge , Qcloud , and Qforward are presented
noise.
in Lemmas 9, 10, and 11. In this work, the computational
The decline in loss with respect to the number of optimized
complexity of the quantum operation is analyzed based on the
weights is presented as follows. The training process is pre-
number of operation sequences.
Specifically, an operation of
sented in Algorithm 1. Fig. 10 demonstrate the loss decline
N subsequent gates (e.g., N n=1 R Z (θj )) yields O(N ). On the
of Scheme A, in which the quantum teleportation operation
hand, the operation consisting of N parallel gates processed
Qtele is employed to convey information between different QNN
using different qubits (e.g., N n=1 R Z (θj )) yields O(1). As pre-
models. For comparison, Fig. 10 also presents the training
sented in the table, Scheme A yields a lower complexity for per-
loss of Scheme B, which requires each edge unit to conduct
forming a feed-forward inference. Moreover, the complexities
quantum measurements to obtain the output of its QNN model. [m]
As demonstrated in Fig. 10, Scheme B exhibits a slightly better of Qedge and Qcloud in both schemes are similar. For additional
loss performance compared to Scheme A. Fig. 10 shows that the comparison, the complexity of classical-based scheme is also
proposed FT-QNN has the ability to learn from the given dataset, exhibited in Table IV. The complexity of the classical-based
in particular using the user data discussed in Section IV-C, and scheme is analyzed in Appendix C. As shown in the table, the
gradually refine its ability to estimate solutions through the classical-based scheme yields a higher complexity compared to
training phase, guided by the training loss that has been defined FT-QNN.
according to the objective function (cf. (28) and (22)), with the Compared to the contrasting method that requires state mea-
decoding process designed to ensure the constraint to be satisfied surement, i.e., Scheme B, the proposed method with quantum
(cf. (27) and (22)). teleportation, i.e., Scheme A, has inherent advantages due to the
direct state transmission facilitated through the use of quantum
teleportation protocol. Firstly, Scheme B relies on classical
B. Complexity Analysis
communication channels and classical data storage to handle
This study compares the proposed FT-QNN (utilizing quan- the transmission of the output of the edge QNN to the cloud,
tum teleportation) and a distributed QNN scheme without which potentially poses an additional burden on the existing
quantum teleportation. For convenience, FT-QNN and the dis- classical communication channels (cf. Fig. 9). In particular,
tributed without quantum teleportation are referred as “Scheme if conventional wireless communication channels are used to
Fig. 7. The quantum circuit implemented during simulation [43], [44], [46], [47].
Fig. 8. The achievable average user rate of the proposed FT-QNN for NOMA Fig. 10. Loss decline of FT-QNN with respect to the number of optimized
power allocation with respect to the transmit SNR ρ. Additionally, the notation weights.
NOMA NOMA
of “strong” and “weak” refer to user rate calculations of Rstr,1 and Rweak,1 of
(32), respectively.
B can expose the system to different security risks such as

spoofing attempts and man-in-the-middle attacks. Specifically,
an attacker may attempt a data poisoning attack by feeding
false data to the cloud QNN with the aim of causing inaccurate
estimations [62].
VI. CONCLUSION
This work proposed the FT-QNN scheme, which integrates
quantum teleportation and QNNs, for a decentralized wireless
resource allocation. In the proposed scheme, a central cloud
QNN is connected to edge QNNs via quantum teleportation. As
Fig. 9. Comparison between Scheme A and Scheme B. Scheme A employs
quantum teleportation to send quantum-based information [58], whereas Scheme
a particular wireless optimization case, FT-QNN is employed to
B utilizes classical information, which can be conveyed through the control plane optimize NOMA power allocation, which is crucial for power-
of a conventional network [59], [60]). domain NOMA as each user is being served using a different
transmit power ratio.
This study highlights the benefits of using FT-QNN for wire-
transmit information from the edge to the cloud, challenges such less resource allocation; FT-QNN yields lower computational
as potential interference and resource allocation may need to complexity while having a similar achievable sum rate (as shown
be addressed. Secondly, sending the output of the edge QNN in Section V), compared to a distributed QNN scheme without
via classical means, as in Scheme B, requires additional com- quantum teleportation protocol.
putational memory at the side of the cloud, which may pose a In the future, the FT-QNN is expected to solve more complex
problem if high-dimensional data is transmitted [61]. In addition optimization problems (for example, optimizing massive MIMO
to these issues, sending classical-valued outputs as in Scheme with NOMA). As this work can considered as an initial effort
#
to utilize quantum teleportation in an FT-QNN, the following 2
topics should be investigated for future work. The utilization 1+ hdata,i
m,n,str ρ −1
of FT-QNN in a network with a massive number of edges and ⇔ 2
≤ λdata,i
m,n,str (34)
hdata,i
m,n,str ρ
users can be inspected. Training protocols for FT-QNN can be
further applied. Different resource allocation scenarios can also Therefore, to satisfy the constraints in (33) and (34)

be investigated. 2
1+hdata,i
m,n,weak ρ −1
Besides, different challenges associated with quantum com- λdata,i λdata,i
(i.e., m,n,str ≤ 2 and m,n,str ≥
puting can be mitigated by considering the following methods. hdata,i
m,n,weak ρ

2
Firstly, the integration of hybrid classical-quantum processing 1+hdata,i
m,n,str ρ −1
can compensate for the lack of quantum computational resources 2 , respectively), the condition of
hdata,i
m,n,str ρ

on the network side. [63], [64]. In addition, the low number of 2
1+hdata,i
m,n,weak ρ −1
quantum bits on current system can be mitigated by reducing data,i
λm,n,str = are required to be satisfied.
2
the input dimension (for instance, via the use of prior autoen- hdata,i
m,n,weak ρ
coder [65]). Moreover, as quantum decoherence (which can

prevent a qubit to maintain superposition for a long period) is still APPENDIX B
prominent [66], protocols to mitigate the impact of decoherence THE QUANTUM OPERATION OF THE SCHEME B
can be explored. Furthermore, the utilization of quantum error
In the following, the quantum circuit for edge QNN in Scheme
correcting methods can be investigated to mitigate the error
B will be discussed.
caused by noise in current quantum system [67]; in particular,
Definition 11 (The m–th edge QNN in Scheme B): The quan-
several methods are available to counter the impact of noise in
tum circuit of the m–th edge QNN is defined as15
quantum-based communication networks [68], [69].

[m] [m] [m] [m] [m] [m]
Qedge M qout CZ qout q [m] [m,l] Uedge Θedge
Nlayer ,Nneuron
APPENDIX A
⎛ ⎛ [m] ⎞ ⎞
POWER ALLOCATION USED FOR REFERENCE Nlayer Nneuron
[m,l]

⎜ [m] ⎜ [m] ⎟ [m] ⎟
Based on [53], the reference power allocation is presented as ⎝Uencode (ℵm) ⊗ ⎝ H ql,r ⎠ ⊗ H qout ⎠,
follows. l=2 r=1
Lemma 8: To satisfy the objective function given in (22), the (35)
optimal powercoefficient for a strong

user Um,n,str was analyzed

2 [m]
1+hdata,i
m,n,weak ρ −1 where Θedge is the set of weight values used for the m–th edge
as λdata,i
m,n,str = 2 . [m]
hdata,i
m,n,weak ρ QNN. Let θl,n be the weight value for the n–th neuron in the
Proof: The optimal power coefficient follows derivation lth layer.
OMA OMA [m,n] [m,1]
in [53]. Let Rm,n,str and Rm,n,weak be the achievable orthog- Lemma 9: We assume that Nuser = Nneuron = . . . =
onal multiple access (OMA) rates of strong and weak users, [m,N
[m]
] [m]
respectively. The following constraints are employed to ensure Nneuronlayer = Nlayer , m ∈ {1, . . . , Nedge }. The complexity of
that strong and weak users achieve higher rates using NOMA. [m] [m] [m] 2
Qedge in Scheme B can be illustrated as Qedge ∈ O((Nlayer ) ).
First, for the strong user [53]: [m]
Proof: As M(·), CZ(·), and Uedge yield O(1), O(1), and
[m] 2 [m] 2
NOMA
Rm,n,weak ≥ Rm,n,weak
OMA O((Nlayer ) ), respectively, the first part yields O((Nlayer ) ).
[m] [m,n] [m]
⎛ ⎞ Considering Lemma 2, Uencode yields O(Nuser ) ≈ O(Nlayer ).
2
hdata,i data,i
m,n,weak (1 − λm,n,str ) ⎠ A single H(·) operation yields O(1). The parallel H(·)
⇔ log2 ⎝1 + [m]
hdata,i
2 data,i
−1 operations yield O(1). Hence, the second part yields O(Nlayer ).
m,n,weak λm,n,str + ρ
[m] [m] [m] 2
2
In conclusion, Qedge yields O(Nlayer ) + O((Nlayer ) ) ≈
≥ (1/2) log2 1 + hdata,i
m,n,weak ρ [m] 2
O((Nlayer ) ).
# [m]
2 The decoding operation of the Qedge in Scheme B is defined
1 + hdata,i
m,n,weak ρ −1
as follows.
⇔ 2
≥ λdata,i
m,n,str (33) [m]
hdata,i ρ Definition 12 (Decoding operation of Qedge in Scheme B,
m,n,weak
[m] [m]
Udecode ): Consider that Qforward is a single feed-forward opera-
[m] [m]
Second, for the weak user [53]: tion of Qedge (35). The operation to decode the output of Qforward
2

NOMA
Rm,n,str ≥ Rm,n,str
OMA
⇔ log2 1 + hdata,i
m,n,str ρλm,n,str
15 The qubits are set to |0 for initialization. Therefore, this study considers
1 2 [m] [m,l]
≥ log2 (1 + hdata,i
m,n,str ρ)
[m] ⊗N N
Qedge |0 layer neuron for implementation.
2
[m]
is defined as where on r ∈ {0, 1} is the rth measurement result of the m–th
Nshot [m]
1 edge, m ∈ {1, . . . , Nedge }, n ∈ {1, . . . , Ngroup }.
ȯ[m] r ,
[m] [m]
Udecode : Qforward ⇔ oedge
m = (36)
Nshot r=1

[m]
APPENDIX C
where ȯ r
∈ {0, 1} is the rth measurement result of the m–th THE ASSUMED CLASSICAL FEDERATED NEURAL NETWORK
edge.
[m] For the sake of comparison, a classical-computation-based
The decoding output of Qedge , i.e., oedgem , is then used as the federated neural network can be presented as follows. In each
parameter of a RZ (·) gate in Qcloud . feed-forward inference, the m–th edge and cloud is involved.
Definition 13 (The circuit of cloud QNN in Scheme B): The [m] [m,l]
For the m–th edge, Nlayer is the number of layers and Nneuron
operation of the cloud QNN is expressed as16 cloud
⎛ ⎞ is the number of neuron in each lth layer. For the cloud, Nlayer
[m]
Nedge Ngroup
cloud,l
is the number of layers and Nneuron is the number of neuron in
Qcloud ⎝ M qN cloud
cloud ,r X qN cloud ,r H
cloud cloud
qN cloud ,r
⎠
each lth layer. The operation of the classical neural network that
layer layer layer
m=1 n=1
is processed by the m–th edge is given by
⎛ ⎞ ⎛ [m,l−1] ⎞
Nedge
edge cloud
[m]
Nlayer Nneuron
[m,l]
Nneuron
Ucloud (Θcloud ) ⎝ RZ om ; q1,m ⎠ [m]
Fclassical = factive ⎝
[m,l] [m,l−1] ⎠
ωi→j yj , (39)
m=1
i=1 i=1 j=1
⎛ ⎞
Nlayer
cloud cloud,l
N neuron [m,l]
where factive (·), ωi→j , and yj
[m,l−1]
, are the activation function,
⎝ cloud ⎠
H(ql,r ) . (37)
weight factor, and output from jth neuron, lth layer, respectively.
l=1 r=1
Additionally, the classical neural network that is processed by
cloud,1 cloud,N cloud the cloud can be defined as
Lemma 10: Assuming that Nneuron = . . . = Nneuron layer = ⎛ cloud,l−1 ⎞
cloud Nlayer Nneuron
cloud cloud,l
Nneuron
Nlayer , the complexity analysis of Qcloud in Scheme B can be
conveyed as Qcloud ∈ cloud 2
O((Nlayer ) ).
cloud
Fclassical = factive ⎝ cloud,l cloud,l−1 ⎠
ωi→j yj ,
i=1 i=1 j=1
Proof: First, parallel operations of M(·)X(·)H(·) ∈ O(1)
(40)
yield O(1). Next, as examined in Lemma 5, Ucloud yields cloud,l
cloud 2
where ωi→j , and yjcloud,l−1 , are the weight factor and output
O((Nlayer ) ). Additionally, parallel operations of RZ (·) yield from jth neuron, lth layer, respectively. The complexity of
O(1). Finally, parallel operations of H(·) yield O(1). In con- cloud [m]
cloud 2
Fclassical and Fclassical can be investigated as follows.
clusion, Qcloud yields O(1) + O((Nlayer ) ) + O(1) + O(1) ≈ Lemma 12: Given similar models at cloud and edge,
2
O((Nlayer
cloud
) ). that is, Fclassical = Fclassical
cloud [m]
= Fclassical , Nlayer = Nlayer
cloud [m]
= Nlayer ,
Consider Qforward as a single feed-forward operation of Qcloud cloud,l [m,l]
Nneuron = Nneuron = Nneuron , the complexity of Fclassical can be
(cf. (37)). The complexity of a single feed-forward inference in
analyzed as Fclassical ∈ O(Nlayer (Nneuron )2 ).
Scheme B is presented as follows.
edge Proof: Let us assume that each multiplication operation and
Lemma 11: Considering Nlayer cloud
= Nlayer = Nlayer , the com-
activation function yield O(1). Considering (40), each layer
putational complexity of Qforward can be investigated as
of Fclassical yields O((Nneuron )2 ). Subsequently, Fclassical yields
Qforward ∈ O((Nlayer )2 Nshot ).
[m] Fclassical ∈ O(Nlayer (Nneuron )2 ).
Proof: From Lemma 9, consider Qedge ∈ O((Nlayer )2 ).
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pp. 480–486, Aug., 2020. interests include 5G/6G wireless communications
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Conf. Quantum Comput. Eng., 2022, pp. 341–352.
[69] L. Gyongyosi and S. Imre, “Advances in the quantum internet,” Commun.
ACM, vol. 65, no. 8, pp. 52–63, 2022.

Federated Quantum Neural Network With Quantum Teleportation For Resource Optimization in Future Wireless Communication

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Federated Quantum Neural Network With Quantum Teleportation For Resource Optimization in Future Wireless Communication

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 72, NO.

11, NOVEMBER 2023 14717

Federated Quantum Neural Network With Quantum

Abstract—The following study introduces FT-QNN, a federated TABLE I

To date, initial experiments have been made on quantum

TABLE III [m]

[m] [m] cloud,[m]

[1] [N ] In our ideal scenario, the edges perform quantum teleportation

[m,n] [m,1] considering

b=1 Ngroup if m > 1, for implementation.

Ucloud (Θcloud ) ⎝ cloud ⎠

Further details regarding (25) are provided in Appendix A.

During the training phase, the maximization of sum rate in (22)

cation of the weak user can simply be calculated as λdata,i

The simulation was conducted as follows. The quan-

presented in Fig. 8. As presented in Fig. 6, Schemes A and B

B can expose the system to different security risks such as

coder [65]). Moreover, as quantum decoherence (which can

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