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arXiv:1906.06374 · International Conference on Micro- and Nano-Electronics (ICMNE)

Representation of Boolean functions in terms of quantum computation

Dmitriy V. Fastovets, Yu. I. Bogdanov, Nadezhda A. Bogdanova, Vladimir F. Lukichev

Published Mar 15, 2019• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1117/12.2522053

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•Mar 1, 2019•0 citations
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 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 (kkk-value) logic when k=pk = pk=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 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.
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.
This paper derives a simple Boolean logic represented as logic circuits. The logic is then extended to selfreferential circuits and shown to generate logic values, analogs to imaginary numbers in ordinary algebra. The generalized logic can represent both combinatorial and sequential circuits, and also demonstrates interesting properties suggestive of both classical and quantum physics, including superposition, interference, and quantum coherence. A solution is given to the Square Root of NOT problem, a function not realizable with isolated classical circuits, and it is shown how this circuit may also be interpreted to represent the equivalent quantum computer circuit. Possible interpretations of the logic for physical systems are discussed.
Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli Z operators (Ising spin operators) with the terms of the sum corresponding to the function’s Fourier expansion. For many classes of Boolean functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation, i.e., as hard as computing its number of satisfying assignments. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses each acting on a fixed number of bits as is common in constraint satisfaction problems. We show composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks, which are particularly suitable for direct implementation as classical software. We further apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results to quantum algorithms for optimization. A goal of this work is to provide a design toolkit for quantum optimization which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to provide a unified framework for the various constructions appearing in the literature.
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.