The intricacies of using S-word arithmetic, the influence of the value of the parameter S on the estimation of the rounding error are analyzed; what are high-precision calculations and where they are used. The problems of two-key cryptography, computer steganography and the problem of transcomputational complexity are considered as areas of application of S-word arithmetic. For the development of S-word arithmetic algorithms, sequential, parallel, quantum computing models are used, and systems of residual classes are used. The architectural features of the computer system for the implementation of an effective algorithm in various models of calculations are considered. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Peculiarities of transferring algorithms to another computing model are considered. An analysis of the complexity of implementing S-word arithmetic operations in sequential, parallel, and quantum computing models is carried out. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Information is provided about the ongoing scientific forum «Calculation optimization issues», the subject of which is closely related to the topic (1969-2023)
The intricacies of using S-word arithmetic, the influence of the value of the parameter S on the estimation of the rounding error are analyzed; what are high-precision calculations and where they are used. The problems of two-key cryptography, computer steganography and the problem of transcomputational complexity are considered as areas of application of S-word arithmetic. For the development of S-word arithmetic algorithms, sequential, parallel, quantum computing models are used, and systems of residual classes are used. The architectural features of the computer system for the implementation of an effective algorithm in various models of calculations are considered. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Peculiarities of transferring algorithms to another computing model are considered. An analysis of the complexity of implementing S-word arithmetic operations in sequential, parallel, and quantum computing models is carried out. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Information is provided about the ongoing scientific forum «Calculation optimization issues», the subject of which is closely related to the topic (1969-2023)
Introduction. The complexity of computational algorithms for solving typical problems of computational, applied, and discrete mathematics is analyzed from the perspective of the theory of computation, depending on the computer architecture and the used computing model: single-processor, multiprocessor, and quantum. The following classes of problems are considered: systems of linear algebraic equations, the Cauchy problem for systems of ordinary differential equations, numerical integration, boundary value problems for ordinary differential equations, factorization of numbers, finding the discrete logarithm of a number in multiplicative integer groups, searching for the necessary record in an unordered database, etc. The purposes of the paper are: 1. To investigate how the computational complexity depends on the computer architecture and the computational model. 2. To show that the construction of the computational process under the given conditions of calculations is related to the solution of the following problems: – the existence ε-solution to the problem; – the existence of T-effective computing algorithms; – the possibility of building a real computing process under the given computing conditions. 3. To investigate the effect of rounding numbers on computational complexity (especially when solving problems of transcomputational complexity). 4. To give the complexity estimates and total error of the computational algorithm for a number of typical problems of computational, applied, and discrete mathematics. The results. The complexity estimates of computational algorithms of the listed classes of problems for single-processor, multiprocessor and quantum computing models are given. The main focus is on high-performance computing: using the principles of parallel data processing and quantum mechanics. Conclusions. The connection of complexity estimates of computational algorithms with the architecture of computers and models of calculations is demonstrated. The characteristics of the first quantum computers (2016 – 2022), which have gone beyond laboratory research, are given. Keywords: computer technologies, rounding error, sequential, parallel and quantum computing models, complexity estimate.
Context. Nature is the relation among processes and phenomena. Nothing exists in the universe without relations. Computer is transactions of relations between data with the help of control and execution mechanisms. Quantum relations are a superposition of particles and their states. Superposition and entanglement are equivalent concepts. Entanglement is a non-local superposition of deterministic states. A quantum computer is unconditional transactions of relations between qubit data. Quantum computer is an analog device for parallel solution of combinatorial problems. Practically oriented definitions of the quantum computer concepts are the path to development of scalable quantum parallel algorithms for combinatorial problems solving. Any algorithm can be reduced to a sequence of operations without conditions, because any truth table is a collection of a complete system of conditions-states. Any sequence of actions can always be reduced to one parallel operation. Conditions and sequences arise only when the developer wants to use previously created primitive constructs to build an always non-optimal computing unit. The paradigm of quantum computer creation is determined through the use of photonic transactions on the electrons of an atom may exclude the use of quantum logic. The evolutionary path of a quantum computer from the classical one: “memory-address-transaction” (MAT) → “electron-addresstransaction” → “electron-address-quantaction” (EAQ) → state-superposition-logic. The meeting point of classical and quantum computers is photon transactions on the structure of electrons. Everything that is calculated on a quantum computer can be calculated in parallel on a classical one on account of memory redundancy. The given example is a memory-driven algorithm for modeling digital products based on qubit-vector forms of functionality description for significant performance boost of computing processes by parallel execution of logical operations. Objective. Simulation of the correct SoC-component behavior based on vector representation of the logic. Formation of the triggering development of a computing based on the superposition of the classical, quantum and analog computing process, which in its development should be based on technological qubit, tabular and vector data structures for the parallel solution of combinatorial problems. Method. MAT-computing implements any algorithms on account of transactions (read-write) in memory. Qubit-vector models for describing functionalities, which differ from known truth tables in compactness of description and manufacturability for the implementation of parallel algorithms of the synthesis and analysis of digital devices and SoC-components. Results. 1) The metric of the technological data structures, focused on parallel troubleshooting in digital systems based on the usage of two logical vector operations, was proposed for the first time. 2) The metric of relations between the individual components of QC, allowing organizing a quantum deterministic computer, has been further developed. 3) Quantum architectural solutions, that allow solving coverage problems in a quasi-parallel mode, were proposed for the first time. 4) Architectural solutions based on an analog-to-digital computing, which can be used to solve the problems of the digital systems parallel analysis, have been further developed. 5) Vector-qubit structures of the logic data, that allow a quasi-parallel simulation of digital circuits, were proposed. Conclusions. Qubit models, quantum methods and combinatorial algorithms for technical diagnostics of digital devices have been implemented, which can significantly (up to 25%) reduce the time of test synthesis, deductive modeling of faulty and correct behavior, search for defective states by introducing an innovative idea of using qubit-vector data structures for describing logical components. Comparative assessments of qubit models and methods usage show an increase in the efficiency of algorithms for modeling digital devices compared to tabular ones. The superposition of a classical, quantum and analog computer is integrally represented, which allows to find the best solutions for recognition and decision making.
Context. Nature is the relation among processes and phenomena. Nothing exists in the universe without relations. Computer is transactions of relations between data with the help of control and execution mechanisms. Quantum relations are a superposition of particles and their states. Superposition and entanglement are equivalent concepts. Entanglement is a non-local superposition of deterministic states. A quantum computer is unconditional transactions of relations between qubit data. Quantum computer is an analog device for parallel solution of combinatorial problems. Practically oriented definitions of the quantum computer concepts are the path to development of scalable quantum parallel algorithms for combinatorial problems solving. Any algorithm can be reduced to a sequence of operations without conditions, because any truth table is a collection of a complete system of conditions-states. Any sequence of actions can always be reduced to one parallel operation. Conditions and sequences arise only when the developer wants to use previously created primitive constructs to build an always non-optimal computing unit. The paradigm of quantum computer creation is determined through the use of photonic transactions on the electrons of an atom may exclude the use of quantum logic. The evolutionary path of a quantum computer from the classical one: “memory-address-transaction” (MAT) → “electron-addresstransaction” → “electron-address-quantaction” (EAQ) → state-superposition-logic. The meeting point of classical and quantum computers is photon transactions on the structure of electrons. Everything that is calculated on a quantum computer can be calculated in parallel on a classical one on account of memory redundancy. The given example is a memory-driven algorithm for modeling digital products based on qubit-vector forms of functionality description for significant performance boost of computing processes by parallel execution of logical operations. Objective. Simulation of the correct SoC-component behavior based on vector representation of the logic. Formation of the triggering development of a computing based on the superposition of the classical, quantum and analog computing process, which in its development should be based on technological qubit, tabular and vector data structures for the parallel solution of combinatorial problems. Method. MAT-computing implements any algorithms on account of transactions (read-write) in memory. Qubit-vector models for describing functionalities, which differ from known truth tables in compactness of description and manufacturability for the implementation of parallel algorithms of the synthesis and analysis of digital devices and SoC-components. Results. 1) The metric of the technological data structures, focused on parallel troubleshooting in digital systems based on the usage of two logical vector operations, was proposed for the first time. 2) The metric of relations between the individual components of QC, allowing organizing a quantum deterministic computer, has been further developed. 3) Quantum architectural solutions, that allow solving coverage problems in a quasi-parallel mode, were proposed for the first time. 4) Architectural solutions based on an analog-to-digital computing, which can be used to solve the problems of the digital systems parallel analysis, have been further developed. 5) Vector-qubit structures of the logic data, that allow a quasi-parallel simulation of digital circuits, were proposed. Conclusions. Qubit models, quantum methods and combinatorial algorithms for technical diagnostics of digital devices have been implemented, which can significantly (up to 25%) reduce the time of test synthesis, deductive modeling of faulty and correct behavior, search for defective states by introducing an innovative idea of using qubit-vector data structures for describing logical components. Comparative assessments of qubit models and methods usage show an increase in the efficiency of algorithms for modeling digital devices compared to tabular ones. The superposition of a classical, quantum and analog computer is integrally represented, which allows to find the best solutions for recognition and decision making.
A methodical analysis of the basic problem related to quantum calculations of parameters of physical systems was made. Emphasis is placed on the physical principles of the operation of a quantum computer, with an emphasis on the fact that simultaneous access to all quantum states is important in quantum computing, which allows the simultaneous change of the quantum state from all superpositions in the qubit system. Emphasis is placed on the fact that in quantum algorithms the Fourier transform and the Hadamard transform are the basic operations - as a simple discrete Fourier transform. The reader's attention is drawn to the fact that quantum computing is primarily implemented in quantum objects with the properties of elementary NOT gates and controlled CNOT, which can be implemented on a Mach-Zehnder interferometer using the phenomena of photon interference and rotation of its polarization vector. Despite the progress of conventional computers, the need for the development of quantum computing is due to the technological limitation due to the dimensional quantization of the electronic spectrum and the exponential increase in the time of calculations by classical algorithms when the volume of data increases. However, the widespread use of quantum computers is limited by a number of problems. This is, first of all, insufficient accuracy and high sensitivity to external influences that can destroy the quantum state. Therefore, to increase the accuracy of calculations on a quantum computer, the calculation algorithm must be repeated a certain number of times, and to avoid the destruction of the quantum states of the qubit, low temperatures are used.
Quantum computing (Q-computing) is a revolutionary technology that applies quantum mechanics (QM) to implement calculations significantly quicker than traditional computers. Q-computing employs Q-bits (qubits) rather than classical bits, which can take on a superposition state, making it possible to represent both 0 and 1 at once. This distinctive property allows qubits to perform multiple calculations simultaneously, making Q computers exponentially more powerful as more qubits are added, compared to the linear scaling of classical computers. However, challenges such as qubit instability and error rates remain significant obstacles to fully realizing this potential. A vital aspect of this technology is the development of Q arithmetic circuits, which act as the foundational blocks for efficiently executing Q algorithms. Q half-adder and full-adder circuits are integral to performing fundamental arithmetic operations within Q computers. In contrast to classical adders, Q adders rely on Q gates such as Controlled NOT (CNOT) and Toffoli gates. These gates exploit Q phenomena like entanglement and superposition, allowing for more efficient computations. The Q half-adder consists of two qubits and adds two single bits, producing a carry and sum output. It utilizes the CNOT gate to compute the sum and the Toffoli gate (controlled-controlled-NOT [CCNOT]) to compute the carry. The Q full-adder builds on this, adding three qubits to produce a sum and a carry output, further increasing computational capacity. Through the entanglement of qubits, these circuits allow for more parallelism and speed compared to classical adders. Q carry-look-ahead adders (QCLAs) and Q ripple-carry adders (QRCAs) are used to perform more complex arithmetic operations based on these half-adder and full-adder structures. Q arithmetic circuits have broad applications, boosting secure key generation in cryptography, speeding up matrix operations for faster model training in machine learning, and enabling more precise, efficient image transformations in Q image processing, benefiting fields like medical imaging and computer vision.
The article deals with new approaches in studying the features of the structure and principles of functioning of quantum computers, as one of the components of the elemental base of modern computer electronics, due to more efficient layout and supply of the corresponding lecture material using the multimedia presentation system. It is shown that at the present stage of the creation of quantum computers goes in two directions: specialized quantum computers that are aimed at solving one particular specific task, such as optimization tasks (quantum computers D-Wave); and universal quantum computers that are capable of implementing arbitrary quantum algorithms (Shore, Simon, Grover) - in implementing from IBM, Google, etc. The work of quantum registers is analyzed. When processing this material, students show that the ability of devices that are based on the classical principles of computing machines, in recent years reached a level in which most tasks can be solved relatively quickly and quite accurately. However, in various areas of knowledge, there are problems to decide which, including using the best supercomputers of the world, is not possible. After all, modern transistors have already reached a few nanometers, and in the future, electronic semiconductor components cannot be reduced infinitely, because their elements will consist of one atom. The development of classical computers in the form of cloud computing capacities is also limited in the area and consumed energy. In addition, not each algorithm is subject to parallelization. Effective processing of large volumes of data requires high-performance computations in the field of artificial intelligence, science-intensive industrial areas, modeling of chemical and physical phenomena and processes, that is, where it ceases to have the capabilities of modern supercomputers. Solving this problem is possible to apply the latest technologies of quantum computers. It has been shown that the quantum computer is a computing device that uses the phenomenon of quantum superposition (parallelism) and quantum confusion to transmit data and data processing, that is, the basis of its central processor is laid logic of quantum mechanics using special quantum algorithms. Such a computer is fundamentally different from a traditional computer that perceives any information in the form of a binary code, which may be in one of two possible states: 0 or 1. Any classic processor represents these two states of low or high voltage levels inside the semiconductor transistor. Its task to select zero or unit. Quantum processors are made according to the same principle, but a radical cancellation that instead of a transistor, they use kubit (Qubit) – quantum bits.