Quantum Computing: An Applied Approach
Abstract
This book integrates the foundations of quantum computing with a hands-on coding approach to this emerging field; it is the first to bring these elements together in an updated manner. This work is suitable for both academic coursework and corporate technical training.
The second edition includes extensive updates and revisions, both to textual content and to the code. Sections have been added on quantum machine learning, quantum error correction, Dirac notation and more. This new edition benefits from the input of the many faculty, students, corporate engineering teams, and independent readers who have used the first edition.
This volume comprises three books under one cover: Part I outlines the necessary foundations of quantum computing and quantum circuits. Part II walks through the canon of quantum computing algorithms and provides code on a range of quantum computing methods in current use. Part III covers the mathematical toolkit required to master quantum computing. Additional resources include a table of operators and circuit elements and a companion GitHub site providing code and updates.
Jack D. Hidary is a research scientist in quantum computing and in AI at Alphabet X, formerly Google X.
... Quantum computing has attracted the attention of both industry and academia in recent years, especially due to its capabilities to solve problems previously considered unmanageable for classical computers in areas like molecular simulations, cybersecurity, finance, and logistics. The main concept behind a quantum computer is to leverage the specific properties described by quantum mechanics to perform computation [1]. ...
... These patterns can be illustrated by a few examples taken from the tests created for the Variational Quantum Eigensolver (VQE) algorithm and detailed in the next subsections. The VQE is a hybrid quantum-classical algorithm that uses a variational technique to find the minimum eigenvalue of a given Hamiltonian operator H [1]. ...
... The test then checks the correctness of this intermediate result, which is needed for further iterations of the algorithm. 1 def test_gradient_calculation(self): Another example of gray-box testing in Qiskit Algorithms is the test of the callback functions of the iterative, hybrid algorithms. For instance, the method test_callback in the TestVQD class (file test/eigensolvers/test_vqd.py) ...
Although classical computing has excelled in a wide range of applications, there remain problems that push the limits of its capabilities, especially in fields like cryptography, optimization, and materials science. Quantum computing introduces a new computational paradigm, based on principles of superposition and entanglement to explore solutions beyond the capabilities of classical computation. With the increasing interest in the field, there are challenges and opportunities for academics and practitioners in terms of software engineering practices, particularly in testing quantum programs. This paper presents an empirical study of testing patterns in quantum algorithms. We analyzed all the tests handling quantum aspects of the implementations in the Qiskit Algorithms library and identified seven distinct patterns that make use of (1) fixed seeds for algorithms based on random elements; (2) deterministic oracles; (3) precise and approximate assertions; (4) Data-Driven Testing (DDT); (5) functional testing; (6) testing for intermediate parts of the algorithms being tested; and (7) equivalence checking for quantum circuits. Our results show a prevalence of classical testing techniques to test the quantum-related elements of the library, while recent advances from the research community have yet to achieve wide adoption among practitioners.
... Charge qubits, spin qubits, light qubits, and quantum dots are among the most commonly explored structures. Charge qubits typically use superconducting materials to manipulate charge states, whereas spin qubits rely on the manipulation of the spin of a single electron, often in a semiconductor setting [40]. Light qubits, or photonic qubits, utilize photon polarization, where quantum information can be encoded in multiple ways, including the polarization of photons (horizontal or vertical) or their paths of travel (path encoding) [1,40,41]. ...
... Charge qubits typically use superconducting materials to manipulate charge states, whereas spin qubits rely on the manipulation of the spin of a single electron, often in a semiconductor setting [40]. Light qubits, or photonic qubits, utilize photon polarization, where quantum information can be encoded in multiple ways, including the polarization of photons (horizontal or vertical) or their paths of travel (path encoding) [1,40,41]. Quantum dots, on the other hand, can trap single electrons and are used as a platform for both charge and spin qubits [41][42][43]. ...
... For further details on the different physical structures used to construct qubits, we refer the reader to Refs. [1,40,41]. ...
Grover’s search algorithm (GSA) offers quadratic speedup in searching unstructured databases but suffers from exponential circuit depth complexity. Here, we present two quantum circuits called HX and Ry layers for the searching problem. Remarkably, both circuits maintain a fixed circuit depth of two and one, respectively, irrespective of the number of qubits used. When the target element’s position index is known, we prove that either circuit, combined with a single multi-controlled X gate, effectively amplifies the target element’s probability to over 0.99 for any qubit number greater than seven. To search unknown databases, we use the depth-1 Ry layer as the ansatz in the Variational Quantum Search (VQS), whose efficacy is validated through numerical experiments on databases with up to 26 qubits. The VQS with the Ry layer exhibits an exponential advantage, in circuit depth, over the GSA for databases of up to 26 qubits.
... The superposition principle states that the linear combination of two or more state vectors is another state vector in the same Hilbert space and describes another system state [16]. This is why every quantum circuit corresponds to a particular unitary operator U in the Hilbert space that transforms an initial quantum state into another quantum state, meeting the criteria shown in Equation (3) for maintaining the superposition principle [15]. ...
... In addition to the superposition principle, the Born rule states that the modulus square of the amplitude of a state is the probability of that state's resulting after measurement [16]. This implies that the coefficient present in Equation (2) can be operated to obtain the likelihood of the states | 0⟩ and | 1⟩, and because all the possible states are present in Equation (2), they must meet a normality condition where the sum of the probabilities of the states is equal to 1, as shown in Equation (4). ...
... After having a superposition step, the oracle marks the desired state of the search problem, which the diffusion operator later amplifies. A measurement procedure is conducted after a series of iterations, making the probability of collapsing the superposition to our desired result higher [16]. In this case, at most, we must iterate π 4 √ N times [18], where N is the number of search terms available, to obtain a better approximation of the desired search state. ...
The search for information in a system has been a continuous problem for a computer. This has resulted in the construction of a set of classical algorithms that can search for a set of data. This is why search systems can be divided into the type of information being searched, the number of solutions to find, and even the terms used for searching. With the emergence of quantum computing, new algorithms have been generated for this type of process. An example is the Grover algorithm, which performs theoretically better than traditional algorithms. This is why there has been research on optimizing it, applying it to new fields, and making it more accessible to industry users. Even if the algorithm is a promising alternative, one of the disadvantages of Grover’s algorithm is the use of an oracle function that must be generated for every set of search data. This review describes three sets of methodologies for generating quantum circuits that can be applied to constructing this oracle quantum circuit.
... B EING one of the most thriving topics for solving large-scale problems, quantum computing (or quantum computation) has attracted increasing attention from the computational community in recent years [1], [2], [3]. In certain cases, quantum algorithms provide exponential speedups over their classical counterparts running on classical computers. ...
... respectively, where notation "|⟩" is the standard notation (called Dirac notation) to describe the state of qubits in quantum mechanics [2]. For example, a column vector and a row vector are written in Dirac notation as |u⟩ and ⟨u|, respectively. ...
... A global phase change has no impact on quantum measurements, which means that the measurement statistics obtained by measuring two states with a global phase difference are exactly identical. In such cases, we say that the two states are equal, up to global phase [2]. ...
Applying quantum computation to solve electromagnetic (EM) problems is still at an early age. Recently, an initial study on applying quantum computation to solve the finite element method (FEM) equations in the EM domain has been made. This article makes a further development beyond the initial study. Specifically, we first develop an approach to systematically prepare the quantum state to represent the right-hand side (RHS) vector of the finite element equation in EM problems. Then, to reduce the number of gates needed in the quantum state preparation process, we propose a quantum-gate-reduction method, which explores the fact that the FEM cells in the input port are a small portion of the total cells in the 3-D EM structure and that the number of gates needed for state preparation depends on the RHS vector’s sparsity pattern. Based on the proposed quantum-gate-reduction method, we further derive the upper and lower bounds analytically for the number of gates needed in the quantum state preparation circuit. Furthermore, to deal with the large condition number of the finite element matrix in EM, we leverage a matrix preconditioner to modify the original linear equations, so as to reduce the number of qubits required in using quantum computation to solve EM problems. Two EM examples are used to illustrate how the proposed quantum computing method can be used to find solutions to EM problems.
... Other simulation techniques have been employed as simulation backends due to their efficacy in memory space [25]. For instance, in case of circuits that have separable states (no two-qubit states), all the quantum gates can be separately applied without the need to construct a unitary matrix for each step [28]. Another optimization upon the unitary matrix technique is to overcome the construction of a unitary matrix, but rather apply the quantum gates directly on the full state vector [9,28]. ...
... For instance, in case of circuits that have separable states (no two-qubit states), all the quantum gates can be separately applied without the need to construct a unitary matrix for each step [28]. Another optimization upon the unitary matrix technique is to overcome the construction of a unitary matrix, but rather apply the quantum gates directly on the full state vector [9,28]. This paper gives more emphasis on the unitary matrix simulation, thus other simulation techniques are discussed in Appendix A.1.1. ...
... A backend can be a simulation model for a quantum circuit or a connector to a real quantum computer [31]. The API that allows users to easily define quantum circuits is usually implemented in a high-level and widely used language [28], whereas the quantum simulation backends are typically implemented in low-level languages due to the prospect of obtaining higher performance [26,50]. ...
In this article, we present TornadoQSim, an open-source quantum circuit simulation framework implemented in Java. The proposed framework has been designed to be modular and easily expandable for accommodating different user-defined simulation backends, such as the unitary matrix simulation technique. Furthermore, TornadoQSim features the ability to interchange simulation backends that can simulate arbitrary quantum circuits. Another novel aspect of TornadoQSim over other quantum simulators is the transparent hardware acceleration of the simulation backends on heterogeneous devices. TornadoQSim employs TornadoVM to automatically compile parts of the simulation backends onto heterogeneous hardware, thereby addressing the fragmentation in development due to the low-level heterogeneous programming models. The evaluation of TornadoQSim has shown that the transparent utilization of GPU hardware can result in up to 506.5x performance speedup when compared to the vanilla Java code for a fully entangled quantum circuit of 11 qubits. Other evaluated quantum algorithms have been the Deutsch-Jozsa algorithm (493.10x speedup for a 11-qubit circuit) and the quantum Fourier transform algorithm (518.12x speedup for a 11-qubit circuit). Finally, the best TornadoQSim implementation of unitary matrix has been evaluated against a semantically equivalent simulation via Qiskit. The comparative evaluation has shown that the simulation with TornadoQSim is faster for small circuits, while for large circuits Qiskit outperforms TornadoQSim by an order of magnitude.
... By contrast, quantum computers process data using qubits (quantum bits) that can be in multiple states simultaneously and that permit calculations to be performed much more rapidly due to their potential for superposition (i.e., positional uncertainty) and entanglement (information exchange) [1]. Quantum computers can process extremely large datasets and provide solutions to complicated optimization problems at exponentially more rapid rates [2,3]. Furthermore, while supercomputers perform parallel processing using multiple cores/nodes working in tandem, quantum computers leverage quantum parallelism to explore multiple (in some cases, all) potential solutions simultaneously [2,3]. ...
... Quantum computers can process extremely large datasets and provide solutions to complicated optimization problems at exponentially more rapid rates [2,3]. Furthermore, while supercomputers perform parallel processing using multiple cores/nodes working in tandem, quantum computers leverage quantum parallelism to explore multiple (in some cases, all) potential solutions simultaneously [2,3]. While supercomputers rely on established technology, routine use of quantum computers awaits further advances, for example, the development of superconducting qubits, topological qubits, trapped ions, and, most recently, dipolar interactions between molecules placed in optical tweezer arrays [4,5]. ...
Quantum computing and supercomputing are two distinct approaches that can be used to solve complex computational problems [...]
... Quantum computing (QC) exploits quantum mechanical effects such as superposition and entanglement to perform computing operations [10]. Compared to conventional computers, a quantum computer uses qubits instead of classic bits as the basic unit of information in the quantum processing unit [11]. ...
... In the entanglement, the change of one qubit's state affects the other qubits that are coupled to it [11]. Therefore, n qubits can represent 2 of states, which results in computational advantages over a classical computer [10,11]. ...
... The phenomena of quantum mechanics such as interference, superposition and entanglement can be exploited to create a new paradigm of computation known as quantum computing [2,3]. Quantum computers are devices that perform quantum computations [4] and are shown to have a great potential of fast information processing. For example, the mathematical problem of integer factorisation is shown to be solved efficiently on a quantum computer despite the fact that it is believed to be intractable on a classical computer [5]. ...
... In other words, the quantum state of the system is given by the tensor product of the first qubit's state |ψ u(t) 〉〈ψ u(t) | and the remaining qubits' states Tr 1 [ρ]. After injecting the input u (t) into the first qubit, the quantum-mechanical system continues evolving for a time τ according to the Schrödinger's equation as discussed previously in (2) and (4). The dynamics of the quantum reservoir is governed by (4) during the time τ and the information that was encoded in the first qubit will spread through the quantum reservoir. ...
This paper applies a quantum machine learning technique to predict mobile users' trajectories in mobile wireless networks by using an approach called quantum reservoir computing (QRC). Mobile users' trajectories prediction belongs to the task of temporal information processing, and it is a mobility management problem that is essential for self‐organising and autonomous 6G networks. Our aim is to accurately predict the future positions of mobile users in wireless networks using QRC. To do so, the authors use a real‐world time series dataset to model mobile users' trajectories. The QRC approach has two components: reservoir computing (RC) and quantum computing (QC). In RC, the training is more computational‐efficient than the training of simple recurrent neural networks since, in RC, only the weights of the output layer are trainable. The internal part of RC is what is called the reservoir. For the RC to perform well, the weights of the reservoir should be chosen carefully to create highly complex and non‐linear dynamics. The QC is used to create such dynamical reservoir that maps the input time series into higher dimensional computational space composed of dynamical states. After obtaining the high‐dimensional dynamical states, a simple linear regression is performed to train the output weights and, thus, the prediction of the mobile users' trajectories can be performed efficiently. In this study, we apply a QRC approach based on the Hamiltonian time evolution of a quantum system. The authors simulate the time evolution using IBM gate‐based quantum computers, and they show in the experimental results that the use of QRC to predict the mobile users' trajectories with only a few qubits is efficient and can outperform the classical approaches such as the long short‐term memory approach and the echo‐state networks approach.
... If the stock price ends up above K at maturity, the option will be 'in-the-money' and can be exercised against the seller accordingly. 45 For simplicity, let K = 0 for both options here, which means that they are always exercised at maturity and, importantly, that their PVs are equal to S k,t for all t. With risk factors and instrument values coinciding in this case, the revaluation over time (see step two above) can be avoided. ...
... See, for example, the introductory work in[45] and the state-of-the-art technology summary in[32].2 CPU: Central Processing Unit; GPU: Graphics Processing Unit.3 ...
Quantum computing allows a significant speed-up over traditional CPU- and GPU-based algorithms when applied to particular mathematical challenges such as optimisation and simulation. Despite promising advances and extensive research in hard- and software developments, currently available quantum systems are still largely limited in their capability. In line with this, practical applications in quantitative finance are still in their infancy. This paper analyses requirements and concrete approaches for the application to risk management in a financial institution. On the examples of Value-at-Risk for market risk and Potential Future Exposure for counterparty credit risk, the main contribution lies in going beyond textbook illustrations and instead exploring must-have model features and their quantum implementations. While conceptual solutions and small-scale circuits are feasible at this stage, the leap needed for real-life applications is still significant. In order to build a usable risk measurement system, the hardware capacity—measured in number of qubits—would need to increase by several magnitudes from their current value of about 10 2 . Quantum noise poses an additional challenge, and research into its control and mitigation would need to advance in order to render risk measurement applications deployable in practice. Overall, given the maturity of established classical simulation-based approaches that allow risk computations in reasonable time and with sufficient accuracy, the business case for a move to quantum solutions is not very strong at this point.
... Quantum computing, an emerging technology, utilizes the principles of quantum mechanics to achieve unprecedented computational power [4][5][6][7]. Quantum algorithms operate within an n-qubit Hilbert space of dimension 2 n , potentially offering polynomial to even exponential computational advantage for models involving vast amounts of data. ...
The Schr\"odingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schr\"odinger-type equations with unitary evolution. It does so via the so-called warped phase transformation that maps the original equation into a Schr\"odinger-type equation in one higher dimension \cite{Schrshort,JLY22SchrLong}. We show that by employing a smooth initialization of the warped phase transform \cite{JLM24SchrBackward}, Schr\"odingerization can in fact achieve optimal scaling in matrix queries. This paper presents the detailed implementation of three smooth initializations for the Schr\"odingerization method: (a) the cut-off function, (b) the higher-order polynomial interpolation, and (c) the Fourier transform methods, that achieve optimality for (a) and near-optimality for (b) and (c). A detailed analysis of key parameters affecting time complexity is conducted.
... Eigenvectors are defined for a matrix where it states that when the matrix is transformed using the vector, it only stretches or scales the matrix and does not change the direction. The scalar by which the vector is stretched is called eigen value [31]. ...
The focus of this literature review is to examine the foundational principles and current advancements in the field of quantum computing, showcasing its potential to address challenges faced by traditional systems. This study concentrates on key concepts like superposition and entanglement, leading to an exploration of various quantum algorithms, such as Grover’s algorithm and Shor’s algorithm. By comparing Grover’s Search algorithm with binary search, the study aims to demonstrate quantum computing’s advantages in terms of efficiency and speed, especially for large datasets and unsorted databases. The comparison reveals the current state of quantum hardware and its limitations. Despite the challenges associated with hardware requirements, IBM has developed a quantum machine with a 456- qubit quantum processor, marking a milestone and showcasing rapid evolution in this field. Insights gained from this comparison include the potential for algorithms to handle scaled datasets, various applications in data science, and the capability to solve complex problems.
... On the other hand, this problem would require an exponential number of queries on a classical computer that used an exact model where no probability of failure was allowed. Nevertheless, if a bounded error was permitted in the algorithm, this advantage disappears [5]. ...
We apply quantum homomorphic encryption (QHE) schemes suitable for circuits with a polynomial number of T + T † gates to Grover's algorithm, performing a simulation in Qiskit of a Grover circuit that contains three qubits. The T + T † -gate complexity of Grover's algorithm is also analyzed in order to show that any Grover circuit can be evaluated homomorphically in an efficient manner. We discuss how to apply these QHE schemes to allow for the efficient homomorphic evaluation of any Grover circuit composed of n qubits using n − 2 extra ancilla qubits. We also show how the homomorphic evaluation of the special case where there is only one marked item can be implemented using an algorithm that makes the decryption process more efficient compared with the standard Grover algorithm.
Published by the American Physical Society 2024
... This is a new technology best suited to large unstructured datasets. Some of the traditional methods of validation may not be appropriate for quantum machine learning tools such as ours [15]. ...
Background
Quantum computing and quantum machine learning (QML) are promising experimental technologies that can improve precision medicine applications by reducing the computational complexity of algorithms driven by big, unstructured, real-world data. The clinical problem of knee osteoarthritis is that, although some novel therapies are safe and effective, the response is variable, and defining the characteristics of an individual who will respond remains a challenge. In this study, we tested a quantum neural network (QNN) application to support precision data-driven clinical decisions to select personalized treatments for advanced knee osteoarthritis.
Methodology
After obtaining patients’ consent and Research Ethics Committee approval, we collected the clinicodemographic data before and after the treatment from 170 patients eligible for knee arthroplasty (Kellgren-Lawrence grade ≥3, Oxford Knee Score (OKS) ≤27, age ≥64 years, and idiopathic aetiology of arthritis) treated over a two-year period with a single injection of microfragmented fat. Gender classes were balanced (76 males and 94 females) to mitigate gender bias. A patient with an improvement ≥7 OKS was considered a responder. We trained our QNN classifier on a randomly selected training subset of 113 patients to classify responders from non-responders (73 responders and 40 non-responders) in pain and function at one year. Outliers were hidden from the training dataset but not from the validation set.
Results
We tested our QNN classifier on a randomly selected test subset of 57 patients (34 responders, 23 non-responders) including outliers. The no information rate was 0.59. Our application correctly classified 28 responders out of 34 and 6 non-responders out of 23 (sensitivity = 0.82, specificity = 0.26, F1 Statistic = 0.71). The positive and negative likelihood ratios were 1.11 and 0.68, respectively. The diagnostic odds ratio was 2.
Conclusions
Preliminary results on a small validation dataset showed that QML applied to data-driven clinical decisions for the personalized treatment of advanced knee osteoarthritis is a promising technology to reduce computational complexity and improve prognostic performance. Our results need further research validation with larger, real-world unstructured datasets, as well as clinical validation with an artificial intelligence clinical trial to test model efficacy, safety, clinical significance, and relevance at a public health level.
... In the following, we give a brief introduction to quantum computing in order to make this article as self-contained as possible; for a more in-depth explanation, the interested reader is referred to the textbooks [12,18]. ...
A quantum circuit is often executed on the initial state where each qubit is in the zero state. Therefore, we propose to perform a symbolic execution of the circuit. Our approach simulates groups of entangled qubits exactly up to a given complexity. Here, the complexity corresponds to the number of basis states expressing the quantum state of one entanglement group. By doing that, the groups need neither be determined upfront nor be bound by the number of involved qubits. Still, we ensure that the simulation runs in polynomial time - opposed to exponential time as required for the simulation of the entire circuit. The information made available at gates is exploited to remove superfluous controls and gates. We implemented our approach in the tool quantum constant propagation (QCP) and evaluated it on the circuits in the benchmark suite MQTBench. By applying our tool, only the work that cannot be carried out efficiently on a classical computer is left for the quantum computer, hence exploiting the strengths of both worlds.
... The nascent quantum computing field has a number of theoretical algorithms proposed in recent decades, some with a supreme advantage over conventional computers at their specific tasks. [1][2][3][4] The Shor algorithm [3] that can factor large numbers rapidly, breaking current RSA encryption. RSA encryption relies on current computers, even the world's most powerful, taking far too long, on the timescale of years, to break. ...
This Article demonstrates how an eigendecomposition problem is inputted into a quantum circuit, how gates are applied in the quantum circuit, and how the output measurements are the correct eigenvalues. This process is known as quantum phase estimation (QPE). A quantum harmonic oscillator example, a foundational quantum physical chemistry problem, is demonstrated within the context of QPE. A particle in a box example, another quantum physical chemistry problem, may be solved by QPE with a caveat. These examples are of the limiting cases of diagonal matrices. Future advances in taking matrix inverses for solving linear sets of equations or finding ground state energies in the Schrödinger equation will use the principles implemented in this Article.
... Quantum computing is an emerging technology that harnesses the laws of quantum mechanics to deliver unprecedented computational power [Hid19,NC02,Pre18,RP11]. Quantum algorithms operate on an n-qubit Hilbert space with dimension 2 n , offering vast multidimensional spaces for computational models. ...
Quantum dynamics, typically expressed in the form of a time-dependent Schr\"odinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission of electrons, it is necessary to use artificial boundary conditions (ABC) to confine the computation within a fixed domain. The introduction of ABCs alters the Hamiltonian structure of the dynamics, and existing quantum algorithms can not be directly applied since the evolution is no longer unitary. The current paper utilizes a recently introduced Schr\"odingerisation method (Jin et al. arXiv:2212.13969 and arXiv:2212.14703) that converts non-Hermitian dynamics to a Schr\"odinger form, for the artificial boundary problems. We implement this method for three types of ABCs, including the complex absorbing potential technique, perfectly matched layer methods, and Dirichlet-to-Neumann approach. We analyze the query complexity of these algorithms, and perform numerical experiments to demonstrate the validity of this approach. This helps to bridge the gap between available quantum algorithms and computational models for quantum dynamics in unbounded domains.
... and along with their multiplies of a ∈ {−1, 1, i, −i} constitutes the Pauli group (Hidary, 2019). These gates act on single two-level qubit (see first equation in 6), and allow to reach any point in the geometrical representation of a two-level qubit, called the Bloch sphere (Bloch, 1946), illustrated in Figure 2. It is a convention that the two antipodes of the sphere are the elements of the computational basis (see Equation 5 in section 2). ...
This paper intends to be a starting point for the reader interested in learning the fundamentals of quantum computing. For that purpose, we give high-level explanations of the key quantum properties leveraged by a quantum computer, the quantum mathematical formalism, information about current hardware and software available to run quantum algorithms, and references for further reading.
... The quantum algorithm they use is Grover's Search Algorithm. 6 Another work on quantum computing in industrial process systems was done by You, et al. of Cornell University. 7 Other authors 8,9 have used eld programmable gate arrays (FPGAs) to emulate quantum circuits for modeling chemical phenomena. ...
A quantum circuit method for modeling steady state behavior of homogeneous hydrogen-air combustion is presented. Extensive empirical testing has pinpointed the factor determining accuracy of quantum circuit calculations. Specifically, the accuracy of the Harrow, Hassidim and Lloyd (HHL) algorithm has been benchmarked using Qiskit simulations. For random linear systems that were constricted by various criteria, the quantum solution was compared to the classical solution. The criteria investigated include orthogonality, condition number, orthonormality, angle between vectors within the linear system, diagonality, and the angle between the solution vector and the right-hand side (RHS) vector. The results of these rigorous tests show that the single most powerful factor in accuracy is the angle between the solution vector and the RHS vector. This insight was used to inform the preconditioning of two reduced models of hydrogen-air combustion. These models have been compared in terms of accuracy and degree of utility in terms of the physical system. This application area could exploit a quantum advantage via the poly-logarithmic scaling of HHL.
... Quantum computing is an emerging field with qubits as information processing units. Qubits are fundamentally different from classical bits as quantum-mechanical properties such as superposition and entanglement are exhibited by qubits [20], [21]. Quantum operations can be performed on qubits to process the information. ...
div>Deep learning techniques are very prominent in processing remotely sensed synthetic aperture radar (SAR) images for real-time, high-impact applications, such as image classification, object detection, and semantic segmentation. The accuracy of deep learning models, such as convolutional neural networks (CNNs), depends on the quality of the input data. Compared to the model-centric approach, where the model parameters are optimized during training, the data-centric approach can enhance the performance accuracy as data quality is improved before training the models. Improving the data quality of SAR images is challenging as SAR image properties are different from optical (OPT) images. Image fusion techniques proved to enhance the quality of SAR images when combined with OPT images. Many fusion techniques exist for combining SAR and OPT images in the classical domain. This paper proposes a novel approach to using quantum computing for the image fusion of SAR and OPT images. Eight different quantum processing techniques are used for the fusion of the images. We designed and created a dataset for land-use classification by collecting data using the Google Earth Engine. The quality metric measurements show that the quality of SAR images has improved by using the proposed quantum processing techniques. In addition, performance evaluation of the deep learning CNNs on the dataset was carried out for all quantum processing techniques. Our approach improved the classification accuracy from 82.64%, with only SAR images for training, to 95.36% using the proposed image fusion techniques.</div
The increasing adoption of blockchain technology has presented significant challenges in maintaining the security and reliability of smart contracts. This study addresses the problem of identifying security flaws in smart contracts, which may result in monetary damages and diminished confidence in blockchain systems. A Hybrid Quantum–Classical Deep Neural Network (HQCDNN) approach was proposed, combining quantum computing principles with classical deep learning methods to identify various vulnerability types, including access control, arithmetic, front-running, reentrancy, time manipulation, denial of service, and unchecked low calls. The SmartBugs Wild Dataset was used for training, with TF-IDF employed as a preprocessing technique optimized for hybrid architectures. Experiments were conducted using hybrid architectures with 2-qubit and 4-qubit quantum layers, alongside a classical deep neural network (DNN) model for comparative analysis. The HQCDNN model attained accuracy levels ranging from 96.4% to 78.2% and F1-scores between 96.6% and 80.2%, showcasing enhanced performance compared to the classical and deep learning models referenced in the literature. These results highlight the capability of HQCDNNs to improve the identification of security flaws in smart contracts. Future work could focus on evaluating the model on actual quantum devices and expanding its application to larger datasets for further validation.
In this chapter, a brief overview is given, describing how quantum effects such as superposition and entanglement have stimulated advances in many areas of science and technology. In addition, some fundamental concepts such as the ideas of qubits and uncertainty are introduced.
Quantum computing has existed in the theoretical realm for several decades. Recently, quantum computing has re-emerged as a promising technology to solve problems that a classical computer could take hundreds of years to solve. However, there are challenges and opportunities for academics and practitioners regarding software engineering practices for testing and debugging quantum programs. This paper presents a roadmap for addressing these challenges, pointing out the existing gaps in the literature and suggesting research directions. We discuss the limitations caused by noise, the no-cloning theorem, the lack of a standard architecture for quantum computers, among others. Regarding testing, we highlight gaps and opportunities related to transpilation, mutation analysis, input states with hybrid interfaces, program analysis, and coverage. For debugging, we present the current strategies, including classical techniques applied to quantum programs, quantum-specific assertions, and quantum-related bug patterns. We introduce a conceptual model to illustrate concepts regarding the testing and debugging of quantum programs and the relationship between them. Those concepts are used to identify and discuss research challenges to cope with quantum programs through 2030, focusing on the interfaces between classical and quantum computing and on creating testing and debugging techniques that take advantage of the unique quantum computing characteristics.
The use of quantum computing as an efficient alternative to classical computing is rapidly evolving in diverse engineering applications to solve complex numerical problems. As its application becomes increasingly prominent in the future, its use should be complemented by validation and practical recommendations. This paper provides practical recommendations regarding the accuracy and efficiency of quantum computing for solving diverse types of Bayesian networks for bridge evaluation and maintenance, including basic and fuzzy-based networks. The methodology for solving the considered networks, development of quantum circuits, and accuracy of the results (quantum computing versus simulator and classical computing) were examined. Research findings indicate the feasibility of quantum computing for solving small-scale bridge Bayesian networks (error of 6% approximately), whereas the accuracy of solutions for solving complex networks using available open-access quantum computers is compromised owing to the limited number of available attempts and the compound effect of quantum error (error up to 37% approximately). Practical recommendations were provided to practitioners and future research needs are identified.
Quantum computing represents a revolutionary computational paradigm with the potential to address challenges beyond classical computers’ capabilities. The development of robust quantum software is indispensable to unlock the full potential of quantum computing. Like classical software, quantum software is expected to be complex and extensive, needing the establishment of a specialized field known as Quantum Software Engineering. Recognizing the regional focus on Latin America within this special issue, we have boarded on an in-depth inquiry encompassing a systematic mapping study of existing literature and a comprehensive survey of experts in the field. This rigorous research effort aims to illuminate the current landscape of Quantum Software Engineering initiatives undertaken by universities, research institutes, and companies across Latin America. This exhaustive study aims to provide information on the progress, challenges, and opportunities in Quantum Software Engineering in the Latin American context. By promoting a more in-depth understanding of cutting-edge developments in this burgeoning field, our research aims to serve as a potential stimulus to initiate pioneering initiatives and encourage collaborative efforts among Latin American researchers.
We review the classical definition of the Mathematical Superposition Principle and the Measurable Superposition Effect, relevant to Young’s double-slit (YDS) experiment and the two-beam Mach–Zehnder Interferometer (MZI) and contrasts them with the prevailing interpretations by Quantum Mechanics. YDS is known to be “at the heart of understanding Quantum Mechanics” and MZI is being used to demonstrate quantum entanglement, a step forward to build quantum computers. We would demonstrate that classical interpretations are the correct ones for these two superposition effects. We recognize that cosmic universe is entangled by gravity and electromagnetic fields generated by innumerable stars of all the galaxies. Yet, we have succeeded in developing approximate mathematical theories, which we validate reproducibly by constructing local instruments for local and causal validity. Data in our instruments is generated as some physical transformation in a chosen “known entity” (detector) that exchanges energy with the “unknown entity” (detectee) under study. The energy exchange must always be guided by one of the natural forces of interaction, compatible between the detector and the detectee. This is known as the locality principle. Thus, all data-generating interactions represent locally entangled (interacting) phenomena. This is true irrespective of the expansive quantum-mathematical definition of the word “entanglement”. All useful engineering data is generated in an instrument as some physical transformation in a detector after it interacts and exchanges energy with one or more superposed signals stimulating it. The proper superposition equation, modeling such interaction processes, must incorporate the characteristic interaction parameter of the detector that would multiply all the superposed signals. Then the mathematically allowed normalization procedure of the unmeasurable summed amplitudes becomes constrained against ad hoc normalization. We recommend: (i) We need to underscore the explicit incorporation of the old fashion “Interaction Process Mapping Thinking” and bring back the engineering reality in all of Physics discourse. (ii) Experimental optical physicists should take active roles in the development of fundamental physics.
Kuantum fiziği, elde edilen gelişmeler ile teknoloji alanına hizmet etme kapasitesi olduğunu kanıtlamıştır. Bu alanlardan en önemlilerinden birisi de askeri kullanım alanlarıdır. NATO bünyesinde de bu teknolojinin gelecek muharebe ortamının bir parçası haline gelmesi için müttefikler arasında iş birliği için stratejiler üretilmiştir. Bununla birlikte kuantum teknolojilerinin kullanımında hassasiyet ve altyapı gibi temel kısıtlar bulunmaktadır. Sistemin atom altı boyutta çalışıyor olması dış etkenlere karşı hassasiyeti muharebe ortamında uygulanmasını kısıtlayan bir etken haline gelmiştir. İkinci olarak sistemlerin hassasiyetinden ve uygulamanın güncel teknoloji ile gerçekleşmesinin güç olması sistemlerin kurulumundaki altyapı sorunun oluşmasına neden olmaktadır. Silahlı kuvvetlerin böyle bir değişime adaptasyonunu sağlaması için geçmiş teknolojik gelişmelere karşı yapmış olduğu doktrin ve teşkilat revizyonlarından farklı adımlar atması gerekmektedir. Kuantum revizyonu, silahlı kuvvetlerin adaptasyon sağlaması için kuantum teknolojilerinin askeri kullanım alanlarının araştırılarak bu alanlara karşı değişimde izlenilecek yolun araştırıldığı revizyon konseptidir.
We introduce a method to enforce some symmetries starting from a trial wave-function prepared on quantum computers that might not respect these symmetries. The technique eliminates the necessity for performing the projection on the quantum computer itself. Instead, this task is conducted as a post-processing step on the system’s “Classical Shadow”. Illustrations of the approach are given for the parity, particle number, and spin projectors that are of particular interest in interacting many-body systems. We compare the method with another classical post-processing technique based on direct measurements of the quantum register. We show that the present scheme can be competitive to predict observables on symmetry-restored states once optimization through derandomization is employed. The technique is illustrated through its application to compute the projected energy for the pairing model Hamiltonian.
Quantum computing is a novel approach to computation that can provide a significant speedup of computationally hard tasks by using special quantum algorithms. Significant scaling of the many cohesive components within a quantum computing system is necessary to achieve quantum advantage for practical tasks. An essential element of any quantum computer is the control and measurement system. Classical superconductive electronics and SFQ circuits in particular can be interfaced with superconductive quantum systems. The performance of SFQ control circuits in terms of fidelity and noise is currently approaching the state-of-the-art as set by conventional control methods. Classical superconductive circuits can be instrumental in the transition to larger scale quantum computers. SFQ circuits can enhance these systems by providing fast, repeated measurements, complex processing, and controlled feedback while introducing low noise and heat load.
Quantum computing stands at the forefront of technological advancement, offering promises of unprecedented computational power and efficiency. Among its myriad applications, quantum algorithms for optimization problems have emerged as a beacon of hope for tackling complex challenges that classical computers struggle to address. This Paper delves into the realm of quantum algorithms tailored for optimization problems, exploring their theoretical foundations, practical implications, and potential transformative impact across diverse domains.
With the rising complexity of our electricity infrastructure, smart grid simulations increasingly rely on co-simulation, which involves jointly executing independent subsystem simulations. However, in large-scale simulation scenarios, such as those involving costly power-flow analysis, co-simulation may experience computational-performance issues. Quantum computing offers a potential solution through quantum–classical co-simulation, in which one or more simulators of an otherwise classical co-simulation are executed on quantum hardware. However, there is no practical realization of this concept that establishes its feasibility. To address this gap, we integrate a quantum power flow simulator with a smart grid co-simulation and conduct an exploratory simulation study using a fictitious case-study scenario. The experiments demonstrate the feasibility of quantum–classical co-simulation; at the same time, they highlight four obstacles to the concept’s realization in practice: (1) To use quantum computing for co-simulation, session-based scheduling is required. (2) Distributed simulation limits possible applications and requires proximity of computing resources. (3) For the efficient extraction of classical information from the quantum states, we need carefully designed operators. (4) Current hardware limitations—such as noise susceptibility and the lack of quantum random access memory—limit practical near-term uses of quantum power flow; therefore, attention should be turned to alternative applications that are more promising in the near term. These findings pave the way for future research on quantum–classical co-simulation and its potential applications in smart grids.
¿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.
Quantum Bayesian computation is an emerging field that levers the computational gains available from quantum computers. They promise to provide an exponential speed‐up in Bayesian computation. Our article adds to the literature in three ways. First, we describe how quantum von Neumann measurement provides quantum versions of popular machine learning algorithms such as Markov chain Monte Carlo and deep learning that are fundamental to Bayesian learning. Second, we describe quantum data encoding methods needed to implement quantum machine learning including the counterparts to traditional feature extraction and kernel embeddings methods. Third, we show how quantum algorithms naturally calculate Bayesian quantities of interest such as posterior distributions and marginal likelihoods. Our goal then is to show how quantum algorithms solve statistical machine learning problems. On the theoretical side, we provide quantum versions of high dimensional regression, Gaussian processes and stochastic gradient descent. On the empirical side, we apply a quantum FFT algorithm to Chicago house price data. Finally, we conclude with directions for future research.
Quantum computing and quantum information science is a burgeoning engineering field at the cusp of solving challenging robotic applications. This paper introduces a hybrid (gate-based) quantum computing and classical computing architecture to solve the motion propagation problem for a robotic system. This paper presents the quantum-classical architecture for linear differential equations defined by two types of linear operators: unitary and non-unitary system matrices, thereby solving any linear ordinary differential equations. The ability to encode information using bits or qubits—is essential in any computation problem. The results in this paper also introduce two novel approaches to encoding any arbitrary state vector or any arbitrary linear operator using qubits. Unlike other algorithms that solve ordinary differential equations (ODEs) using purely quantum or classical architectures, the ODE solver presented in this paper leverages the best of quantum and classical computing paradigms.
Our electricity infrastructure is getting more complex and heterogeneous. Holistically analyzing grids is therefore increasingly challenging. Co-simulation, i.e. the coordinated execution of independent subsystem simulators, is inherently well suited to handling these challenges. However, the computational needs of calculating power flows within the simulated grid may limit the scalability for large-scale co-simulations. Recent advances in quantum computing offer a potential solution to these concerns: The computing paradigm‘s potential for exponentially speeding up power flow has been shown. To utilize these capabilities for smart grid simulations, we propose quantum–classical co-simulation: integrating simulators running on quantum hardware with an otherwise classical co-simulation. Specifically, we focus on exploiting quantum power flow in smart grid co-simulations. This concept is promising for applications that require comprehensive grid simulation and whose scalability is impeded by the computational properties of power flow. This paper highlights the concept of quantum–classical co-simulation, and advocates for its criticality and applications in supporting smart grid analytics. We encourage and facilitate research by recommending a five-item research roadmap. We also provide a detailed discussion on the potential obstacles in implementing this concept, to help bring its theoretical value to practice.
With the rapid development of digital technology, all areas of society may accelerate their entry into the virtual world, thus blurring the boundary between the physical and digital worlds and promoting a Metaverse. The more financial opportunities in metaverse offers its members, the more choices they have. Whether you're a visitor or a company that has invested in the metaverse to sell your goods, there are many ways to generate money. The potential of the metaverse to build virtual spaces for people to connect may have a significant negative impact on the financial and banking sector, among other useful applications it offers. For the reason of metaverse, people can interact with cutting-edge technology using extended technology (VR, AR, MR) etc. With these developing ideas rise to the metaverse, a virtual economy where things may be produced and purchased. Metaverse increased the financial trust-ability of the people for its reliable and safe technologies. Metaverse uses digital virtual identifications, extended reality, blockchain decentralization techniques, AI applications, IoT systems, and digital twins for metaverse finances. In this paper, we are going to discuss metaverse applications, their challenges, and a state of art approach for the finance industry.
Driven by the rapid progress in quantum hardware, recent years have witnessed a furious race for quantum technologies in both academia and industry. Universal quantum computers have supported up to hundreds of qubits, while the scale of quantum annealers has reached three orders of magnitude (i.e., thousands of qubits). Quantum computing power keeps climbing. Race has consequently generated an overwhelming number of research papers and documents. This article provides an entry point for interested readers to learn the key aspects of quantum computing and communications from a computer science perspective. It begins with a pedagogical introduction and then reviews the key milestones and recent advances in quantum computing. In this article, the key elements of a quantum Internet are categorized into four important issues, which are investigated in detail: a) quantum computers, b) quantum networks, c) quantum cryptography, and d) quantum machine learning. Finally, the article identifies and discusses the main barriers, the major research directions, and trends.
This work presents a brief review on the modern approaches to data modeling by the methods developed in the quantum physics during the last one hundred years. Quantum computers and computations have already been widely investigated theoretically and attempted in some practical implementations, but methods of quantum data modeling are not yet sufficiently established. A vast range of concepts and methods of quantum mechanics have been tried in many fields of information and behavior sciences, including communications and artificial intelligence, cognition and decision making, sociology and psychology, biology and economics, financial and political studies. The application of quantum methods in areas other than physics is called the quantum-like paradigm, meaning that such approaches may not be related to the physical processes but rather correspond to data modeling by the methods designed for operating in conditions of uncertainty. This review aims to attract attention to the possibilities of these methods of data modeling that can enrich theoretical consideration and be useful for practical purposes in various sciences and applications.
Quantum machine learning is a rapidly growing field at the intersection of quantum technology and artificial intelligence. This review provides a two-fold overview of several key approaches that can offer advancements in both the development of quantum technologies and the power of artificial intelligence. Among these approaches are quantum-enhanced algorithms, which apply quantum software engineering to classical information processing to improve keystone machine learning solutions. In this context, we explore the capability of hybrid quantum-classical neural networks to improve model generalization and increase accuracy while reducing computational resources. We also illustrate how machine learning can be used both to mitigate the effects of errors on presently available noisy intermediate-scale quantum devices, and to understand quantum advantage via an automatic study of quantum walk processes on graphs. In addition, we review how quantum hardware can be enhanced by applying machine learning to fundamental and applied physics problems as well as quantum tomography and photonics. We aim to demonstrate how concepts in physics can be translated into practical engineering of machine learning solutions using quantum software.
The rapid progress in digitalization and automation have led to an accelerated growth in healthcare, generating novel models that are creating new channels for rendering treatment at reduced cost. The Metaverse is an emerging technology in the digital space which has huge potential in healthcare, enabling realistic experiences to the patients as well as the medical practitioners. The Metaverse is a confluence of multiple enabling technologies such as artificial intelligence, virtual reality, augmented reality, internet of medical devices, robotics, quantum computing, etc. through which new directions for providing quality healthcare treatment and services can be explored. The amalgamation of these technologies ensures immersive, intimate and personalized patient care. It also provides adaptive intelligent solutions that eliminates the barriers between healthcare providers and receivers. This article provides a comprehensive review of the Metaverse for healthcare, emphasizing on the state of the art, the enabling technologies to adopt the Metaverse for healthcare, the potential applications, and the related projects. The issues in the adaptation of the Metaverse for healthcare applications are also identified and the plausible solutions are highlighted as part of future research directions.
This paper proposes a quantum computing approach for insurance capital modelling. Using an open-source software development kit, Qiskit, an algorithm for working on a superconducting type IBM quantum computer is developed and implemented to predict the capital of insurance companies in the classical surplus process. With the fundamental properties of quantum mechanics, Dirac notation and Feynman’s path calculation are shown. Furthermore, custom quantum insurance premium and claim gates are investigated in order to build a quantum circuit with respect to initial reserve, premium and claim amounts. Some numerical results are presented and discussed at the end of the paper.
This Resource Letter provides information and guidance for those looking to incorporate computation into their courses or to refine their own computational practice. We begin with general resources, including policy documents and supportive organizations. We then survey efforts to integrate computation across the curriculum as well as provide information for instructors looking to teach a computational physics course specifically. An overview of education research into computation in physics, including materials from beyond Physics Education Research, is then provided, followed by suggestions for tools, languages, and environments. We conclude with some emerging topics for which only preliminary resources exist but represent important topics for future innovation.
In this paper, a protocol for nondestructive parity meter (NPM) of two cat‐state qubits is proposed. The physical model contains two cavities and an auxiliary qubit. For each cavity, the quantum information is encoded on the odd and even cat state, forming a cat‐state qubit. By adjusting the strength of the driving field on the auxiliary qubit, an effective Hamiltonian for cavity‐selective transition is derived. Moreover, reverse engineering based on the effective Hamiltonian is applied to find the evolution path, and the systematic‐error‐sensitivity nullification method is used to select proper parameters of the evolution path. In this way, robust control fields are designed to combat the influence of the systematic errors. Furthermore, the effects of random noise and decoherence on the fidelity of the protocol are considered. Numerical simulations show that the protocol is insensitive to the experimental imperfection, including systematic errors, random noise and decoherence. Therefore, the protocol is expected to expand the vision of realizing NPM. A nondestructive parity meter is designed to determine the parity of the two‐cat‐state‐qubit. The physical model contains two cavities and an auxiliary qubit. Two Kerr‐nonlinear cavities are used to restricted the evolution of the two‐qubit‐cat‐state in the cat‐state subspace. The parity of the two‐cat‐state‐qubit is known by the final state of the auxiliary qubit.
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