need to detect as fast as possible [177] and models require frequent update
[58]. We also suggest that with the advent of adversarial ML, additional
measures such as robustness of features and security of ML model (as suggested
in [178]) be considered.

Summary: Due to skewed data and class distribution, we recommend that
researchers and practitioners use F-score, ROC, confusion matric, G-mean,
prediction time, training time, and feature and model robustness to evaluate
ML-based data exfiltration countermeasures.

###  V-I Need for Real-time Detection of Data Exfiltration Attacks

By real-time attack detection, we mean the data exfiltration detector collects
new and targeted information from both internal (e.g., computing servers) and
external sources (e.g., blacklisted IPs), and process the information almost
instantaneously to generate alerts about a potential data exfiltration attack.
Given the sophisticated data exfiltration attack vector, it is imperative to
detect data exfiltration in real-time or at least near real-time. This is
because the longer an attack goes undetected, the higher is the damage [179].
Given the fact that data exfiltration attacks are highly targeted attacks
(e.g., aiming only to leak top secrets of an enterprise), the exfiltration
process is quite quick especially after a hacker has located the target data.
With real-time detection, the success rate of a data exfiltration attack can
be reduced by around 97% [180]. For instance, in 2019, Wipro – a consultancy
company fell victim to a data exfiltration attack aimed at exfiltrating
customer data [181]. However, the swift response of a company to detect and
mitigate an attack saved a company from a huge loss. In our review, only 12/92
studies report the training and prediction time. Table XVI shows the training
time (for training a detection system) and the prediction time (i.e., time
taken to detect an attack) for these studies. Given the significance of time
for detecting a data exfiltration attack, we believe that all researchers
should consider response time (training and prediction time) as a critical
quality attribute and report it in their studies. Based on Table XVI, the mean
training time is 4.72 min, and the mean prediction time is 8.78 min for the
reviewed ML-based data exfiltration countermeasures. This time is primarily
consumed in the analysis of a large size of data. For example, an organization
as large as HP generates around one trillion security events per day [182].
Therefore, we recommend the incorporation of big data technologies (e.g.,
Apache Spark and Apache Hadoop) in ML-based data exfiltration countermeasures.
It is because big data technologies have proven advantages. For example, a
study [183] by Zions incorporation revealed that the use of Apache Hadoop
reduced the processing time from 20-60 min to 1 min to analyse a month of
security event data.

Summary: Researchers should consider and report response time as an important
quality attribute for ML based data exfiltration countermeasures. Moreover,
researchers should explore the use of big data technologies for engineering ML
based data exfiltration countermeasures.

TABLE XVI: Studies that report the training time and prediction time

Study | S11 | S13 | S25 | S43 | S44 | S51 | S55 | S59 | S62 | S63 | S64 | S65 | S70 | S73 | S78 | S92
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Training Time (sec) |  | 0.3 | 46.13 |  | 45.5 | 3.34 | 0.77 | 4.3 |  |  | 2268 | 7.96 | 39.24 | 964 | 0.05 | 22.72
Prediction Time (sec) | 18 | 0.21 |  | 0.08 | 6180 |  | 0.0023 | 0.016 | 0.0758 | 0.64 | 1 | 0.81 |  | 123 | 0.03 |

###  V-J Under-addressed Quality Attributes of ML-based Data Exfiltration
Countermeasures

We observe that most of the ML based data exfiltration countermeasures are
evaluated using two quality measures i.e., accuracy used by 70% studies and
response time reported by 15% studies. However, we believe that designing and
evaluating ML-based data exfiltration countermeasures only with accuracy and
response time is not enough. Additional quality measures such as
extensibility, customizability, interpretability, interoperability, automation
of model update, security, availability, and reproducibility should be
analyzed to ensure the quality and reliability of the ML-based data
exfiltration countermeasures [184]. For instance, it is crucial to consider
how the data exfiltration countermeasure interoperates in the whole security
spectrum of an organization, i.e., integration of the countermeasure into the
unified security orchestration platform [185]. Similarly, the ML model
employed in the data exfiltration countermeasure should be interpretable to
investigate how and why a data exfiltration attack went undetected through the
countermeasure. Testing and verifying the quality of these ML-based software
systems, including data exfiltration countermeasures is inherently challenging
because traditional software testing methods are not directly applicable to
these systems [186, 184, 187] In this regard, there needs to be close
collaboration between software and ML communities [184], so both work together
to drive the future of ML-based cybersecurity systems such as intrusion
detection system, malware detection system, and data exfiltration detection
systems.

Summary: Both software engineering and Machine learning communities should
work together to integrate quality in ML lifecycle for ML-based data
exfiltration countermeasures.

##  VI Threats to Validity

Whilst we strictly followed the guidelines provided by Kitchenham and Charters
[54] for qualitative analysis and performed quantitative analysis based on
[63], we had similar threats like other SLRs in software engineering and
machine learning domains. The findings of this SLR may have been affected by
the following threats.

  1. 1.

_Search Strategy_ For any SLR, identifying all the primary studies is a
challenging task. To mitigate this threat, we used six data sources which
enabled us to cover a wide variety of publication sources for finding relevant
papers [36]. Furthermore, to address the issue of missing search terms in the
search string, we employed three strategies: i) Search string was improved
iteratively based on pilot searches. ii) We took inspiration from the search
strings used in existing reviews [1, 14] and iii) We performed forward and
backward snowballing to find other relevant papers that might have been missed
by the search string.

  2. 2.

_Bias in Study Selection and Data Extraction_ : This step can be prejudiced to
researchers’ subjective judgement about whether the reviewed papers meet the
inclusion or exclusion criteria reported in SectionIII-D. To address this
issue, the studies were selected through a multi-phase process (Section
III-E). Furthermore, we performed a cross-check using a random number of
selected papers to ensure that our inclusion and exclusion criteria returns
the papers that we already knew to be relevant. Such a process ensures to a
high degree that all relevant studies are identified and selected. In addition
to study selection, the bias in extraction of data from the primary studies
can also affect the findings of this SLR. To mitigate this threat, we
formulated a data extraction form (see Table IV) to consistently extract and
analyse data for answering the research questions.

  3. 3.

_Box Plots_ : The data points in the boxplots reported in this SLR to
synthesize the quantitative data is limited to what is reported in the
reviewed studies. For example, during the investigation of feature selection
in relation to number of features selected, filter method had 14 data rows
while embedded had only two data row. This issue might skew the findings and
so threatens the validity of our findings. To mitigate this threat, we
selected median of data group for quantitative analysis. Similar to [188, 57],
we conclude that it is important to investigate the multiple factors involved
in ML lifecycle to investigate the model performance.

##  VII CONCLUSION

The main goal of this SLR was to analyse and synthesise the existing
literature on ML-based data exfiltration countermeasures in order to identify
what kind of features, datasets, modelling techniques, validation method and
performance evaluation criteria are used in these studies. A systematic
literature review was conducted to fulfil the study goals and answers five
research questions. After a comprehensive research protocol, 92 studies were
selected as primary studies. Besides identifying the research gaps and making
recommendations for future research, the main outcomes of this SLR are
summarised below.

  1. 1.

ML-based data exfiltration countermeasures can be classified into two main
groups, i.e., data-driven and behaviour-driven approaches. Behaviour-driven
approaches result in more (median $>6\%$) false positives but are capable of
detecting complex attack vectors like APT, lateral movement, and insider
threat. In contrast, data-driven approaches attain high performance (median
accuracy and FPR $>95\%$ and $<6\%$). However, these approaches are only used
frequently to detect the _Delivery_ stage of DELC.

  2. 2.

Six types of features are used by the identified countermeasures, i.e.,
statistical, structural, behavioural, spatial, syntactic, and temporal
features. Among these, statistical features are most frequently used. Most of
these features are extracted by using domain knowledge; 27% of the reviewed
studies used automated feature engineering process. Feature selection
techniques are not used frequently by these studies.

  3. 3.

Twelve datasets used by the reviewed studies are publicly available out of
which eight are one decade old.

  4. 4.

Classification is mostly used by the reviewed studies and only 16% studies
used anomaly-based detection. RFT and SVM are the most frequently used
classifiers. Ensemble classifiers RFT, Gradient Boosting and Adaboost,
performed better than all the other classifiers achieving F-Score of 98%.

  5. 5.

K-Fold and Hold Out are the most frequently used validation methods. Accuracy,
Recall, and FPR are most commonly reported performance measures in these
studies. The countermeasures are effective in detecting command and control,
data tunnelling, Phishing, Overt channel and Malicious domain attacks with a
high median accuracy and recall, i.e., 95% and median FPR of less than 2%.

To conclude, the performance and quality of ML-based models for detecting data
exfiltration is driven by the features, datasets, modelling technique and
validation method. These models are still in infancy and many limitations need
to be addressed, in terms of the quality and reliability of these methods. In
this respect, we hope that this SLR will provide a research direction to
researchers and help practitioners to adopt better practices.

##  VIII Acknowledgemets

This work is partially supported by CSIRO’s Data61, Australia and Cyber
Security Research Centre Limited whose activities are partly funded by the
Australian Government’s Cooperative Research Centres Programme.

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TABLE A: Selected studies in the review. Here ID denotes study identification number. ID | Title | Authors | Venue | Year
---|---|---|---|---
S1 |  Automatic malware classification and new malware detection using machine learning |  Liu Liu, Bao-sheng Wang, Bo Yu, Qiu-xi Zhong |  Frontiers of Information Technology & Electronic Engineering |  2017
S2 |  detection of advanced persistent threat using machine learning correlation analysis |  Ibrahim Ghafir, Mohammad Hammoudeh, Vaclav Prenosil, Liangxiu Han, Robert Hegarty, Khaled Rabie, Francisco J.Aparicio-Navarro |  Future Generation Computer Systems |  2018
S3 |  Steganalysis of JPEG ImageBased Steganography with Support Vector Machine |  Satoshi Watanabe ,Kazuki Murakami ,Tomoya Furukawa , Qiangfu Zhao |  International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing (SNPD) |  2016
S4 |  Use of Machine Learning in Big Data Analytics for Insider Threat Detection |  Michael Mayhew , Michael Atighetchi , Aaron Adler, Rachel Greenstadt |  IEEE Military Communications Conference |  2015
S5 |  Phishing detection based Associative Classification data mining |  Neda Abdelhamid, Aladdin Ayesh, Fadi Thabtah |  Expert Systems with Applications |  2014
S6 |  Supervised Learning for Insider Threat Detection Using Stream Mining |  Pallabi Parveen, Zackary R Weger, Bhavani Thuraisingham, Kevin Hamlen and Latifur Khan |  IEEE International Conference on Tools with Artificial Intelligence |  2011
S7 |  A New Take on Detecting Insider Threats:Exploring the use of Hidden Markov Models |  Tabish Rashid, Ioannis Agrafiotis, Jason R.C. Nurse |  International Workshop on Managing Insider Security Threats |  2016
S8 |  NIGHTs-WATCH: a cache-based side-channel intrusion detector using hardware performance counters |  Maria Mushtaq, Ayaz Akram, Muhammad Khurram Bhatti, Maham Chaudhry, Vianney Lapotre, Guy Gogniat |  International Workshop on Hardware and Architectural Support for Security and Privacy |  2018
S9 |  Ensemble Classifiers for Steganalysis of Digital Media |  Jan Kodovsky , Jessica Fridrich ,Vojtěch Holub |  IEEE Transactions on Information Forensics and Security |  2012
S10 |  A DataCentric Approach to Insider Attack Detection in Database Systems |  Sunu Mathew, Michalis Petropoulos , Hung Q. Ngo, Shambhu Upadhyaya |  International conference on Recent advances in Intrusion detection |  2010
S11 |  Identifying APT Malware Domain Based on Mobile DNS Logging |  Weina Niu, Xiaosong Zhang,GuoWu Yang,Jianan Zhu, Zhongwei Ren |  Mathematical Problems in Engineering |  2017
S12 |  Insider-threat detection using Gaussian Mixture Models and Sensitivity Profiles |  Kholood Al-tabash, Jassim Happa |  Computers & Security |  2018
S13 |  Investigation of Malicious Portable Executable File Detection on the Network using Supervised Learning Techniques |  Rushabh Vyas , Xiao Luo , Nichole McFarland , Connie Justice |  IEEE Symposium on Integrated Network and Service Management (IM) |  2017
S14 |  DNS Tunneling Detection Method Based on Multilabel Support Vector Machine |  Ahmed Almusawi and Haleh Amintoosi |  Security and Communication Networks |  2018
S15 |  Detection of Tunnels in PCAP Data by Random Forests |  Anna L. Buczak ,Paul A. Hanke ,George J. Cancro, Michael K. Toma ,Lanier A. Watkins,Jeffrey S. Chavis |  Cyber and Information Security Research Conference |  2016
S16 |  Detection of Malicious Webmail Attachments Based on Propagation Patterns |  Yehonatan Cohen. Danny Hendler, Amir Rubin |  Knowledge-Based Systems |  2017
S17 |  Detection of fraudulent emails by employing advanced feature abundance |  Sarwat Nizamani, Nasrullah Memon,Mathies Glasdam, Dong Duong Nguye |  Egyptian Informatics Journal |  2014
S18 |  Detection of Exfiltration and Tunneling over DNS |  Anirban Das , Min-Yi Shen , Madhu Shashanka , Jisheng Wang |  IEEE International Conference on Machine Learning and Applications (ICMLA) |  2017
S19 |  Classification of Phishing Email Using Random Forest Machine Learning Technique |  Andronicus A. Akinyelu and Aderemi O. Adewumi |  Journal of Applied Mathematics |  2014
S20 |  Multiclass JPEG Steganalysis Using Extreme Learning Machine |  Veenu Bhasin, Punam Bedi |  International Conference on Advances in Computing, Communications and Informatics |  2013
S21 |  Automated feature engineering for HTTP tunnel detection |  Jonathan J.Davis, Ernest Foo |  Computers & Security |  2016
S22 |  Analysis and Detection of Malicious Data Exfiltration in Web Traffic |  Areej Al-Bataineh , Gregory White |  International Conference on Malicious and Unwanted Software |  2012
S23 |  A Support Vector Machine-based Framework for Detection of Covert Timing Channels |  Pradhumna Lal Shrestha ; Michael Hempel ; Fahimeh Rezaei ; Hamid Sharif |  IEEE Transactions on Dependable and Secure Computing |  2015
S24 |  An Unsupervised MultiDetector Approach for Identifying Malicious Lateral Movement |  Atul Bohara ; Mohammad A. Noureddine ; Ahmed Fawaz ; William H. Sanders |  IEEE 36th Symposium on Reliable Distributed Systems (SRDS) |  2017
S25 |  Proposed efficient algorithm to filter spam using machine learning techniques |  Ali Shafigh Aski, Navid Khalilzadeh Sourati |  Pacific Science Review A: Natural Science and Engineering |  2016
S26 |  Detecting Malicious URLs using Machine Learning Techniques |  Frank Vanhoenshoven , Gonzalo Nápoles , Rafael Falcon , Koen Vanhoof ,Mario Köppen |  IEEE Symposium Series on Computational Intelligence (SSCI) |  2016
S27 |  Decision Tree Rule Induction for Detecting Covert Timing Channels in TCP/IP Traffic |  Félix Iglesias,Valentin Bernhardt,Robert Annessi,Tanja Zseby |  International Cross-Domain Conference for Machine Learning and Knowledge Extraction |  2017
S28 |  Detecting Malicious URLs Using Lexical Analysis |  Mohammad Saiful Islam Mamun, Mohammad Ahmad Rathore, Arash Habibi Lashkari,Natalia Stakhanova,Ali A. Ghorbani |  International Conference on Network and System Security |  2016
S29 |  Detecting DNS Tunnel through Binary-Classification Based on Behaviour Features |  Jingkun Liu ; Shuhao Li ; Yongzheng Zhang , Jun Xiao , Peng Chang , Chengwei Peng |  IEEE Trustcom/BigDataSE/ICESS |  2017
S30 |  DTB-IDS: an intrusion detection system based on decision tree using behaviour analysis for preventing APT attacks |  Daesung MoonHyungjin ImIkkyun KimJong Hyuk Park |  Journal of Supercomputing |  2017
S31 |  A Machine Learning based Web Spam Filtering Approach |  Santosh Kumar , Xiaoying Gao , Ian Welch , Masood Mansoori |  International Conference on Advanced Information Networking and Applications (AINA) |  2016
S32 |  DNS tunneling detection through statistical fingerprints ofprotocol messages and machine learning |  M. Aiello M. Mongelli G. Papaleo |  International Journal of Communication Systems |  2014
S33 |  Detection of DNS Tunneling in Mobile Networks Using Machine Learning |  Van Thuan Do,Paal Engelstad,Boning Feng, Thanh van Do |  International Conference on Information Science and Applications |  2017
S34 |  Detecting DNS Tunneling Using Ensemble Learning |  Saeed Shafieian,Daniel Smith,Mohammad Zulkernine |  International Conference on Network and System Security |  2017
S35 |  Uncovering social spammers: social honeypots + machine learning |  Kyumin Lee, James Caverlee, Steve Webb |  Proceedings of the 33rd international ACM SIGIR conference on research and development in information retrieval |  2010
S36 |  Detecting Phishing Attacks Using Natural Language Processing and Machine Learning |  Tianrui Peng , Ian Harris , Yuki Sawa |  International Conference on Semantic Computing (ICSC) |  2018
S37 |  PHISH-SAFE: URL Features-Based Phishing Detection System Using Machine Learning |  Ankit Kumar Jain,B. B. Gupta |  Cyber Security. Advances in Intelligent Systems and Computing |  2018
S38 |  Detecting advanced persistent threats using fractal dimension based machine learning classifcation, |  Sana Siddiqui,Muhammad Salman Khan,Ken Ferens, Witold Kinsner |  International Workshop on Security And Privacy Analytics |  2016
S39 |  PeerRush: Mining for unwanted P2P traffic |  Babak Rahbarinia, Roberto Perdisci,Andrea Lanzi,KangLi |  Journal of Information Security and Applications |  2014
S40 |  EXPOSURE: A Passive DNS Analysis Service to Detect and Report Malicious Domains |  Leyla Bilge, Sevil Sen, Davide Balzarotti, Engin Kirda, Christopher Kruegel |  ACM Transactions on Information and System Security |  2014
S41 |  Detecting Remote Access Trojans through External Control at Area Network Borders |  Shuang Wu , Shengli Liu , Wei Lin , Xing Zhao , Shi Chen |  IEEE Symposium on Architectures for Networking and Communications Systems (ANCS) |  2017
S42 |  An Approach to Detect Remote Access Trojan in the Early Stage of Communication |  Dan Jiang , Kazumasa Omote |  International Conference on Advanced Information Networking and Applications |  2015
S43 |  A network-based framework for RAT-bots detection |  Ahmed A. Awad , Samir G. Sayed , Sameh A. Salem |  IEEE Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON) |  2017
S44 |  A Practical Experiment of the HTTP-Based RAT Detection Method in Proxy Server Logs |  Mamoru Mimura , Yuhei Otsubo , Hidehiko Tanaka , Hidema Tanaka |  Asia Joint Conference on Information Security (AsiaJCIS) |  2017
S45 |  A host-based framework for RAT bots detection |  Ahmed A. Awad , Samir G. Sayed , Sameh A. Salem |  International Conference on Computer and Applications (ICCA) |  2017
S46 |  Towards detection of phishing websites on client-side using machine learning based approach |  Ankit Kumar JainB. B. Gupta |  Telecommunication Systems |  2018
S47 |  Detection of phishing websites using an efficient feature-based machine learning framework |  Routhu Srinivasa Rao, Alwyn Roshan Pais |  Neural Computing and Applications |  2018
S48 |  Malicious sequential pattern mining for automatic malware detection |  Yujie Fan,Yanfang Ye, Lifei Chen |  Expert Systems with Applications: An International Journal |  2016
S49 |  Detection of Mobile Applications Leaking Sensitive Data |  Yavuz Canbay , Mehtap Ulker , Seref Sagiroglu |  International Symposium on Digital Forensic and Security (ISDFS) |  2017
S50 |  Defending Malicious Script Attacks Using Machine Learning Classifiers |  Nayeem Khan,Johari Abdullah, Adnan Shahid Khan |  Wireless Communications & Mobile Computing |  2017
S51 |  Prediction of Cross-Site Scripting Attack Using Machine Learning Algorithms |  B. A. Vishnu,K. P. Jevitha |  International Conference on Interdisciplinary Advances in Applied Computing |  2014
S52 |  Cache-based side-channel attacks detection through Intel Cache Monitoring Technology and Hardware Performance Counters |  Mohammad-Mahdi Bazm , Thibaut Sautereau , Marc Lacoste , Mario Sudholt , Jean-Marc Menaud |  International Conference on Fog and Mobile Edge Computing (FMEC) |  2018
S53 |  Wide scope and fast websites phishing detection using URLs lexical features |  Ammar Yahya Daeef , R. Badlishah Ahmad , Yasmin Yacob , Ng Yen Phing |  International Conference on Electronic Design (ICED) |  2016
S54 |  Boosting the phishing detection performance by semantic analysis |  Xi Zhang , Yu Zeng , Xiao-Bo Jin , Zhi-Wei Yan , Guang-Gang Geng |  IEEE International Conference on Big Data (Big Data) |  2017
S55 |  SQLiGoT: Detecting SQL injection attacks using graph of tokens and SVM |  Debabrata Kar, Suvasini Panigrahi, Srikanth Sundararajan |  Computers & Security |  2016
S56 |  User-profile-based analytics for detecting cloud security breaches |  Trishita Tiwari , Ata Turk , Alina Oprea , Katzalin Olcoz , Ayse K. Coskun |  IEEE International Conference on Big Data (Big Data) |  2017
S57 |  Detecting Information Theft Based on Mobile Network Flows for Android Users |  Zhenyu Cheng , Xunxun Chen , Yongzheng Zhang , Shuhao Li , Yafei Sang |  International Conference on Networking, Architecture, and Storage (NAS) |  2017
S58 |  Phishing website detection using Latent Dirichlet Allocation and AdaBoost |  Venkatesh Ramanathan , Harry Wechsler |  IEEE International Conference on Intelligence and Security Informatics |  2012
S59 |  SQL injection attack classification through the feature extraction of SQL query strings using a gap-weighted string subsequence kernel |  Paul R.McWhirter, Kashif Kifayat,Qi Shi, Bob Askwith |  Journal of Information Security and Applications |  2018
S60 |  Website Defacements Detection Based on Support Vector Machine Classification Method |  Siyan Wu, Xiaojun Tong,Wei Wang, Guodong Xin, Bailing Wang, Qi Zhou |  International Conference on Computing and Data Engineering |  2018
S61 |  Malware Analysis and Detection Using Data Mining and Machine Learning Classification |  Mozammel Chowdhury, Azizur Rahman, Rafiqul Islam |  International Conference on Applications and Techniques in Cyber Security and Intelligence |  2018
S62 |  idMAS-SQL: Intrusion Detection Based on MAS to Detect and Block SQL injection through data mining |  Cristian I.Pinzón, Juan F.De Paz, Álvaro Herrero, Emilio Corchado, Javier Bajo, Juan M.Corchado |  Information Sciences |  2013
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A unifying primary framework for quantum graph neural networks from quantum graph states

Ammar Daskin

Graph states are used to represent mathematical graphs as quantum states on quantum computers. They can be formulated through stabilizer codes or directly quantum gates and quantum states.
In this paper we show that a quantum graph neural network model can be understood and realized based on graph states.
We show that they can be used either as a parameterized quantum circuits to represent neural networks or as an underlying structure to construct graph neural networks on quantum computers.

quantum machine learning, quantum graph states, quantum graph neural networks

§ INTRODUCTION

§.§ Graph Neural Networks [1, 2, 3, 4, 5]

Artificial neural networks are described through neurons (nodes) connected to each other through a multi layered structure.  In neural networks each node is connected to every node in the subsequent layers. This can be depicted as follows:

mynode=[thick,draw=blue,fill=blue!10,circle,minimum size=3]
[x=2.2cm,y=1.4cm, scale=0.5]

ı[evaluate==; =/2-ı+0.5; =int(-1);] in 1,...,
[mynode] (N-ı) at (,) ;
ȷin 1,...,[]
[thick] (N-ȷ) – (N-ı);

On the other hand, graph neural networks [6] (see Ref.[4] for an introductory tutorial or Ref.[7] for a very short overview.) describe neural network structure through specified graphs where the connection between subsequent layers are defined by the connection properties of the graph.
They are also used as building blocks to solve combinatorial optimization problems [8].
In general, graphs represent some real world problems by equating vertices to the features and relationships between features to the edges.
Therefore, for a feature matrix $X=[x_1, \dots, x_{|V|}]$, each vertex represents a feature in the data set.
To build an example graph neural network, we consider the following  graph:

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[vnode]  (B) at (0,3) B;
[vnode]  (C) at (2.5,4) C;
[vnode]  (D) at (2.5,1) D;
[vnode]  (E) at (2.5,-3) E;
[->] (A) edge node  (B)
(B) edge node  (C)
(A) edge node  (D)
(D) edge node  (C)
(A) edge node  (E)
(D) edge node  (E);

[tdplot_main_coords,thick,scale=0.7, every node/.style=transform shape]
(layer1) at  (0,0,0);
[]  at (layer1)layer=white/;

A GNN with this graph can be built by layering this graph as in the neural networks and using the connectivity of the original graph to adjust information flow (the connection) between layers.
An example with three such layers is shown in Fig.<ref> where the information flow between layers is represented for the node $A$ in the first layer. The information of $A$ accumulates through the nodes $E,B, D$ in the second layer since there is an edge starting from $A$ and ending with any of these nodes in the original graph.
Based on the newly obtained information, the state of a node or an edge can be updated via a trainable model.

This construction can be considered as a generalization of the neural networks since we can also describe neural networks by using the above description. Therefore, we can say that graph neural networks are at least as powerful as the neural networks in solving different kinds of problems [9].
On the other hand, since the connectivity in a problem can be represented by different isomorphic graphs, there is no guarantee that all isomorphic graphs would produce the same deep neural networks.
This means the graph neural networks are not permutation invariant [4]. Since there is no certain way to choose the original graph and hence to determine the optimum construct, one can say that the training task becomes more difficult.

§.§.§ Pooling and message passing (update) operations

The operations in GNNs can be summarized as pooling and message passing [10] where either the information is flowed from the edges to the nodes or from the nodes to the edges:
In message passing, the information in nodes are updated by using a differential function applied to the messages sent by their immediate neighbors.
This layer mainly changes the graph into a new representation. That is why it is also called the permutation equivariant layer in the literature. Mathematically, in message passing layer of GNNs, the  message passing for a node $u$ with the neighborhood $N(u)$ can be described as a function of the node and edge features by the followings [5]:

\begin{equation}
\label{eq:classical_msgpassing}
h_u = \phi \left( \bf{x}_u, \bigoplus \psi (\bf{x}_u, \bf{x}_v, \bf{e}_{uv})\right),
\end{equation}

where $v\in N(u)$, $\bf{x}_u, \bf{x}_v,$ and $\bf{e}_{uv}$ are the features for nodes $u$ and $v$ and the edge $(u,v)$.
In addition, $\phi$ and $\psi$ are the differentiable functions commonly used in neural networks such as the ReLU function. These functions are also called update and message passing functions. And $\bigoplus$ represents a permutation invariant aggregation operator such as sum, max, or mean.
Similarly, there are also the attention  and the convolution based update functions which are incorporated with the function definitions of $\phi$ and $\psi$ (see Ref.[7]).

The pooling layer could be local or global. It basically applies pooling operation (the neighborhood aggregation) first to the local neighborhoods and then to the resulting graph globally to obtain a readout value which is used to compare learning outcome.
There are different pooling operations such as top-$k$ pooling[11] or self attention pooling [12].

Here, as it can be observed from the above definitions, the construction of a neural network though a graph gives more freedom in choosing and describing a neural network model. Therefore, one can apply different pooling or message passing strategies based on the problem.

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[vnode]  (B) at (0,3) B;
[vnode]  (C) at (2.5,4) C;
[vnode]  (D) at (2.5,1) D;
[vnode]  (E) at (2.5,-3) E;
[->] (A) edge node  (B)
(B) edge node  (C)
(A) edge node  (D)
(D) edge node  (C)
(A) edge node  (E)
(D) edge node  (E);

[on background layer]
[tape, tape bend height=1cm, fill=#1!30, minimum width=5cm,
fit=(A)(B)(C)(D)(E), rotate=-20, inner sep=5pt,opacity=0.4] #2;

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(layer3) at  (8,0,-1);
(layer2) at  (4,0,-0.5);
(layer1) at  (0,0,0);
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[]  at (layer1)layer=blue/Layer 1;
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edgered/.style=draw=red!90,very thick,dashed,
edgegreen/.style=draw=red!90,very thick,dash dot
[->,edgeblue] (A) edge node  ($(B)+(layer2)$)
(A) edge node  ($(D)+(layer2)$)
(A) edge node  ($(E)+(layer2)$);
[->,edgegreen] ($(D)+(layer2)$) edge node  ($(E)+(layer3)$)
($(D)+(layer2)$) edge node  ($(C)+(layer3)$);
[->,edgegreen] ($(B)+(layer2)$) edge node  ($(C)+(layer3)$);

An example graph neural network: Each node accumulates information through the nodes in the subsequent layers where only nodes in the neighborhood of the node in the original graph are considered. As an example, the vertex A is connected to E, D, and B in the original graph. Therefore, it is updated using these vertices in the previous layer.

§.§ Different Level of Tasks

In summary, a problem can be solved by a GNN in general by the following steps [4]:

* We take a graph as an input to the model. In some GNN models, it takes input from neural networks.

* Apply GNN layers. This can be a single graph convolution network[13, 14] or a multi-layered graph neural network.

* Output a transformed graph

* Then, apply a classification or another network layer to out a prediction.

The details of these steps are adjusted or changed based on the considered graph tasks. In literature,
these tasks are categorized as follows[4, 7]:

§.§.§ Graph-Level Tasks

The graph level tasks can be supervised classification or unsupervised clustering like problems. For instance, it may be related to identifying a given graph based on whether it includes some structures or associating one of the given labels to the graph. The example applications include chemical property prediction such as message passing graph networks applied to quantum chemistry problems [10] and molecular fingerprints that classify molecules for pharmaceuticals[15].

§.§.§ Node-Level Tasks

These tasks involve predicting target labels for each node: e.g. disease prediction [16] or problems related to social networks [17].

§.§.§ Edge-Level Tasks

Similarly to the node level task, in this case we deal with the edges: Either we predict the property of a link or its existence (see Ref.[18, 19] for the definition of the link prediction problem and its solution with GNNs).

§.§ Quantum Graph Neural Network Models

In recent years, quantum research in graph neural networks is as vibrant as classical research.
There are a few quantum graph neural networks models proposed and applied to various problems: This includes Ref.[20] which has described a parameterized circuit with sequence of parameterized quantum Hamiltonians as an equivalent to the classical graph neural network. Using the same remedy, it has also formulated quantum versions of many different classical graph neural network models such as graph recurrent, graph convolution, and spectral graph convolutional networks.
Ref.[21] has proposed a graph convolution network  based on variational quantum circuits where each link between nodes is established by using a controlled rotation $Y$ gate.
Ref.[22] has applied the same proposed design in [21] to the financial fraud detection problem.
In addition,
variational approaches are used for the simulation of particle interactions [23] and material search [24], a hybrid approach is used for particle track reconstruction[25], .
Ref.[26] has described an ancillary based model where graph initializing, convolution, and other layers are parameterized with the controlled rotation gates $X, Y,$ and $Z$.
This model is tested on various real world datasets and has overall performed well in those tests against the chosen classical models.
There are also models such as graph convolutional networks [27, 28, 29] and a quantum feature embedding described for classical graph neural networks[30].

§.§ Motivation and Contribution

The problems solved with graphs are in general difficult which requires processing graphs with enormous sizes. Therefore, graph sizes in real world applications tend to be inherently large and continue to increase every year [4].

As mentioned above, there are already a few quantum graph neural network models.
These models can be summarized as parameterized quantum circuits where some quantum gates are determined from the graphs. Although this encompasses the employment of a given graph into the neural network model, it is immediately not clear how it can be generalized for different graphs and problems using different formulations. In addition, it is not clearly stated how these models are different from the general quantum parameterized circuits used as neural network models.

Quantum graph states, as explained in the following section, are considered the quantum equivalent of mathematical graphs. Although many proposed quantum graph neural networks have some fruits of graph states in their definitions, to the best of our knowledge, they are not directly formulated through the graph states.
Therefore, we believe it is important to have a formalism of graph neural networks based on the graph states.
This makes their implementation general and their design intuition more understandable.

§ QUANTUM GRAPH NEURAL NETWORK MODEL BASED ON GRAPH STATES

§.§ Quantum Graph States

Graph states [31, 32]   are instances of stabilizer codes [33] which are used to design error correcting codes.
They can be used to study entanglement in mathematical graphs and formulate quantum solutions to the problems represented by these graphs(see Ref.[34]).
Given a graph $G=(V, E)$ with the sets of vertices $V =\{v_1, \dots, v_n\}$ and edges $E \subseteq \{ (u, v) | (u, v) \in V^2$ and $u\neq v\} $:

\begin{equation}
\ket{G} = \prod _{(u, v)  \in E} Uz(u,v) \ket{+}^{\otimes n},
\end{equation}

where $Uz(u,v)$ is a $Z$ gate that is controlled by the qubit $u$ and applies $Z$ to the qubit $v$. This basically establishes an entanglement between qubits $u$ and $v$ when there is an edge $(u, v)$ in graph $G$.
One can also specify the amount of the entanglement by using a general phase gate instead of a $Z$ gate:

\begin{equation}
\label{eq:uzuvw}
Uz(u, v, w) = e^{-iw\sigma_z^u\sigma_z^v},
\end{equation}

where $\sigma_z$ is one of the Pauli matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ and $w$ corresponds to the weight on the edge $(u, v)$ of the graph. Here note that, the resulting weighted graph states |G⟩ can be decomposed concisely into the standard basis with $N$ number of $W_i$ vectors:

\begin{equation}
\begin{split}
\ket{G} = &\frac{1}{\sqrt{N}} \sum_{W\subseteq V} \prod_{u, v\in E} Uz(u, v, w_{uv}) \ket{W}_z  \\
=   & \frac{1}{\sqrt{N}} \sum_{W\subseteq V} e^{i\frac{1}{2} W\cdot A \cdot W} \ket{W}_z,
\end{split}
\end{equation}

where $A$ is the adjacency matrix for the graph, $W$ is the matrix of the basis set, and $\ket{W}_z = \ket{W_1}\dots \ket{W_N}$.

This entanglement for a given set of pairs of qubits can be also established with the stabilizer codes. For each vertex $v$, we define a stabilizer operator with the Pauli matrices $\sigma_{x,y,z}$:

\begin{equation}
S_v = \sigma_x^{(v)} \prod_{u\in N(v)} \sigma_z^{(u)},
\end{equation}

where $N(v)$ is the neighborhood of $v$. The graph state $\ket{G}$ is the common eigenvector to the set of independent commuting operators $\mathcal{S} = \{S_{v_1}, \dots, S_{v_n}\}$. That means for any $S_{v} \in \mathcal{S}$,

\begin{equation}
S_v\ket{G} = \ket{G}.
\end{equation}

When the state |G⟩ is measured according to the pattern $S_v$ on $x-$direction and $z-$direction with the outcomes $m_x^v=\pm 1$ and $m_z^u=\pm 1$, $S_v$ puts the following constraint to the outcome:

\begin{equation}
m_x^v\prod m_z^u = 1.
\end{equation}

Here, note that the generalization to weighted graphs is not possible in the scope of Stabilizer formalism within the Pauli group, however, the amount of the entanglement can still be related to weights in the graphs (See Ref.[34] for more properties.).

In this paper, we will use the graph state |G(W)⟩ which represents a graph with the weights given by the matrix $W$.

§.§ Incorporating Data in |G⟩

Quantum machine learning models are based on variational (parameterized) quantum circuits.
These models either feed data into the quantum gates or uses a quantum state vector |x⟩ representing normalized data vector $\bf{x}$ or uses a single qubit
and its state |x_i⟩ to represent the $i^\text{th}$ feature in vector $\bf{x}$.
Note that one can also prepare the data matrix as a density operator by using $\sum_i \ket{\bf{x_i}}\bra{\bf{x}}$.

Since we describe GNNs through quantum graph states, the gates $U(u,v,w)$ describes the operation for the edge ($u$,$v$) with the weight $w$ and each qubit state |+⟩ describes a node of the graph.
To incorporate data into these states, we will use the above-mentioned formalism where |x_i⟩ describes the qubit state for the $i^\text{th}$ feature.
In this case, instead of |+⟩, each qubit state would be the following:

\begin{equation}
\ket{x_i} = \alpha_i \ket{0} +\beta_i \ket{1}.
\end{equation}

As a result, the graph state |G⟩ reads as:

\begin{equation}
\ket{G} = \prod _{(u, v)  \in E} Uz(u,v,w_{uv})  \bigotimes_i^n \ket{x_i}.
\end{equation}

Note that the dimension of |x_i⟩ may be different based on the considered data. In that case one can use multiple qubits for each feature (or vertex in the graph).

Also note that we can further rewrite this by using a parameterized single quantum gate $R_y(\theta_i)$ on each qubit $i$ with an angle value $\theta_i$:

\begin{equation}
\ket{G} = \prod _{(u, v)  \in E} Uz(u,v,w_{uv})  \bigotimes_i^n R_y(\theta_i) \ket{0}.
\end{equation}

In this formalism, by using the above equation, one can try to classically optimize parameters $\theta_i$s and the weights $w_{uv}$ as done in the standard quantum machine learning models based on parameterized quantum circuits.

§.§ Representing Layers

The layers in classical GNNs can be considered as the number of updates applied to the graph before getting the final outcome.
In quantum version, we have multiple choices to design equivalent formulation:

* One can represent this whole process as a quantum state with a fixed number of layers.

\begin{equation}
\label{eq:gnn1}
\ket{GNN_1} = \frac{1}{\sqrt{m}}\sum_i^m \ket{G_i},
\end{equation}

where |G_i⟩ represents the $i^\text{th}$ layer in the defined $m-$layer GNN.
Then by applying some operations this whole |GNN⟩ can be turned into a desired neural network which represents the trained model.

* We can also use the following:

\begin{equation}
\label{eq:gnn2}
\ket{GNN_2} =  \ket{G_1}\dots \ket{G_m}.
\end{equation}

In this case, the trained model can be obtained by controlled operations between subsequent registers: i.e., |G_i⟩ is updated by using $G_{i-1}$ and so on. Then, |G_m⟩ becomes the final solution.

* One can also start with a graph |G_1⟩ and  change this into a desired graph |G_m⟩ by applying quantum operations adjusted and chosen based on some measurement results:

\begin{equation}
\label{eq:gnn3}
\ket{GNN_3} =  \ket{G_1} \rightarrow \ket{G_2} \rightarrow \dots \rightarrow \ket{G_m}.
\end{equation}

In this case |G_m⟩ represents the final trained neural network.

Although the above definitions seem different, one can show that they are equivalent in the formalism of quantum information which allows measurements along with computations. However, the number of computations and their computational complexities are different since they require different numbers of qubits and operations.

In the following subsections, we are going to define the pooling and message passing  operations used in classical GNNs.
Here note that we will describe these operations in general terms since their specifications can be adjusted based on the considered graph problem which can be either an edge, a node, or a global graph level task.

§.§ Quantum Pooling

§.§.§ Pooling with Measurements and the State Collapsing
In |GNN⟩, a qubit represents the state of a vertex in the graph.
Therefore, the pooling (a type of accumulation) operation can be implemented by using various measurements on the qubits.
If we have $n$-qubit state, the qubits first can be put into groups based on their neighborhoods.
Then starting from measuring the local neighborhood, successive measurements at the end yield a readout vector of dimension determined by the final number of qubits (if we are trying to predict a single global binary value, then the last qubit output determines the readout.).

Here, the important point is that if the local neighborhood requires measuring some $\tilde{n}$ number of qubits,
it requires $O(2^{\tilde{n}})$ complexity to obtain a tomography of the
$\tilde{n}-$qubit state with accuracy  $O(2^{-\tilde{n}})$.
Therefore,  the size of the local neighborhoods for pooling should be kept as small as possible with this pooling method.

The second important point is that classical pooling methods can be also based on nonlinear functions such as the sigmoid function.
Since |G⟩ is designed in Hadamard bases with the phases determined by the weights.
The difference in the measurement outcome of |0⟩ and |1⟩ on any qubit is determined by the cosine of the phases.
This provides a periodic activation function (e.g. see Ref.[35]) in contrast to the aperiodic functions like the sigmoid or step functions.
By feeding the local measurement results into the chosen aperiodic function, one can classically compute the nonlinear aperiodic functions from the same measurement results.

§.§.§ Pooling with the phase estimation

Instead of measurement one can use an ancilla register to compute the phases in the local neighborhoods.
In this case, each $Uz$ operation is controlled by the qubit representing $Uz$'s local group in the ancilla register.
A neural network model is also described based on this idea [35].

§.§.§ Pooling with Controlled Rotation or NOT Gates

A controlled rotation operation on any qubit involves their amplitudes $\alpha_i$, $\beta_i$s, and their phases.
If it is not necessary to output a single value from the whole graph, then we can describe pooling as a rotation operation.
Then, any measurement on the qubit involves all the amplitudes in a summation which can be considered as the final step of the pooling operation.

§.§ Quantum Message Passing

As in the classical case given in Eq.(<ref>), the quantum message passing can be designed through quantum operations. However, since the constructed graph |G⟩ represents an entangled quantum state any operation on any qubit may potentially affect the other qubits.

Since the whole quantum state |G⟩ involves not only |x_i⟩s but also $w_{uv}$s, an operation only on the $z-$axis potentially changes only the initial weights. If we apply controlled-$Z$ gates, then this operation involves the amplitudes $\alpha_i$s and $\beta_i$s in |x_i⟩s, and changes the weights, i.e. the connection (entanglement) between qubits.
Using this intuition, we can describe the following operation as quantum message passing or update operation:

\begin{equation}
Uz(u,N(u), w_{uN(u)}),
\end{equation}

where $N(u)$ is the local neighborhood of $u$ and $w_{uN(u)}$ can be -1 for a simple $Z$ gate or a different phase which may be related to the problem type.
Note that this controlled $Z-$ gate can be applied also to the determined local graph invariants.

§.§ Formalism for Solving Different Graph Tasks

§.§.§ Node Level Tasks

We prepare each qubit in |x_i⟩ state with the amplitudes $\alpha_i$ and $\beta_i$.
These amplitudes change during the graph evolution with the operations such as pooling and message passing.
Therefore, any measurement on the $y-$ axis gives us the qubit state |0⟩ or |1⟩ which can be used for binary classification.
For non-binary cases, one needs to use more than one qubit: For four node classes, we use two qubits for each node.

§.§.§ Edge Level Tasks

Since the edge weights are created by using the controlled rotation $Z$ gates ($U_z$ in (<ref>)) dimension, the information of the updated weights are related to the phases.
This information can be obtained by measuring qubits on the $z-$ axis: i.e. using a measurement operator as a product of Pauli Z gates.

In addition, using an ancilla register in the Hadamard bases, and controlling related $U_z$ operations by this ancilla, we can obtain the updated phases (direct phase estimation) which can be used for link based predictions.

§.§.§ Graph Level Tasks

Graph states describe a graph.
Therefore, the similarity between quantum states can be obtained through swap-tests.
If we have multiple classes and if we need to determine the class of a given graph state (i.e. global classification of the graphs), we can use a superposition of the classes, and obtain a superposition of the swap tests.
The measurement result yields the class of the given graph with a probability which can be high if the euclidean separation of the classes are wide enough.

§.§ Computational Complexity

We have presented three different formalism in Eq.(<ref>), (<ref>), and (<ref>) for representing GNN layers as |GNN_1⟩, |GNN_2⟩, and |GNN_3⟩, respectively.
In terms of complexity, the formalism |GNN_3⟩ is the simplest since we have a single network, for the representation, it requires less number of qubits than |GNN_2⟩. And since it is not a superposition state, obtaining node probabilities or edge values from this state are easier than obtaining from |GNN_1⟩.

Operations such as pooling and message passing involves two or more qubits and as discussed before, they can be implemented through controlled $Z$ and $Y$ gates.
However, in |GNN_1⟩, if these operations are global, then they can be applied to all layers in superposition.
One can also simplify these operations in |GNN_2⟩.

|GNN_3⟩ can be considered a classical equivalent where we process graphs from one layer to another layer in a sequential manner.
Therefore, in terms of complexity, it cannot provide any significant advantage over the classical models. However, the entanglement between qubits can perfectly mimic the connections in real world data.
Therefore, this may improve the accuracy of the models.

§.§ Other Operations on Graphs

Similar to convolutions in neural networks, there are polynomial filters and Chebnet [36] which are also used building blocks for the graph neural networks [37].
The polynomial filters are described through the graph Laplacian $L$:

\begin{equation}
L = D-A,
\end{equation}

where $A$ is the adjacency matrix and $D$ is the diagonal degree matrix.
Polynomials of the Laplacian operator is a general $d$-degree polynomial of the matrix $L$::

\begin{equation}
p_\bf{w}(L) = w_0I+w_1L^1+w_2L^2+\dots w_dL^d = \sum_{i=0}^{d} w_iL^i,
\end{equation}

where $\bf{w}$ is a vector of coefficients $w_0, \dots, w_d$. If we know the eigendecomposition of $L$ or if $L$ is a unitary matrix, we can generate the above polynomial easily on quantum circuits.
by using a unitary gate [38]:

\begin{equation}
\label{eq:UL}
U_L = \left(\begin{matrix}
I & 0\\
0 & L
\end{matrix}\right).
\end{equation}

The circuit for construction of the polynomial of the Laplacian operator. $U_L(i)$ represents the $i^\text{th}$ power of the matrix given in Eq.(<ref>).

If we put this gate on a circuit as depicted for four qubits in Fig.<ref>, we can obtain the following matrix representation:

\begin{equation}
P(A) =
\left(\begin{matrix}
\end{matrix}\right).
\end{equation}

We can include the $w$ vector as in the initial state of this matrix. Therefore, we can implement a quantum filter.
This operation can be directly applied to the models |GNN⟩ we have described before.

§ CONCLUSION

In this paper, we go through all the necessary steps to define quantum graph neural networks through quantum graph states.
We show how graph neural networks can be represented by using different formulations and discuss how pooling and message passing operations can be applied in these formulations.
Since graph states are well understood within the quantum information community, they can play a unifying framework and be building blocks to study quantum graph neural networks and understand their power.

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