Introduction to Quantum Computing
Abstract
This book provides a self-contained undergraduate course on quantum computing based on classroom-tested lecture notes. It reviews the fundamentals of quantum mechanics from the double-slit experiment to entanglement, before progressing to the basics of qubits, quantum gates, quantum circuits, quantum key distribution, and some of the famous quantum algorithms. As well as covering quantum gates in depth, it also describes promising platforms for their physical implementation, along with error correction, and topological quantum computing. With quantum computing expanding rapidly in the private sector, understanding quantum computing has never been so important for graduates entering the workplace or PhD programs. Assuming minimal background knowledge, this book is highly accessible, with rigorous step-by-step explanations of the principles behind quantum computation, further reading, and end-of-chapter exercises, ensuring that undergraduate students in physics and engineering emerge well prepared for the future.
... Les portes de Pauli, qui tirent leur nom du physicien Wolfgang Pauli, sont des éléments essentiels en informatique quantique. Elles se composent des portes X, Y et Z. Chacune de ces portes de Pauli réalise une rotation de 180 degrés autour de son axe respectif (X, Y ou Z) sur la sphère de Bloch [24]. ...
... -La porte Y Elle échange les coefficients des états |0⟩ et |1⟩ tout en introduisant une phase de i (l'unité imaginaire) pour l'un des états. Cette opération correspond à une combinaison d'une inversion de bits (similaire à une porte X) et d'une déphasage [24]. La représentation matricielle de Y est donnée par : ...
... Ces mouvements sont dépeints dans la figure 1.10 qui illustre comment la porte de Hadamard agit sur un état de superposition spécifique lorsqu'il est représenté sur la sphère de Bloch. [24] : ...
Ce mémoire se concentre sur la simulation de systèmes quantiques à l’aide d’ordinateurs quantiques, avec une attention particulière portée au modèle de Heisenberg,
qui décrit les interactions magnétiques entre spins. L’objectif principal est de démontrer comment l’Hamiltonien du modèle de Heisenberg peut être simulé sur la
plateforme Qiskit, tout en explorant l’influence de la constante de couplage (J) et du
champ magnétique externe (h).
Le premier chapitre établit les fondations du calcul quantique en introduisant les
concepts fondamentaux tels que les qubits, les portes logiques quantiques et les circuits
quantiques. Il souligne également les avantages de cette approche pour résoudre des
problèmes physiques complexes, souvent inaccessibles aux méthodes classiques.
Le deuxième chapitre approfondit la notion d’Hamiltoniens, qui déterminent l’énergie des systèmes quantiques, ainsi que les algorithmes emblématiques du calcul quantique, tels que ceux de Shor et de Grover. Ce chapitre présente également quelques
applications concrètes en calcul quantique, illustrant la supériorité de ces méthodes
face aux approches traditionnelles.
Le troisième chapitre se concentre sur l’application de la simulation de l’Hamiltonien du modèle de Heisenberg. Il décrit en détail les techniques de décomposition des
Hamiltoniens et leur traduction en circuits quantiques, représentant les interactions
entre qubits. Grâce à Qiskit, divers scénarios sont analysés en faisant varier les paramètres J (constante de couplage) et h (champ magnétique externe). Ces variations
sont modélisées sous forme d’angles, θ = 2∆th/ℏ et ϕ = 2∆tJ/ℏ, utilisés pour configurer les portes de rotation et les portes CNOT dans les circuits quantiques. Cette
méthodologie permet de concevoir des circuits adaptés à l’étude des propriétés du
système dans différents régimes physiques.
ii
En conclusion, ce mémoire illustre le potentiel des ordinateurs quantiques pour
modéliser des systèmes physiques complexes, en particulier les interactions magnétiques du modèle de Heisenberg. Les simulations effectuées avec Qiskit mettent en
évidence l’impact des variations des paramètres physiques et ouvrent de nouvelles
perspectives pour la recherche en calcul quantique et en physique des systèmes magnétiques.
... Quantum computers can achieve algorithmically superior scaling for certain problems, particularly complex linear algebra and matrix exponential [9]. An application with proven quantum advantage is solving linear equations [10][11][12][13]. ...
... 3.1 Overview of quantum computing 3.1.1 Qubits and vector representation: Quantum computers represent information (digital bits, 0 and 1) using two different states of a quantum system [9], e.g., two different quantized energy levels (0 as the ground state and 1 as an excited state of an electron) or two spin states of the particle (0 as electron spins "down" and 1 as electron spins "up"). Thus, a qubit depicted in Figure 7 can be considered as the supposition of two quantum states. ...
... 3.1.2 Time evolution of a quantum system: the time evolution of quantum state |ψ⟩ at time t under the following time-independent Schrödinger equation i ∂ ∂t |ψ(t)⟩ = H |ψ(t)⟩ is |ψ(t)⟩ = e −iHt |ψ(0)⟩ where H is the Hamiltonian operator (a Hermitian matrix acting on n qubits) [9]. The exponential component e −iHt can be expressed using a Taylor expansion as follows: ...
Power system dynamics are generally modeled by high dimensional nonlinear differential-algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system nonlinearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, i.e. Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.
... QCs power arises from the exponential growth in the size of the quantum state with increasing numbers of qubits combined with the ability to apply an operation simultaneously to all quantum states leading to an exponential speedup for certain types of computations. Many quantum programs begin by placing individual qubits into states of superposition with operations like the Hadamard gate [13] as shown in Fig. 5a. Next, the quantum program operates in parallel on all quantum basis states using a function U f that is a unitary implementation of some classical function f . ...
... The impact of this type of computational model is profound and captures the essence of the power of QC: we can evaluate some f (x) over all 2 n quantum states of n entangled qubits in a single operation. Not shown in Fig. 6a is how the quantum program will create a state of entanglement between one or more qubits through the use of unitary operations like the Controlled NOT gate (CNOT) [13] which conditionally inverts the state of one qubit based upon the state of another. ...
... One of the foundational quantum algorithms is Grover's Search (c.f. [13]) for finding a item or items in an unordered list of length N. A classical implementation of an unstructured search requires on average O(N/2) steps since on average about half of the items in the set would need to be examined. The circuit for Grover's Search is shown in Fig. 6a for a 3-qubit system. ...
The focus of this Visualization Viewpoints article is to provide some background on Quantum Computing (QC), to explore ideas related to how visualization helps in understanding QC, and examine how QC might be useful for visualization with the growth and maturation of both technologies in the future. In a quickly evolving technology landscape, QC is emerging as a promising pathway to overcome the growth limits in classical computing. In some cases, QC platforms offer the potential to vastly outperform the familiar classical computer by solving problems more quickly or that may be intractable on any known classical platform. As further performance gains for classical computing platforms are limited by diminishing Moore's Law scaling, QC platforms might be viewed as a potential successor to the current field of exascale-class platforms. While present-day QC hardware platforms are still limited in scale, the field of quantum computing is robust and rapidly advancing in terms of hardware capabilities, software environments for developing quantum algorithms, and educational programs for training the next generation of scientists and engineers. After a brief introduction to QC concepts, the focus of this article is to explore the interplay between the fields of visualization and QC. First, visualization has played a role in QC by providing the means to show representations of the quantum state of single-qubits in superposition states and multiple-qubits in entangled states. Second, there are a number of ways in which the field of visual data exploration and analysis may potentially benefit from this disruptive new technology though there are challenges going forward.
... Grover's algorithm [2,10,11] consists of three components, termed as blocks, which are: (i) Block1 initializes n input qubits to a uniform distribution using Hadamard (H) gates; note that the H gates impose all the input qubits into uniform superposition states to generate a complete quantum search space of {|0⟩,|1⟩} ⊗n for Grover's algorithm to solve and search for solutions, i.e., the H gates are considered as an implicit generator, (ii) Block2 consists of an oracle that adds negative phases to solutions, i.e., inverts the phases of input qubits as the first rotation of solutions over the complete quantum search space; note that such phases inversion occurs due to the phase kickback for a quantum Boolean-based gate from the functional qubit (fqubit) of |-⟩ state to the input qubits for a Boolean oracle, or due to the effect of quantum phase-based gates on the input qubits for a Phase oracle, and (iii) Block3 consists of Grover diffusion operator that performs the second rotation (H gates), the conditional phase shift (X gates), and the phase inversion (Z0 gate) of solutions, i.e., amplifying the amplitudes of the inverted phases of input qubits (solutions); for this reason, Block3 is called the amplitude amplification. In addition, Grover diffusion operator has also termed the diffuser or conventional diffuser. ...
... The total number of utilized qubits decreased for Phase oracles and increased for Boolean oracles. Depending on the design implementation, Phase oracles are mainly utilizing n input qubits with almost no ancillae, while Boolean oracles are utilizing n input qubits, m ancillae, and one fqubit, where n ≥ 1 and m ≥ 0. The CUs operator follows the same structural design of Grover's algorithm [2,10,11] of the aforementioned three blocks, except for the fqubit is initially set to the |0⟩ state since the phase kickback is not utilized, and Fig. 1(d) illustrates the workflow of Grover's algorithm of the CUs operator. Moreover, the components and functionalities of the CUs operator are demonstrated in Fig. 1(e). ...
... The CUs operator successfully finds all true solutions for the Boolean oracle, as illustrated in Fig. 6(f), while the Us operator fails and outputs false solutions, i.e., the solutions for this application when its Boolean oracle is inverted, for the Boolean oracle, as shown in Fig. 6(e), and partial true solutions for Phase oracle, as illustrated in Fig. 6(d). Note that the input qubits are in Dirac notation [2,11] of |MSQ LSQ⟩; such that, number0 = |a1 a0⟩ and number1 = |b1 b0⟩. For this first application, Boolean and Phase oracles are generated using the Boolean structure in (4). ...
A controlled diffuser is designed as a new approach for Grover’s algorithms to search for solutions for arbitrary Boolean oracles, since the conventional diffuser is not capable of searching for solutions for arbitrary Boolean and Phase oracles. This controlled diffuser relies on the states of functional (output) qubit as the reflection of Boolean decisions from a Boolean oracle, without relying on the phase kickback. This article discusses the problems that are designed as Boolean and Phase oracles using the structures of POS, SOP, ESOP, digital logic circuits, and CSP-SAT. Our work concludes that the conventional diffuser only finds the solutions for these problems when their collector gates are in the form of a Boolean AND gate (Toffoli) for Boolean oracles or a multiple-controlled Z gate for Phase oracles, while the controlled diffuser successfully finds the solutions for all Boolean oracles regardless of different Boolean gates of their collector gates, in Grover iterations of O times. In this article, we proposed new terms of Collector Gate, Calibration Test, and calculated Quantum Cost for this controlled diffuser.
... Quantum computers provide a complementary computing platform for such challenging problems. Quantum computers perform computation differently to achieve algorithmically superior scaling for certain problems, particularly complex linear algebra and matrix exponential [8]. An application with proven quantum advantage is solving linear equations with a quantum solver. ...
... A. Overview of Quantum Computing 1) Qubits and vector representation: Quantum computers represent information (digital bits, 0 and 1) using two different states of a quantum system [8]. Specifically, it represents 0 and 1 using two different quantized energy levels (e.g., 0 as the ground state and 1 as an excited state of an electron) or using the spin of the particle (e.g., 0 as electron spin "down" and 1 as electron spins "up"). ...
Power system dynamics are generally modeled by high dimensional nonlinear differential-algebraic equations due to a large number of generators, loads, and transmission lines. Thus, its computational complexity grows exponentially with the system size. In this paper, we aim to evaluate the alternative computing approach, particularly the use of quantum computing algorithms to solve the power system dynamics. Leveraging a symbolic programming framework, we convert the power system dynamics' DAEs into an equivalent set of ordinary differential equations (ODEs). Their data can be encoded into quantum computers via amplitude encoding. The system's nonlinearity is captured by Taylor polynomial expansion and the quantum state tensor whereas state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the dynamics of the power system with high accuracy whereas its complexity is polynomial in the logarithm of the system dimension.
... Quantum computing involves parallel computing in which the values of the required function could be calculated at all points of its argument before the superposition is transferred to a fixed base state by measurement [2] with assigning a measured value to a physical quantum computing, thus, giving its estimation, and requires significant repetition of the calculations to obtain estimates of the event probabilities. Quantum computing can be implemented in the gated interpretation [3,4], by creating a cluster state with subsequent measurement in an adjustable basis [5,6], as well as nonuniversal calculations based on sampling model [7,8] when the probability to measure a specific photon pattern on output of 2 −interferometer with input photons depends upon the permanent of some submatrix. The mobility and bosonic nature of photons can be extremely resource-efficient to solve sampling problem that is considered to be classically hard [9]. ...
... (2) ( 1 , ) = (2 −1) ( 1 ) − 2 (1 + 2(2 +1) ) ( 1 (2 ) ( 1 )) + (1 + 2(2 +1) ) ( 1 1 ) 2 ( 1 (2 ) ( 1 )) + 4 (2 ) 2 (2 +1) 2 ( 1 1 ) 3 ( 1 (2 ) ( 1 )) = (2 −1) ( 1 ) + 2 +1 ...
We propose a model for computing of a certain set of analytical functions based on estimating the output distribution of multiphoton outcomes in an optical scheme with an initial single-mode squeezed vacuum (SMSV) state and photonic states measuring the number of photons in one of the output modes of the beam splitter (BS) by photon number resolving (PNR) detector. The set of considered analytical functions is polynomial expressions including arbitrary derivatives of certain functions which can take on very large values even on small interval in their argument and small values of the parameter indicating the number of the subtracted photons. The large values that the analytic functions can take are offset by a very small term including the factorial of the number of subtracted photons, which guarantees an output normalized distribution of multiphoton measurement outcomes. The quantum computing algorithm makes it possible to find the values of the analytical functions for each number of extracted photons after a sufficiently large number of trials that would allow replacing the measurement repetition rate of multiphoton events by their probabilities. Changing the initial parameters (squeezing amplitude of the SMSV state and BS parameter) makes it possible to implement calculations of the functions over the entire (or, at least, significant) continuous interval of alteration in their argument. The potential of optical quantum computing based on nonclassical states of a certain parity can be expanded both by adding new optical elements such as BSs, and by using other continuous variable (CV) states of definite parity.
... The Paul trap apparatus [in case of both 2D linear ion traps (LIT) and 3D versions] has been developed and refined for high finesse quantum engineering experiments, high precision spectroscopy [40,41], along with classical mass spectrometry (MS) [42][43][44][45][46][47] or chemical analysis [48], including the detection of aerosols and chemical warfare [49][50][51][52][53][54][55][56][57]. Besides, ion traps also enable exceptional control in preparing and manipulating atomic quantum states [58][59][60][61][62][63], which is why their wide area of applications also includes quantum logic [64][65][66][67][68], quantum sensing [69][70][71][72], quantum metrology [73,74] and even time fractals [75] or time crystals [76]. To these one adds high accuracy optical frequency standards [77][78][79][80], which are amongst the most sensitive quantum sensors [81,82] used to perform searches for physics beyond the Standard Model (BSM) [83][84][85] or to disseminate atomic time scales and redefine the SI unit of time, the second [86][87][88][89][90]. ...
The stability properties of the Hill equation are discussed, and especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu equation for a trapped ion are characterized by using the Floquet theory and Hill's Method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters a and q that are real. Characteristic curves are introduced naturally by the Sturm-Liouville problem for the well known even and odd Mathieu equations and . We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for axial and radial motion. In case of a Paul trap the stable solution corresponds to a superposition of harmonic motions. The problem of evaluating the maximum amplitudes of stable oscillations for the ideal conditions (taken into consideration) is also approached. Anharmonic corrections are discussed within the frame of the perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter. The results apply to 2D and 3D ion traps used for different applications in quantum engineering, among which optical clocks, quantum logic and quantum metrology, but not restricted only to these.
... Apart from that, ITs also enable exceptional control in preparing and manipulating atomic quantum states [53][54][55][56][57][58] because they make possible applications such as quantum logic [59][60][61][62][63], quantum sensing [64][65][66][67], quantum metrology [68,69] and even time fractals [70] or time crystals [71]. To these, one adds high-accuracy optical frequency standards [72][73][74][75], which are amongst the most sensitive quantum sensors [76,77] used to perform searches for physics beyond the Standard Model (BSM) [78][79][80] or to disseminate atomic time scales and redefine the SI unit of time, the second [81][82][83][84][85]. ...
The stability properties of the Hill equation are discussed, especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu-Hill equation for a trapped ion are characterized by employing the Floquet theory and Hill’s method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters a and q that are real. Characteristic curves are introduced naturally by the Sturm–Liouville problem for the well-known even and odd Mathieu equations cem(z,q) and sem(z,q). In the case of a Paul trap, the stable solution corresponds to a superposition of harmonic motions. The maximum amplitude of stable oscillations for ideal conditions (taken into consideration) is derived. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for both axial and radial motion, where the former is described by the canonical Mathieu equation. Anharmonic corrections for nonlinear Paul traps are discussed within the frame of perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter and we demonstrate they are shifted towards negative values of the a parameter. The applications of the results include but are not restricted to 2D and 3D ion traps used for different applications such as mass spectrometry (including nanoparticles), high resolution atomic spectroscopy and quantum engineering applications, among which we mention optical atomic clocks and quantum frequency metrology.
... Quantum computing as a novel computation method enables superior scaling with certain complex problems, such as linear algebra and matrix exponential calculations [3]. One application with proven quantum advances is solving linear equations. ...
This paper presents a quantum computing framework to solve a system of nonlinear ordinary differential equations (ODEs) used in the power system dynamic analysis. The framework exploits the linearization of power system dynamics' nonlinear ODEs at particular points of state variable vector to construct a system of linear ODEs (LDE) modeling the system dynamics around considered points within a small time interval. The analytical solution of this LDE system in the matrix exponential form can be simulated and transformed into quantum states using the popular design of a Variational Quantum Circuit (VQC) acting as a quantum LDE solver oracle. The oracle can be used repeatedly to construct the trajectory of the original nonlinear power system dynamics along the time evolution. Obtained numerical results using Julia-based simulatable quantum circuits demonstrate that we can tailor and leverage recent advances in quantum computing algorithms, originally designed for linear systems, to model nonlinear power system dynamics with high accuracy.
... Nuclear magnetic resonance (NMR) is a technique used to visualize nuclear spin states of a given sample. When an external magentic field is applied to a given sample, nuclear spin degeneracy is broken, and the splitting of nuclear spin states occurs, known as nuclear Zeeman splitting [10,11]. Spectrum of nuclear Zeeman sublevels is called NMR spectrum. ...
Quantum computers are information processing devices which rely on quantum parallelism. Various physical systems such as NMR and ion traps are employed for realizing this parallelism. Specific algorithms which utilize this parallelism are in place. These algorithms make quantum computers outperform their classical counterparts in computational performance for certain class of problems. As and when efficient quantum algorithms are developed, and with a reliable physical system in place, quantum computer are destined to become universal computing platforms in decades to come. This review sheds light on some of the fundamental aspects of quantum computing along with the physical systems which implement them in an unambiguous way.
... The secret can be recovered by the participants using the secret reconstruction strategy only if they cooperate. However, due to the fundamental limitations imposed by the No Cloning Theorem [4], only one participant can reconstruct the secret. Hence, other participants must assist by providing their shares to the unanimously chosen participant. ...
Proposed by Hillery et al., Quantum Secret Sharing (QSS) is a technique used to break a quantum secret into multiple pieces (called Shares), such that any proper subset of the pieces does not reveal any information about the original secret. The secret can be reconstructed only when all of the pieces are combined together. However, most of the QSS protocols assume that the shares are untampered, hence fail to regenerate the original secret if some of the shares are damaged due to several reasons like cheating participants, eavesdropping, etc. In order to tackle this practical challenge, Resilient Quantum Secret Sharing (RQSS) protocols are required. In this paper, we propose an RQSS protocol that uses Quantum Error Correcting Codes (QECC) for share generation and secret reconstruction. Our protocol generates n shares of a m-qubit quantum secret, owned by the dealer, and distributes it among nparticipants; moreover, it can regenerate the original secret even if k<n shares are damaged. To the best of our knowledge, no such generalized protocol exists in the available literature.
... Quantum correlations, inherent relationships between subsystems, quantify both local and non-local information encoded in quantum states [23,24]. Entanglement is a primary notion in quantum communication theory, such as the quantum coding, cryptography, and teleportation [25]. ...
Many quantum algorithms operate on classical data, by first encoding classical data into the quantum domain using quantum data encoding circuits. To be effective for large data sets, encoding circuits that operate on large data sets are required. However, as the size of the data sets increases, the encoding circuits quickly become large, complex and error prone. Errors in the encoding circuit will provide incorrect inputs to quantum algorithms, making them ineffective. To address this problem, a formal method is proposed for verification of encoding circuits. The key idea to address scalability is the use of abstractions that reduce the verification problem to bit‐vector space. The major outcome of this work is that using this approach, the authors have been able to verify encoding circuits with up to 8191 qubits with very low memory (85 MB) and time (0.29s), demonstrating that the proposed approach can easily be employed to verify even much larger encoding circuits. The results are very significant because, traditional verification approaches that rely on modelling quantum circuits in Hilbert space have only demonstrated verification scalability up to 250 qubits. Also, this is the first approach to tackle the verification of quantum encoding circuits.
This paper presents a method for teaching quantum computing for engineering students of the Master's degree in Quantum Computing at the International University of La Rioja. The approach is designed so that students can acquire the basic concepts of quantum computing in a context in which they have no knowledge of quantum mechanics. The focus is therefore on the mathematical, physical and, above all, computational knowledge acquired by the student in engineering. Theoretical and practical concepts of quantum computation are introduced, as well as the most important quantum algorithms. The student is also taught to work with a real quantum device. The method was successfully tested using the IBM Quantum platform, which consists of real quantum devices and simulators where students were able to test the acquired knowledge and implement the learned algorithms. In a context where quantum computing‐related degrees and subjects are starting to emerge, the aim of this work is to help not only future generations of engineers to study quantum computing, but also those responsible people for implementing such studies.
Quantum computing is a rapidly emerging and promising field with the potential to transform various research domains including drug design, network technologies, and sustainable energy solutions. Due to the inherent complexity and divergence from classical computing, several major quantum computing libraries have been developed to implement quantum algorithms, namely IBM Qiskit, Amazon Braket, Cirq, PyQuil, and PennyLane. These libraries enable quantum simulations on classical computers and execution on corresponding quantum hardware, such as Qiskit programs on IBM quantum computers. Despite the variations among these platforms, the core concepts remain the same. One notable challenge is the absence of a Python-based quantum interpreter to connect these five frameworks, a gap that remains to be fully addressed. In response, our work introduces a tool called Qinterpreter, accessible through a user-friendly web interface, the Quantum Science Gateway QubitHub, which operates alongside Jupyter Notebooks. Built using the Python Object-Oriented Programming System, Qinterpreter unifies the five well-known quantum libraries into a single framework. Designed as an educational tool for students and researchers entering the quantum domain, Qinterpreter enables the straightforward development and execution of quantum circuits across such platforms. This work highlights the quantum programming versatility and accessibility of Qinterpreter and underscores our ultimate goal of pervading Quantum Computing through younger, less specialized, and diverse cultural and national communities.
A controlled-diffusion operator for Boolean oracles is designed as a new approach for Grover’s algorithm to search for solutions for arbitrary logical structures of such oracles, since the Grover diffusion operator is not able to find correct solutions for some logical structures of Boolean oracles. We also show that the Phase oracles do not work sometimes correctly using the Grover diffusion operator. Our proposed controlled-diffusion operator relies on the states of output qubit, as the reflection of Boolean decisions from a Boolean oracle without relying on the phase kickback. We prove that on many examples of Boolean and Phase oracles the Grover diffusion operator is not working correctly. The oracles in these examples are constructed using different structures of POS, SOP, ESOP, CSP-SAT, and XOR-SAT. Our mathematical models and experiments prove that the proposed controlled-diffusion operator successfully searches for all solutions for all Boolean oracles regardless of their different logical structures.
The Fourth Industrial Revolution (4IR) aims at improving the manufacturing cost and quality by inter-connection of Industrial Internet of Things (IIoT) devices. In the 4IR, IIoT-based machines need to collaborate securely and autonomously through the insecure Internet channel. Although various blockchain-based protocols have been introduced for securing cross-industry Machine-to-Machine (M2M) communications, they, are insecure to some cyberattacks, rely on some intermediary servers for successful M2M authentication, are not efficient enough to be employed in the context of IIoT environments, and are not resilient to quantum computers. To address these weaknesses, this paper suggests a novel quantum-resilient self-certified authentication protocol for cross-industry communications using the cutting-edge blockchain technology. Thanks to the blockchain ledger, the cross-industry connections have been facilitated and thanks to the self-certified approach, the blockchain referrals have been reduced to only one ledger reading. In our protocol, no intermediary server needs to involve and IIoT machines can directly authenticate each other. The advantages and efficiency enhancements of our protocol, compared to top-related ones, are shown, respectively, by comparative property and performance analyses. Specifically, our protocol indicates 13% and 91% improvements in communication and computation overheads over the most efficient protocol.
The internet of things (IoT) has been used in a wide range of applications since its emergence, including smart cities, intelligent systems, smart homes, smart agriculture, and healthcare. IoT systems rely on information processing and sharing, where data leakages may jeopardize their security and privacy. On the other hand, quantum computers are poised to solve complex problems that traditional computers cannot. However, due to the fact that the majority of cyber algorithms are based on significant computational complexity, quantum computing poses a substantial threat to the cyber security of global digital infrastructure, including IoT networks, smart cities, banking, and intelligent infrastructure. This chapter discusses potential security and privacy measures for a post-quantum world against threats posed by quantum computing, including post-quantum cryptography, quantum software testing, post-quantum blockchain technology, and architectural considerations for creating post-quantum secure IoT systems.
This study introduces a novel Quantum Wasserstein Generative Adversarial Network approach with a Gradient Penalty (QWGAN-GP) model that leverages a quantum generator alongside a classical discriminator to synthetically generate time series data. This approach aims to accurately replicate the statistical properties of the S&P 500 index. The synthetic data generated by this model were compared to the original series using various metrics, including Wasserstein distance, Dynamic Time Warping (DTW) distance, and entropy measures, among others. The outcomes demonstrate the model’s robustness, with the generated data exhibiting a high degree of fidelity to the statistical characteristics of the original data. Additionally, this study explores the applicability of the synthetic time series in enhancing prediction models. An LSTM (Long-Short Term Memory)-based model was developed to evaluate the impact of incorporating synthetic data on forecasting accuracy, particularly focusing on general trends and extreme market events. The findings reveal that models trained on a mix of synthetic and real data significantly outperform those trained solely on historical data, improving predictive performance.
We study a hybrid structure of a ferromagnetic insulator and a superconductor connected by a weak link, which accommodates Andreev bound states whose spin degeneracy is lifted due to the exchange interaction with the ferromagnet. The resultant spin-resolved energy levels realize a two-state quantum system, provided that a single electron is trapped in the bound state, i.e., an Andreev spin qubit. The qubit state can be manipulated by controlling the magnetization dynamics of the ferromagnet, which mediates the coupling between external fields and the qubit. In particular, our hybrid structure provides a simple platform to manipulate and control the spin qubit using spintronic techniques. By employing a modified Hahn spin echo protocol for the magnetization dynamics, we show that our Andreev spin qubit can realize a nonadiabatic geometric gate.
The focus of this Visualization Viewpoints article is to provide some background on quantum computing (QC), to explore ideas related to how visualization helps in understanding QC, and examine how QC might be useful for visualization with the growth and maturation of both technologies in the future. In a quickly evolving technology landscape, QC is emerging as a promising pathway to overcome the growth limits in classical computing. In some cases, QC platforms offer the potential to vastly outperform the familiar classical computer by solving problems more quickly or that may be intractable on any known classical platform. As further performance gains for classical computing platforms are limited by diminishing Moore’s Law scaling, QC platforms might be viewed as a potential successor to the current field of exascale-class platforms. While present-day QC hardware platforms are still limited in scale, the field of quantum computing is robust and rapidly advancing in terms of hardware capabilities, software environments for developing quantum algorithms, and educational programs for training the next generation of scientists and engineers. After a brief introduction to QC concepts, the focus of this article is to explore the interplay between the fields of visualization and QC. First, visualization has played a role in QC by providing the means to show representations of the quantum state of single-qubits in superposition states and multiple-qubits in entangled states. Second, there are a number of ways in which the field of visual data exploration and analysis may potentially benefit from this disruptive new technology though there are challenges going forward.
This chapter presents the beginnings of homomorphic encryption, definitions, and types. Also, an entire section is focused on fully homomorphic encryption, the most interesting and promising type of homomorphic encryption. Further, the chapter presents recent advancements in homomorphic encryption discussing the advantages, disadvantages, performance, security requirements and presents case studies and practical applications in different domains (for example, Internet-of-Things, Machine Learning, Artificial Intelligence, etc.) Lastly, the chapter presents the challenges and research directions for homomorphic encryption.
¿Cuáles son los principios subyacentes a toda herramienta en programación? Si quiere conocer los ocho principios, técnicos y conductuales, que dan respuesta a esta pregunta, ha llegado al libro indicado.
En una época donde cada día surgen nuevas tecnologías, el beneficio de conocer conceptos transversales a todas ellas no solo es imprescindible, sino también necesario. Además, con la llegada de sofisticadas aplicaciones de inteligencia artificial, la pregunta ya no reside en qué herramienta aprender, sino en qué tienen en común para poder integrarlas.
Gracias a la lectura de este libro, descubrirá los cinco tomos que lo componen y que dan soporte a la nueva forma de entender la programación.
In the field of cryptography, many algorithms rely on the computation of modular multi-plicative inverses to ensure the security of their systems. In this study, we build upon our previous research by introducing a novel sequence, (z j) j≥0 , that can calculate the modular inverse of a given pair of integers (a, n), i.e., a −1 ; mod, n. The computational complexity of this approach is O(a), which is more efficient than the traditional Euler's phi function method, O(n, ln, n). Furthermore, we investigate the properties of the sequence (z j) j≥0 and demonstrate that all solutions of the problem belong to a specific set, I, that only contains the minimum values of (z j) j≥0. This results in a reduction of the computational complexity of our method, especially when a ∼ n and it also opens new opportunities for discovering closed-form solutions for the modular inverse.
There exist multiphoton states that cannot be expressed as a product of individual photon states. These states are called entangled states. Einstein used hidden variables in an attempt to explain the probabilities of quantum mechanics. John Bell proposed a test, using entangled states, showing that quantum mechanics cannot be explained by hidden variables.
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