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A Review on Quantum Machine Learning and Quantum Cryptography

Una revisión sobre el aprendizaje automático cuántico y la criptografía cuántica

Uma revisão sobre aprendizado de máquina quântica e criptografia quântica

Mauricio Solar

,(*), Felipe Cisternas Alvarez

, Jean-Pierre Villacura

, Liuba Dombrovskaia

Recibido: 10/10/2024 Aceptado: 10/10/2024

Summary. - This article corresponds to an extensive review of Quantum Computers. We chose to consider topics

relevant to quantum computing, such as machine learning, and the deepening of other issues related to cybersecurity.

We introduce the reader to the basic concepts of quantum computing so that they can easily understand the terms

mentioned in this review. We analyze different state of the art articles, and we give a summary of the contributions

made. Finally, we conclude with the analysis of the bibliography, the research centers, the current state of the art,

surprising results and conclusions.

Keywords: Quantum Machine Learning; Quantum Key Distribution (QKD); Quantum Cryptography.

(*) Autor de correspondencia.

Académico, Universidad Técnica Federico Santa María, mauricio.solar@usm.cl, ORCID iD: https://orcid.org/0000-0002-4433-4622

Estudiante postgrado, Universidad Técnica Federico Santa María, felipe.cisternasal@sansano.usm.cl,

ORCID iD: https://orcid.org/0009-0000-7029-8736

Estudiante postgrado, Universidad Técnica Federico Santa María, jean-pierre.rojas@sansano.usm.cl,

ORCID iD: https://orcid.org/0009-0002-8611-1408

Académica, Universidad Técnica Federico Santa María, liuba@inf.utfsm.cl, ORCID iD: https://orcid.org/00000001-6572-9765

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Resumen. - Este artículo corresponde a una revisión extensa (no exhaustiva) de Computación Cuántica. Se eligió

considerar temas relevantes para la computación cuántica, como el aprendizaje automático, y la profundización de

otros temas relacionados con la ciberseguridad. Se presenta los conceptos básicos de la computación cuántica para

comprender los términos mencionados en esta revisión. Se analiza diferentes artículos sobre el estado del arte y se

entrega un resumen de los aportes realizados. Finalmente, se presentan las conclusiones sobre el análisis de la

bibliografía, los centros de investigación, el estado actual del arte y resultados.

Keywords: Aprendizaje Automático Cuántico; Distribución de Claves Cuánticas (QKD); Criptografía Cuántica.

Resumo. - Este artigo corresponde a uma revisão extensa (não exaustiva) da Computação Quântica. Optou-se por

considerar temas relevantes para a computação quântica, como aprendizado de máquina, e aprofundar outros temas

relacionados à segurança cibernética. Os conceitos básicos da computação quântica são apresentados para

compreender os termos mencionados nesta revisão. São analisados diferentes artigos sobre o estado da arte e

fornecido um resumo das contribuições realizadas. Por fim, são apresentadas as conclusões sobre a análise da

bibliografia, os centros de investigação, o estado atual da arte e os resultados.

Palavras-chave: Aprendizado de Máquina Quântica; Distribuição Quântica de Chaves (QKD); Criptografia

Quântica.

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1. Introduction. - With certainty it can be stated that today’s computers are much faster than the computers of 70 years

ago. The computers of that time were large, heavy, with a very limited capacity and processing speed compared to what

is the standard now a day. We could consider quantum computers to be in this same state, as an emerging technology

that is still expensive, bulky and with a lot of research potential (15).

The theory of quantum computing points out that its processing speed can be much faster than even the fastest

supercomputer today. Examples such as Shor’s algorithm with its potential ability to factor large prime numbers in a

matter of seconds, as opposed to the thousands of years that classical computing could take, are considered signs of the

advances and development that is to come with quantum computing (14).

This paper explores a range of subjects concerning quantum computing, including quantum computers and

technologies. It is structured to provide readers with a comprehensive understanding, starting from the basics of

quantum computing and progressing to cover a wide range of proposed models for quantum computers. Additionally,

the paper delves into the future prospects and developments of the fields Quantum Machine Learning (QML) and

Quantum Cryptography, highlighting the immense potential of quantum computing and discussing current

advancements.

The structure of this paper after this introduction includes a brief overview of Quantum Computing (Quantum

Computers and Technologies, Quantum Data, Quantum Gates, Noise, Quantum Error Correction Quantum

Cybersecurity, and Quantum Machine Learning). In the section State of the Art we show the methodology, we describe

the new works and research, we show a comparative analysis of the latest advances, a bibliographic discussion and we

show a state of the art timeline jointly with expected or surprising results. The final section summarizes the conclusions

of this work.

2. Brief overview of Quantum Computing. –

2.1. Quantum Computing. - Quantum computing relies on properties of quantum mechanics to compute problems

that would be out of reach for classical computers. A Quantum Computer (QC) uses qubits. Qubits are like regular bits

in a classical computer, but with the added ability to be put into a superposition state and share entanglement with one

other (26).

A QC works using quantum principles. Quantum principles require a new dictionary of terms to be fully understood,

terms that include superposition, entanglement, and decoherence. Let’s explain these principles below.

 Superposition: Superposition states that, much like waves in classical physics, you can add two or more

quantum states and the result will be another valid quantum state. Conversely, you can also represent

every quantum state as a sum of two or more other distinct states. This superposition of qubits gives QCs

their inherent parallelism, allowing them to process millions of operations simultaneously.

 Entanglement: Quantum entanglement occurs when two systems link so closely that knowledge about

one gives you immediate knowledge about the other, no matter how far away they are. Quantum

processors can draw conclusions about one particle by measuring another one, Quantum entanglement

allows QCs to solve complex problems faster. When a quantum state is measured, the wavefunction

collapses and you measure the state as either a zero or a one. In this known or deterministic state, the qubit

acts as a classical bit. Entanglement is the ability of qubits to correlate their state with other qubits.

 Decoherence: Decoherence is the loss of the quantum state in a qubit. Environmental factors, like

radiation, can cause the quantum state of the qubits to collapse. A large engineering challenge in

constructing a QC is designing the various features that attempt to delay decoherence of the state, such as

building specialty structures that shield the qubits from external fields.

The current state of quantum computing is referred to as the Noisy Intermediate-Scale Quantum (NISQ) era (8),

characterized by quantum processors containing 50–100 qubits which are not yet advanced enough for fault-tolerance

or large enough to achieve quantum supremacy, the term NISQ was coined by (32). These processors, which are

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sensitive to their environment (noisy) and prone to quantum decoherence, are not yet capable of continuous quantum

error correction. This intermediate scale is defined by the quantum volume, which is based on the moderate number of

qubits and gate fidelity.

Classical computers perform deterministic classical operations or can emulate probabilistic processes using sampling

methods. By harnessing superposition and entanglement, QCs can perform quantum operations that are difficult to

emulate at scale with classical computers. Ideas for leveraging NISQ quantum computing include optimization,

quantum simulation, cryptography, and Machine Learning (ML).

Notably, QCs are believed to be able to solve many problems quickly that no classical computer could solve in any

feasible amount of time—a feat known as quantum supremacy.

2.2. QCs and technologies. - A QC is a computer that exploits quantum mechanical phenomena. At small scales,

physical matter exhibits properties of both particles and waves, and quantum computing leverages this behavior using

specialized hardware. Classical physics cannot explain the operation of these quantum devices, and a scalable QC

could perform some calculations exponentially faster than any modern classical.

No one has shown the best way to build a fault-tolerant QC, and multiple companies and research groups are

investigating different types of qubits. We give a brief example of some of these qubit technologies below.

 Gate-based ion trap processors: Trapped ion QCs implement qubits using electronic states of charged

atoms called ions. The ions are confined and suspended above the microfabricated trap using

electromagnetic fields. Trapped ion-based systems apply quantum gates using lasers to manipulate the

electronic state of the ion (41). Trapped ion qubits use atoms that come from nature, rather than

manufacturing the qubits synthetically (31).

 Gate-based superconducting processors: Superconducting quantum computing is an implementation

of a QC in superconducting electronic circuits. Superconducting qubits are built with superconducting

electric circuits that operate at cryogenic temperatures (22).

 Photonic processors: A quantum photonic processor is a device that manipulates light for computations.

Photonic QCs use quantum light sources that emit squeezed-light pulses, with qubit equivalents that

correspond to modes of a continuous operator, such as position or momentum (23).

 Neutral atom processors: Neutral atom qubit technology is similar to trapped ion technology. However,

it uses light instead of electromagnetic forces to trap the qubit and hold it in position. The atoms are not

charged, and the circuits can operate at room temperatures (42). QuEra has a publicly accessible neutral-

atom computer

. It is a 256-qubit QC based on programmable arrays of neutral Rubidium atoms, trapped

in vacuum by tightly focused laser beams.

 Rydberg atom processors: A Rydberg atom is an excited atom with one or more electrons that are further

away from the nucleus, on average. Rydberg atoms have a number of peculiar properties including an

exaggerated response to electric and magnetic fields, and long life. When used as qubits, they offer strong

and controllable atomic interactions that you can tune by selecting different states (21).

 Quantum annealers: Quantum annealing uses a physical process to place a quantum system’s qubits in

an absolute energy minimum. From there, the hardware gently alters the system’s configuration so that

its energy landscape reflects the problem that needs to be solved. The advantage of quantum annealers is

that the number of qubits can be much larger than those available in a gate-based system. In fact, Quantum

annealing is implemented in D-Wave’s generally available QCs, such as the Advantage™

, enabling the

creation of Quantum Processing Units (QPUs) with more than 1200 qubits, far beyond the state of the art

for gate-model quantum computing. However, their use is limited to specific cases only (1).

https://www.quera.com

https://www.dwavesys.com

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2.3. Quantum Data. - Quantum data is any data source that occurs in a natural or artificial quantum system. Quantum

data exhibits superposition and entanglement, leading to joint probability distributions that could require an exponential

amount of classical computational resources to represent or store. The quantum supremacy experiment showed it is

possible to sample from an extremely complex joint probability distribution of 253 Hilbert space.

The qubit serves as the basic unit of quantum information. It represents a two-state system, just like a classical bit,

except that it can exist in a superposition of its two states. In one sense, a superposition is like a probability distribution

over the two values. However, a quantum computation can be influenced by both values at once, inexplicable by either

state individually. In this sense, a superposed qubit stores both values simultaneously. When measuring a qubit, the

result is a probabilistic output of a classical bit. If a QC manipulates the qubit in a particular way, wave interference

effects can amplify the desired measurement results.

The quantum data generated by NISQ processors are noisy and typically entangled just before the measurement occurs.

Heuristic ML techniques can create models that maximize extraction of useful classical information from noisy

entangled data.

The following are examples of quantum data that can be generated or simulated on a quantum device:

 Chemical simulation: Extract information about chemical structures and dynamics with potential

applications to material science, computational chemistry, computational biology, and drug discovery (10).

 Quantum matter simulation: Model and design high temperature superconductivity or other exotic states

of matter which exhibits many-body quantum effects (25).

 Quantum control: Hybrid quantum-classical models can be variationally trained to perform optimal open or

closed-loop control, calibration, and error mitigation. This includes error detection and correction strategies

for quantum devices and quantum processors (17).

 Quantum communication networks: Use ML to discriminate among non-orthogonal quantum states, with

application to design and construction of structured quantum repeaters, quantum receivers, and purification

units (3).

 Quantum metrology: Quantum-enhanced high precision measurements such as quantum sensing and

quantum imaging are inherently done on probes that are small-scale quantum devices and could be designed

or improved by variational quantum models (24).

2.4. Quantum Gates. - The state of qubits can be manipulated by applying quantum logic gates, analogous to how

classical bits can be manipulated with classical logic gates. Unlike many classical logic gates, quantum logic gates are

reversible (6). Quantum logic gates are represented by unitary matrices, a gate which acts on n qubits is represented

by 2n × 2n unitary matrix.

Quantum states are typically represented by kets, from a notation know as bra-ket, the vector representation of a single

qubit is shown in equation 1.

Here v0 and v1 are the complex probability amplitudes of the qubit, these values determine the probability of measuring

a 0 or a 1, when measuring the state of the qubit. The value zero is represented by the ket in equation 2, and the value

one is represented by the ket in equation 3.

The tensor product denoted by the symbol ⊗, is used to combine quantum states. The action of the gate on a specific

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quantum state is found by multiplying the vector |φ1i which represents the state by the matrix U representing the gate,

thus the result is a new quantum state |φ2i shown in equation 4.

There exist many numbers of quantum gates, below we review some of the most often used in the literature:

 NOT Gate: This gate is widely known as X-Pauli gate, as this particular quantum gate transforms the

existing state of the qubit to be rotated around the X-axis. As the name suggests, the NOT gate would

convert a qubit from its initial state to its complement state. This quantum gate is represented by the

matrix in equation 5 and operates as shown in equation 6.

 Y-Pauli Gate: The Y-Pauli gates are capable of rotating the input qubit around the Y-axis. This quantum

gate is represented by the matrix in equation 7 and operates as shown in equation 8.

 Z-Pauli Gate: The Z-Pauli or phase flip gate are capable of rotating the input qubit around the Z-axis.

This quantum gate is represented by the matrix in equation 9.

Pauli Z leaves the biasis state |0i unchanged and maps |1i to −|1i as shown in equation 10.

 Controlled NOT Gate: The Controlled NOT (CNOT) gate acts on 2 (or more) qubits and performs the

NOT operation on the second (or more) qubit only when the first qubit is |1i, this gate is represented by

the Hermitian unitary matrix (equation 11), and operates as in equation 12.

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 Hadamard gate: The Hadamard gate, acts on a single qubit and creates an equal superposition state

given a basis state. The Hadamard gate performs a rotation of√ π around the axis (xˆ + zˆ)/ 2 at the

Bloch Sphere (Figure I). This gate is represented by the Hadamard matrix (equation 13) and operates as

in equation 14.

2.5. Noise. - Noise is present in modern day QCs. Qubits are susceptible to interference from the surrounding

environment, imperfect fabrication, TLS and sometimes even gamma rays. Until large scale error correction is

reached, the algorithms of today must be able to remain functional in the presence of noise. This makes testing

algorithms under noise an important step for validating quantum algorithms and quantum models.

Figure I. Bloch sphere to represent a qubit.

2.6. Quantum Error Correction (QEC). - QEC is used in quantum computing to protect quantum information from

errors due to decoherence and other quantum noise. Traditional error correction employs over repetition. The

repetition code is the simplest but most inefficient way. The idea is to store the information multiple times and take a

larger vote in the event that these copies are later found to differ. Copying quantum information is not possible due to

the no-cloning theorem. This theorem seems to present an obstacle to formulating a theory of QEC, nevertheless, it is

conceivable to transfer the logical information of a single qubit to a highly entangled state of several physical qubits

(33).

2.7. Quantum Cybersecurity. –

2.7.1. Quantum Cryptography. - Cryptography has had an important development since 1975 with the establishment

of the Data Encryption Standard (DES) algorithm for file encryption while computing was emerging. Since then,

several algorithms have been developed to fulfill this cryptographic function, the best-known being RSA (Rivest-

Shamir-Adleman) and Advanced Encryption Standard (AES).

With the emergence of quantum computing, several researchers comment on the risk that the rise of this paradigm

may threaten encryption algorithms based on classical computing, which are considered secure due to the amount of

time it takes to test all combinations, easily 50 years using supercomputers. Quantum computing threatens the security

of these algorithms by being able to perform calculations much faster. Being able to solve operations that in the

classical paradigm would take about 50 years using supercomputers in a matter of seconds.

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The state of the art of quantum computing proposes modifications to algorithms RSA (7) and AES (19):

 (19) proposes a modified AES algorithm comparing different methods of random number generation,

resulting in the use of Quantum Random Walk (QRW) the best encryption performance. It is proposed

the modifying the Shift row operation introducing random movements using QRW, making it difficult

to predict the correct order during the decryption process. This adds an additional layer of complexity

and makes attempts to decrypt encrypted information difficult without proper knowledge of the correct

sequence.

 (7) proposes that the little investigation about RSA algorithm in Quantum computing is due to actual

limits of Shor’s algorithm. They propose a Quantum ring algorithm: GEECM (Grover plus Elliptic-Curve

factorization Method using Edwards curves), using pre-quantum algorithm to find small primers and

accelerate it with quantum techniques (7).

2.7.2. Quantum Key Distribution (QKD). - This topic is closely related to cryptography, since the security of the

data depends on the transmission of the key previously generated by a cryptographic algorithm.

There are dangers associated with the transmission of the key that were partly solved with the introduction of the

asymmetric key (whose most famous algorithm is RSA).

Quantum computing proposes new solutions in this aspect by being able to transmit the same key to two recipients

(Alice and Bob), with a very high certainty of corroborating that there is no third person obtaining this key. This is

possible due to quantum properties such as entanglement and non-cloning. With entanglement it is possible to know

the state of both photons (i.e. direction of spin) and non-cloning allows to detect if there is any intruder in the system,

since it would yield a common key different from Alice and Bob (Figure II).

Figure II: Quantum Key Distribution (QKD)

Since Alice and Bob have the same common key, Alice can encrypt her message and send it to Bob, who has the key

to decrypt and therefore read the message. We assume noise-free channel for this situation.

Quantum computing can also contribute to the ’one-time-pad’ problem of classical computing by providing random

keys due to the Heisenberg uncertainty principle. Note that in this type of problem security depends to some degree

on the randomness of the key.

2.8. Quantum Machine Learning (QML). - One of the most successful technologies of this century is ML, a subset

of Artificial Intelligence (AI) that focuses on developing algorithms and models that enable computers to learn from

data and make predictions or decisions without being explicitly programmed.

Like other classical theories, ML and learning theory can in fact be embedded into the quantum mechanical formalism.

Formally speaking, this embedding leads to the field known as QML which aims to understand the ultimate limits of

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data analysis allowed by the laws of physics. While there are similarities between classical and QML, there are also

some differences. Because QML employs QCs, noise from these computers can be a major issue.

In ML we have different paradigms that also applies to QML:

 Supervised Learning (Task-based)

 Unsupervised Learning (Data-based)

 Reinforcement Learning (Reward-based) and there is a bunch of

algorithms of QML being researched:

 Quantum Neural Networks (QNNs)

 Quantum Kernels (QKs)

 Variational Quantum Algorithms (VQAs)

2.8.1. Quantum Neural Networks (QNNs). - A QNN is used to describe a parameterized quantum computational

model that is best executed on a QC. This term is often interchangeable with Parameterized Quantum Circuit (PQC).

These involve a sequence of unitary gates acting on the quantum data states |ψji, some of which have free parameters

θ that will be trained to solve the problem.

QNNs are employed in all three QML paradigms mentioned above. For instance, in a supervised classification task,

the goal of the QNN is to map the states in different classes to distinguishable regions of the Hilbert space, in the

unsupervised learning scenario of a clustering task is mapped onto a MAXCUT problem (27) and solved by training

a QNN to maximize distance between classes. Finally, in the reinforced learning task of a QNN can be used as the Q-

function approximator (36), which can be used to determine the best action for a learning agent given its current state.

As in classical neural networks, there are different types of networks such as convolutional networks, recurrent

networks, etc. their quantum variants have been researched, such as quantum convolutional neural networks (13) and

quantum recurrent neural networks (4).

2.8.2. Quantum Kernels (QKs). In ML, a kernel is a function that defines the similarity or distance between pairs of

data points in a high-dimensional feature space. QK methods consider the computation of kernel functions using QCs.

There are many possible implementations. For example, consider a reproducing kernel Hilbert space equal to the

quantum state space, which is finite dimensional. In simpler terms, we can think of the quantum state space as a finite-

dimensional space (34). By using this approach, we can calculate kernel functions within this finite-dimensional space.

Another approach involves studying a reproducing kernel Hilbert space that is infinite-dimensional. In this case, we

are transforming classical vectors (which represent data points) using a QC. The QC helps us map these classical

vectors into infinite-dimensional vectors, an infinite-dimensional space allows for more complex representations and

calculations.

2.8.3. Variational Quantum Algorithms (VQAs). - VQAs are a hybrid quantum-classical optimization algorithm in

which an objective function is evaluated by quantum computation, and the parameters of this function are updated

using classical optimization methods (12).

The variational method in quantum theory is a classical method for finding low energy states of a quantum system.

The idea of this method is that one defines a wave function (called an ansatz) as a function of some parameters, and

then one finds the values of these parameters that minimize the expectation value of the energy.

It has been realized that QCs can mimic the classical technique and that a QC does so with certain advantages (29),

(40), when one applies the classical variational method to a system of n qubits, an exponential number of complex

numbers is necessary to generically represent the wave function of the system. However, with a QC, one can directly

produce this state using a PQC with less than exponential parameters.

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2.8.4. Inductive Bias. - Inductive bias means that any model, can only represent a subset of all possible functions and

is naturally inclined towards certain types of functions. These functions relate the input features to the output

predictions.

Inductive bias encompasses the assumptions and restrictions made in the model design and optimization process,

shaping the search space for potential models. The choice of model parameterization or embedding, as well as

techniques like regularization and learning rate modulation, contribute to the inductive bias.

To achieve quantum advantage with QML, we aim for QML models that have an inductive bias that is difficult to

simulate efficiently using classical models. Recent research has shown that it is possible to construct QKs with this

property, although there are some complexities regarding their trainability.

3. State of the Art. –

3.1. Methodology. - To gather new information on Quantum Cybersecurity it was used the classical search method

with the following set:

 Relevant Topic: Quantum Cybersecurity. Format: Investigation and State of art. Specialized Authors:

Abd El-Latif, Ahmed. Time Frame: 2021-2023.

 Keywords: Quantum, post-quantum, Cybersecurity, encryption, Key-Distribution, Authentication,

Digital signature, IoT. BDB: Web of Science, Springerlink.

Meanwhile, to gather more information on QML, it was used the Snowball methodology, reading the citations of the

most recent papers on QML.

3.2. Related works and Research to QML and Cybersecurity. - The related works on cybersecurity reported in the

literature is summarized below.

In (19), the authors propose a modified AES algorithm and use quantum computing to encrypt/decrypt AES image

files using IBM Qiskit for performance evaluation. They show that AES algorithm can be implemented using quantum

gates and suggest that AES be implemented with random number generation.

AES is combined with the use of random number generation in the process. In the traditional implementation of the

AES algorithm, the Shift Row operation moves the data to align them at certain encryption steps. Since the decryption

process can reverse the order of these steps, it becomes predictable. To address this vulnerability, the author suggests

modifying the performance of the Shift Row operation to introduce random movements using Quantum Random Walk

(QRW), making it difficult to predict the correct order during the decryption process, achieving greater security than

classical approach.

In (37), the authors focus on analyzing characteristics of the quantum cryptography and exploring of the advantages

of it in the future internet. They analyze the QKD protocol in the noise-free channel by making measurements of

different variables. Probability of the eavesdropper being detected v/s number of photons measured in a noise-free

channel and 30% noise. Also analyzes the probabilities of errors in the receiver v/s probability of eavesdropper to

eavesdrop on the channel.

In (39), the authors make a contribution in the state of the art of cybersecurity from wide perspectives. They give an

overview of quantum computing and how it can affect cybersecurity issues. Also demonstrate solutions in quantum

computing to problems in classical computing paradigm related to cybersecurity and relates how quantum computing

could be used in the future to make cybersecurity solutions better.

In (7), the authors make a contribution proposing parameters and changes to RSA to make Key-Generation, encrypt

and decryption, signatures and verification feasible in actual computing and, at the same time, protected against

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quantum computing attacks. They propose a new quantum algorithm to generate factor numbers, GEECM faster than

Shor and algorithms of classic paradigm.

In (36), the authors introduce a new training method for PQCs that can be used to solve Reinforcement Learning (RL)

tasks for discrete and continuous state spaces based on the deep Q-learning algorithm. They adapt the Deep Q-Network

(DQN) algorithm to use a PQC as its Q-function approximator instead of a Neural Network (NN). For this, they use

a hardware-efficient ansatz, a target network, an -greedy policy to determine the agent’s next action and experience

replay to draw samples for training the Q-network PQC. The Q-network then is Uθ(s) parametrized by θ and the target

network PQC is Uˆθδ(s), where θδ is a snapshot of the parameters θ which is taken after fixed intervals of episodes δ

and the circuit is otherwise identical to that Uθ(s).

Depending on the state the authors distinguish between two different types of space states: Discrete and Continuous.

The Q-values of the quantum agent are computed as the expectation values of a PQC that is fed a state s according to

equation 15.

where Oa is an observable and n the number of qubits, and the model outputs a vector including Q-values for each

possible Oa.

In (11), the authors demonstrate the out-of-distribution generalization for the task of learning in QML, where the

training and testing data are drawn from a different distribution. The authors consider the QML task of learning an

unknown n-qubit unitary U ∈ U(C2n). The goal is to use training states to optimize the classical parameters α of V (α),

an n-qubit unitary QNN, such that for the optimized parameters αopt,V (αopt) well predicts the action of U on previously

unseen test states. The prediction performance of the trained QNN V (αopt) can be quantified in terms of the average

distance between the output state predicted by V (αopt) and the true output state determined by U.

They provide numerical evidence to support analytical results showing that out-of-distribution generalization is

possible for the learning of quantum dynamics. They focused on the task of learning the parameters of an unknown

target Hamiltonian by studying the evolution of product states under it. The authors work establishes that for learning

unitaries, QNNs trained on quantum data enjoy out-of-distribution generalization between some physically relevant

distributions if the training data size is roughly the number of trainable gates.

In (2), the authors propose a new strategy for reducing the number of measurements in variational quantum-classical

algorithms (VQCAs) needed for convergence. VQCAs efficiently evaluate a cost function on a QC while optimizing

the cost value using a classical computer. Certain issues arise in VQCAs that are not common in classical algorithms,

implying that standard off-the-shelf classical optimizers may not be best suited to VQCAs. For example, multiple runs

of quantum circuits are required to reduce the effects of shot noise on cost evaluation. An additional complication is

that quantum hardware noise flattens the training landscape.

The authors have recently investigated shot-frugal gradient descent for VQCAs, introduced an optimizer, called

iCANS (individual Coupled Adaptive Number of Shots), which outperformed offthe-shelf classical optimizers such

as Adam for variational quantum compiling and Variational Quantum Eigensolver (VQE) tasks. The key feature of

iCANS is that it maximizes the expected gain per shot by frugally adapting the shot noise for each individual partial

derivative. In VQE and other VQCAs, it is common to express the cost function C = hHi as the expectation value of

a Hamiltonian H that is expanded as a weighted sum of directly measurable operators {hi}i according to equation 16.

then C is computed from estimations of each expectation hhii, which is obtained from many shots. The author’s

proposal is to randomly assign shots to the hi operators according to a weighted probability distribution proportional

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to |ci|, they prove that this leads to an unbiased estimator of the cost C, even when the number of shots is extremely

small like a single shot. This allows one to unlock a level of shot-frugality for unbiased estimation that simply cannot

be accessed without operator sampling. In addition, the randomness associated with operator sampling can provide a

means to escape from local minima of C.

A combination of the new sampling strategy with iCANS leads to the main result, which is an improved optimizer for

VQCAs that they call Rosalin (Random Operator Sampling for Adaptive Learning with Individual Number of shots).

Rosalin retains the crucial feature of maximizing the expected gain per shot. the authors analyze the potential of

Rosalin by applying it to VQE for three molecules, namely H2, LiH, and BeH2, and compare its performance with that

of other optimizers. In cases with more than a few terms in the Hamiltonian, Rosalin outperforms all other optimizer

and sampling strategy combinations considered.

In (20), the authors generalize a quantum natural gradient to consider arbitrary quantum states to significantly

outperform other VQAs. Quantum Fisher information in the context of general VQCs is a measure that quantifies how

much and in what way changing parameters in a quantum circuit affects the underlying quantum state.

The aim of the authors is to minimize the expectation value E(θ) = Tr[ρ(θ)H] of a Hermitian observable H over the

parameters θ using a VQC that depends on these parameters, this circuit produces the quantum states ρ(θ) = Φ(θ)ρ0

via mapping and might involve non-unitary transformations due to experimental imperfections or indeed intentional

non-unitary elements, such as measurements.

The authors propose a natural gradient update rule, where the quantum Fisher information matrix Fq corrects the

gradient vector gk to account for the dependent and non-uniform effect of the parameters on an arbitrary quantum state

ρ(θ) mixed or pure. their method also applies to infinite-dimensional quantum states as continuous-variable systems.

The natural gradient descent proposed by the authors in principle allows for improvements relative to imaginary time

evolution and the pure-state variant of natural gradient. First even when the objective function is generated by an

observable as E(θ) = Tr[ρ(θ)H], their approach allows for general non-unitary elements as Completely Positive Trace-

Preserving (CPTP) maps which in principle enable the manipulation of exponentially more degrees of freedom.

Second the expected value E(θ) := Tr[ρ(θ)H] is a mapping that is linear in quantum state, their results shown that are

well-defined for more general objective functions and its convergence is guaranteed even in the presence of shot noise.

When compared to previous studies, the new approach has the advantage that it explicitly takes into account

imperfections of the VQC.

In (16), the authors provide an accessible introduction to Quantum Embedding Kernels (QEKs) and then analyze the

practical issues arising when realizing them on a noisy near-term QC. QEKs are a subclass of quantum kernel methods

where a PQC is used to embed datapoints into the Hilbert space of quantum states. QEKs have certain appealing

properties that make them attractive for use, like they limited depth does not require long coherence times, another

strong point is that noisy PQCs still lead to well-defined QEKs. The authors propose a series of improvements. First,

to use the kernel-target alignment as a cost function to train parameters of the QEK to increase its performance on

particular datasets. Second, they propose a mitigation strategy tailored for the QEKs that exploits the kernel’s

definition to infer the underlying noise levels. Lastly, they propose a strategy for alleviate the influence of noise on

the kernel matrix based on a semi-definite program.

The QEK is defined as the inner product between quantum states, which is given by the overlap shown in equation

17.

Associated to the quantum feature map |φ(x)i, but we are not able to avoid noise, which means that we cannot assume

that the embedded quantum state is pure, then the quantum embedding is realized by a data-dependent density matrix

ρ(x) which for pure states reduces to ρ(x) = |φ(x)ihφ(x)|, with this modification the inner product is given by equation

18.

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This inner product is also known as the Hilbert-Schmidt inner product for matrices. In summary, any quantum feature

map induces a QEK. We can use this kernel as a subroutine in a classical kernel method, for example the Support

Vector Machine (SVM), which yields a hybrid quantum-classical approach.

To be able to use QEKs in this way, is needed to evaluate the overlap of two quantum states on near-term hardware.

There are a number of advanced algorithms to estimate the overlap of two quantum states. All these algorithms work

for arbitrary states, and so they are agnostic to how the states were obtained by necessity. By exploiting the structure

and specifics of QEKs, though. The authors propose a better way to do this overlap, for unitary quantum embeddings

they construct the adjoint of the data-encoding circuit U†(x). Another approach proposed is the SWAP test, based on

the SWAP trick, a mathematical gimmick that allow to transform the product of the density matrices into a tensor

product (9).

Finally, the authors have performed various numerical experiments that showed improvement in classification

accuracy after training. They have also investigated noise mitigation techniques and proposed device noise mitigation

techniques specific for kernel matrices and combined them with regularization methods. Lastly, they tested a large set

of combinations, both on simulated depolarizing noise as well as on data from a real quantum processing unit.

3.3. Comparative analysis of the latest advances. - Table I shows the advantage (column “Comparative advantage”)

of the contribution made (2nd column) by the reference in column “Ref”.

3.4. Bibliographic Discussion. - From the literature about the problems and the context of development of the search,

it has been possible to delve into the new contributions and their functionalities.

We have noticed that in the QML field, researchers opted for different algorithms competing against each other to see

which one gives the best results, QNNs vs. QVAs vs. QKs, each one with its own pros and cons. It will be necessary

to observe how these algorithms evolve with the passage of time and the advancement of technology.

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Ref

Contribution made

Comparative advantage

(36)

New training method for PQCs that can be used to solve

Reinforcement learning tasks for discrete and continuous

states spaces based on the deep Q-learning algorithm

Training method for discrete and continuous

state spaces for quantum circuits

(11)

Demonstration the Out-of-Distribution generalization for

the task of learning in QML where the training and testing

data are drawn from different distributions

Ability to extrapolate from training data to

unseen data with the potential of QML

methods to outperform classical ML

(2)

New strategy for reducing the number of measurements with

an adaptive optimizer to construct an improved optimizer

called Rosalin that implements stochastic gradient descent

while adapting the shot noise for each partial derivative and

randomly assigning the shots according to a weighted

distribution

Rosalin outperforms other optimizers in the

task to find the ground states of molecules

H2, LiH, and BeH2 without and with

quantum hardware noise

(20)

Generalization of quantum natural gradient to consider

arbitrary quantum states via completely positive maps, thus

the circuits can incorporate both imperfect unitary gates and

fundamentally non-unitary operations such as measurements

Demonstration in numerical simulations of

noisy quantum circuits the practicality of the

new approach and confirm it can

significantly outperform other variational

techniques

(16)

An accessible introduction to quantum embedding kernels,

a analysis of the practical issues arising when realizing them

on a noisy near-term QC, and a strategy to mitigate these

detrimental effects which is tailored to quantum embedding

kernels

Improvement in classification accuracy after

training, noise mitigation techniques and

regularization methods for specific kernel

matrices

(19)

Propose of AES algorithm for Quantum Computing with

improved Security using QRW

Propose of Quantum version of AES

algorithm with improvement against

Quantum attacks

(37)

Explication of QKD and experiments with Quantum Noise

State of art about QKD and experiments with

eavesdropper

(7)

Propose parameters and changes to RSA,on QC, to make

feasible in actual

Proposes a GEECM, faster algorithm than

Shor and experiments with eavesdropper

(39)

Give an overview of QC related to Cybersecurity presenting

several Quantum solutions and show how can be used in

future to make the area better than now

Proposes a state of art of Quantum attacks,

and existing Quantumbased approaches for

Cybersecurity

Table I. Comparative Table

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3.5. State of the Art Timeline. - Table II shows a timeline of quantum computing major advances.

Date

Quantum Computing Major Advances

1970

James Park articulates the no-cloning theorem (28)

1973

Alexander Holevo articulates the Holevo’s Theorem and Charles H. Bennet shows that computation

can be done reversibly (6)

1980

Paul Beinoff describes the 1st quantum mechanical computer model (5), Tomasso Toffoli introduces

the Toffoli Gate (38)

1985

David Deutsch describes the 1st universal QC

1992

David Deutsch and Richard Jozsa propose a computational problem that can be solved efficiently with

the Deutsch–Jozsa algorithm on a QC

1993

Dan Simon invents an oracle problem, for which a QC would be exponentially faster than a

conventional computer

1994

Peter Shor publishes the Shor’s Algorithm

1995

Peter Shor proposes the 1st schemes for quantum error correction (35)

1996

Lov Grover invents the quantum DB search algorithm

2000

Arun K. Pati and Samuel L. Braunstein proved the quantum no-deleting theorem

2001

First execution of Shor’s algorithm

2003

Implementation of the Deutsch–Jozsa algorithm on a QC

2006

First 12 qubit QC benchmarked

2007

D-Wave Systems demonstrates use of a 28-qubit annealing QC

2009

First electronic quantum processor created

2010

Single-electron qubit developed

2014

Scientists transfer data by quantum teleportation over a distance of 3 m with 0% error rate (30)

2017

IBM unveils 17-qubit QC

2018

Google announces the creation of a 72-qubit quantum chip

2019

IBM reveals its biggest QC yet, consisting of 53 qubits

2020

Google engineers report the largest chemical simulation on a QC

2021

IBM claims that it has created a new 127 quantum bit processor

2022

Researchers at Google Quantum AI Team Make Traversable Wormhole with a QC

2023

Researchers of Innsbruck have entangled two ions over a distance of 230 m

Table II. Timeline of Quantum Computing Major Advances

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3.6. Expected or surprising results. - While we look for information for this survey, we read a diversity of articles,

where some of them appears to be science fiction or coming out from a movie, but they are real scientific investigations

like (18) in Nature. This article caused some controversy and what we least want is to get into controversy, but we

must not forget that it is an amazing experiment and discovery.

Another point that caught the attention is the rapid advance of quantum computing. It is a subject that is not heard as

much as AI or neural networks, but it is a field that is advancing by leaps and bounds. So we are happy to contribute

with this survey.

4. Conclusions. - Quantum Computing is still in its early stages, and building a functional and efficient QC with

enough qubits will take years.

QCs have the ability to simulate molecular behavior at a fundamental level, making them valuable for various

industries. Automakers like Volkswagen and Daimler use QCs to analyze and improve the composition of electric

vehicle batteries. Pharmaceutical companies also utilize QCs to study chemicals and explore new possibilities for

medicine development. Quantum computing has the potential to revolutionize society, with its ability to solve

optimization problems quickly by evaluating numerous solutions. Airbus employs QCs to determine fuel-efficient

flight paths, while Volkswagen has developed a tool for optimizing bus and taxi routes to reduce traffic congestion.

Some scientists believe that QCs could accelerate advancements in AI. However, the full extent of quantum

computing’s potential may take many years to realize.

The QML domain should also target designing new quantum learning models that will observe patterns under quantum

mechanics schemes, not classical statistical theory. This will provide an opportunity to explore new model

architectures that might overcome classical machine learning limitations.

The development of post-quantum cryptography is crucial to mitigate the cybersecurity risks posed by quantum

computing. Post-quantum cryptography refers to algorithms that are resistant to attacks from QCs. It not only improves

database search capabilities but also addresses optimization problems in various business domains such as data

analytics, logistics, and medical research.

Discovering better algorithms to work with quantum computing is still an open area of research. Finally, this study

gives an overview of QML, quantum cybersecurity and recent studies in quantum computations with its possible

applications.

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Author contribution:

1. Conception and design of the study

2. Data acquisition

3. Data analysis

4. Discussion of the results

5. Writing of the manuscript

6. Approval of the last version of the manuscript

MS has contributed to: 1, 2, 3, 4, 5 and 6.

FCA has contributed to: 1, 2, 3, 4, 5 and 6.

JPV has contributed to: 1, 2, 3, 4, 5 and 6.

LD has contributed to: 1, 2, 3, 4, 5 and 6.

Acceptance Note: This article was approved by the journal editors Dr. Rafael Sotelo and Mag. Ing. Fernando A.

Hernández Gobertti.