Abstract In the field of quantum computing, the Deutsch–Jozsa problem serves as a classical example illustrating the computational advantages of quantum algorithms over classical counterparts in specific scenarios. The problem involves efficiently determining the nature of a multi-valued function, showcasing how quantum parallelism can accelerate problem-solving. The Deutsch–Jozsa algorithm provides a framework for exploring these quantum advantages and offers insights into optimization tasks such as the minimum spanning tree (MST) problem in network design. Applications include network engineering, routing optimization, and resource allocation, where quantum-enhanced approaches can potentially outperform classical methods. This study examines the implementation of the Deutsch–Jozsa algorithm on quantum simulators and its adaptation in quantum versions of classical MST algorithms, such as the Ambainis–Spalek quantum variant of Edmonds–Karp. The analysis highlights the role of Grover’s search as a foundational component, integrating it with quantum MST techniques. Practical simulations using IBM Quantum platforms illustrate the algorithm’s potential for network optimization and broader information security applications.
Abstract In the field of quantum computing, the Deutsch–Jozsa problem serves as a classical example illustrating the computational advantages of quantum algorithms over classical counterparts in specific scenarios. The problem involves efficiently determining the nature of a multi-valued function, showcasing how quantum parallelism can accelerate problem-solving. The Deutsch–Jozsa algorithm provides a framework for exploring these quantum advantages and offers insights into optimization tasks such as the minimum spanning tree (MST) problem in network design. Applications include network engineering, routing optimization, and resource allocation, where quantum-enhanced approaches can potentially outperform classical methods. This study examines the implementation of the Deutsch–Jozsa algorithm on quantum simulators and its adaptation in quantum versions of classical MST algorithms, such as the Ambainis–Spalek quantum variant of Edmonds–Karp. The analysis highlights the role of Grover’s search as a foundational component, integrating it with quantum MST techniques. Practical simulations using IBM Quantum platforms illustrate the algorithm’s potential for network optimization and broader information security applications.
This article analyzes mathematical models of quantum parallelism based on the transfer of a quantum system into a superposition state. The principles of quantum parallelism and its mathematical foundations are substantiated. To demonstrate the quantum advantage, mathematical models of fundamental quantum algorithms Deutsch-Joža and Grover are analyzed, which illustrate the efficiency of quantum computing compared to classical methods. The Deutsch-Joshy algorithm is considered, which generalizes the problem of determining the type of a binary function by extending it to a set of arguments written in a quantum register. This algorithm allows determining whether a function is constant or balanced in a single quantum call, while in classical calculations this requires exponential time. It is noted that although the Deutsch-Joshy algorithm demonstrates the possibilities of parallel quantum computing, it is of mainly theoretical importance. Grover’s algorithm implements quantum parallel computing for the practical problem of searching for an element in an unordered database. It is based on iterative amplification of the amplitude of the state corresponding to the searched element and provides quadratic speedup compared to classical algorithms. In addition to database search, Grover’s algorithm can be adapted to solve other optimization problems. The work of the considered algorithms is illustrated by numerical examples, which simplifies the understanding of their principles and contributes to the further methodological development of parallel quantum computing. The article also outlines the advantages of quantum parallelism over classical computing and identifies the prospects for the application of parallel quantum algorithms.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch–Jozsa and Simon’s problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch–Jozsa problem with probability 1 using only one oracle query, and Simon’s problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch–Jozsa and Simon’s problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms and calls for further study of oracle separation between quantum and classical computation.
This paper introduces a comprehensive quantum calculator using Qiskit that integrates quantum arithmetic and logic functions into a single calculator. The calculator relies on quantum Fourier transforms (QFT), controlled-phase gates, and quantum modular arithmetic to perform basic arithmetic operations such as addition, subtraction, and multiplication. These arithmetic techniques used quantum parallelism and interference to perform calculations more efficiently, particularly for tasks involving large numerical inputs. Numerical values are stored in qubit registers and processed using QFT-based circuits, which enable efficient and reversible mathematical computations. In addition to arithmetic operations, the calculator also implements quantum logic algorithms, including Grover’s Search and the Deutsch-Jozsa algorithm. Grover’s algorithm is an unstructured search algorithm that operates by constructing an oracle and iteratively amplifying the probability amplitude of the target solution. This amplitude amplification significantly increases the likelihood of measuring the desired outcome. Due to this property, Grover’s algorithm has practical relevance in applications such as database searching and solving combinatorial optimization problems. The Deutsch-Jozsa algorithm determines whether or not a certain Boolean function is constant or balanced and offers an exponential speed up compared to classical methods for this class of problem. A key fundamental function of the calculator is the ability to generate hybrid quantum circuits that support both arithmetic and logical functions. That allows the exploration of simple as well as complex problem-solving strategies that take advantage of both quantum arithmetic and decision-making algorithms. For instance, combining Grover’s amplification with function characterizations from the Deutsch-Jozsa algorithm provides not only interesting circuit configurations but also adds value to the possibilities of the algorithms. This research offers a modular idea and allows exploring the fundamental principles of quantum computing generally speaking and will be a valuable resource for educational as well as research purposes in the rapidly growing field of quantum algorithm development.
Quantum computation transformed our thinking about computational complexity by showing that quantum mechanical principles can be used to reduce the number of queries needed to solve some problems. The work constitutes a critical analysis of the evolution of the query complexity through three fundamental quantum algo rithms: the Deutsch algorithm, the Deutsch–Jozsa algorithm and Simon’s algorithm. Together, these algorithms illustrate the emergence of quantum advantage where quantum systems outperform classical deterministic and probabilistic approaches by exploiting superposition, interference, entanglement and unitary evolution.The study begins with the Deutsch algorithm, the first demonstration that a quantum computer can compute a global property of a Boolean function with fewer oracle queries than any deterministic classical algorithm. We further extend the analysis to the Deutsch–Jozsa algorithm, which generalizes this speedup to the exponential level by distinguishing between balanced and constant Boolean functions with one quantum query. Then we study Simon’s Algorithm, a milestone that achieved an exponential separation between classical randomized and quantum query complexity. The groundwork for many later advances, such as Shor’s algorithm for factoring integers, was established by this body of work. The discussion continues with an emphasis on Linear Algebra and the use of Hilbert spaces, Tensor Products, Unitary Transformations, and Fourier based interference methods. As the report describes, significant advancements toward reducing the time it takes for quantum systems to perform calculations are made possible through use of Oracle construction, Spectral Decomposition, and Mea surement Theory. This comparative analysis explores the use of computational queries in deterministic, probabilistic, and quan tum models while examining the transition from classical decision tree complexities into quantum computation paradigms. Ultimately, the results of this analysis show that the early quantum algorithms must be recognized not merely as isolated computational oddities, but rather as crucial components of the development of quantum information sci ence. They provided a conceptual and mathematical framework for the modern understanding of what constitutes a quantum advantage; Moreover, they also provide valuable insight into how quantum mechanical systems influ ence both computational efficiency and complexity theory, specifically as well as future information processing system design.
Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N – 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithm’s known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch-Jozsa and Simon's problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch-Jozsa problem with probability 1 using only one oracle query, and Simon's problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch-Jozsa and Simon's problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms, and calls for further study of oracle separation between quantum and classical computation.