Abstract The fundamental relationship between quantum physics and discrete mathematics is examined. A method for representing Boolean functions in the form of unitary transformations is described. The question of the connection of Zhegalkin polynomials defining the algebraic normal form of a Boolean function with quantum circuits is considered. It is shown that the quantum information language provides a simple algorithm for constructing the Zhegalkin polynomial based on the truth table. The developed methods and algorithms are generalized to the case of an arbitrary Boolean function with a multibit domain of definition and a multibit set of values, as well as to the case of multivalued ($$k$$-value) logic when $$k = p$$ is a prime number. The developed approach is important for the implementation of quantum computer technologies and is the foundation for the transition from classical computer logic to quantum hardware.
Abstract The fundamental relationship between quantum physics and discrete mathematics is examined. A method for representing Boolean functions in the form of unitary transformations is described. The question of the connection of Zhegalkin polynomials defining the algebraic normal form of a Boolean function with quantum circuits is considered. It is shown that the quantum information language provides a simple algorithm for constructing the Zhegalkin polynomial based on the truth table. The developed methods and algorithms are generalized to the case of an arbitrary Boolean function with a multibit domain of definition and a multibit set of values, as well as to the case of multivalued (k-value) logic when k=p is a prime number. The developed approach is important for the implementation of quantum computer technologies and is the foundation for the transition from classical computer logic to quantum hardware.
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal form of Boolean function, and quantum logic circuits is described. It is shown that quantum information approach provides simple algorithm to construct Zhegalkin polynomial using truth table. Developed methods and algorithms have arbitrary Boolean function generalization with multibit input and multibit output. Such generalization allows us to use many-valued logic (k-valued logic, where k is a prime number). Developed methods and algorithms can significantly improve quantum technology realization. The presented approach is the baseline for transition from classical machine logic to quantum hardware.
Abstract Quantum computing has gained attention in recent years due to the significant progress in quantum computing technology. Today many companies like IBM, Google and Microsoft have developed quantum computers and simulators for research and commercial use. The development of quantum techniques and algorithms is essential to exploit the full power of quantum computers. In this paper we propose a simple visual technique (we call Q-Map) for quantum realization of classical Boolean logic circuits. The proposed method utilizes concepts from Boolean algebra to produce a quantum circuit with minimal number of quantum gates.
We first propose herein a novel parallel computation, even though today's algorithm methodology for quantum computing, for all of the combinations of values in variables of a logical function. Our concern so far has been to obtain an attribute of some function. In fact such a task is only for one task problem solving. However, we could treat positively the plural evaluations of some logic function in parallel instead of testing the function for finding out its attribute. In fact, these evaluations of the function are naturally included and evaluated, in parallel, in normal quantum computing discussing a function in a Boolean algebra stemmed from atoms in it. As is naturally understandable with mathematics, quantum computing with qubit systems naturally is included and exemplified by a Boolean algebra, which treats only both 0 and 1. Namely, the theory in this paper is quite natural in logical sense even though physics domain. Therefore, quantum computing has an ability to solve some mathematical problems described in a Boolean algebra. The reason why we positively introduce a Boolean algebra here is because we have multiple evaluations of a function in quantum computing general.
Digital computers are based on devices that can take on only two states, one of which is denoted by 0 and the other by 1. By concatenating several 0s and is together, 0-1 combinations can be formed to represent as many different entities as desired. A combination containing a single 0 or 1 is called a bit. In general, n bits can be used to distinguish among 2 n distinct entities and each addition of a bit doubles the number of possible combinations. Computers use strings of bits to represent numbers, letters, punctuation marks, and any other useful pieces of information. In a classical computer, the processing of information is done by logic gate. A logic gate maps the state of its input bits into another state according to a truth table. Quantum computers require quantum logic, something fundamentally different to classical Boolean logic. This difference leads to a greater efficiency of quantum computation over its classical counterpart. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Quantum computing represents one of the most significant paradigms shifts in the history of computation, promising to solve classes of problems that remain intractable for classical machines. Unlike classical computing, which is built on Boolean algebra and binary logic, quantum computing is fundamentally a mathematical discipline rooted in linear algebra, complex vector spaces, and probability theory. This article examines the mathematical structures that underpin quantum computation, including qubits, superposition, entanglement, unitary transformations, and quantum gates, and explores how these structures enable algorithms such as Shor's and Grover's to outperform their classical counterparts. The discussion further extends to quantum error correction, the challenges of decoherence, and the interdisciplinary convergence of mathematics and computer science that is shaping next-generation computing architectures. The article concludes with a reflection on open mathematical problems and the future trajectory of quantum information science.
Quantum computing represents one of the most significant paradigms shifts in the history of computation, promising to solve classes of problems that remain intractable for classical machines. Unlike classical computing, which is built on Boolean algebra and binary logic, quantum computing is fundamentally a mathematical discipline rooted in linear algebra, complex vector spaces, and probability theory. This article examines the mathematical structures that underpin quantum computation, including qubits, superposition, entanglement, unitary transformations, and quantum gates, and explores how these structures enable algorithms such as Shor\\\'s and Grover\\\'s to outperform their classical counterparts. The discussion further extends to quantum error correction, the challenges of decoherence, and the interdisciplinary convergence of mathematics and computer science that is shaping next-generation computing architectures. The article concludes with a reflection on open mathematical problems and the future trajectory of quantum information science.