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.

中文翻译：

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布尔代数与量子信息学的关系

摘要

研究了量子物理学与离散数学之间的基本关系。描述了一种以unit变换形式表示布尔函数的方法。考虑了定义布尔函数的代数范式与量子电路的Zhegalkin多项式的连接问题。结果表明，量子信息语言为基于真值表的Zhegalkin多项式构造提供了一种简单的算法。所开发的方法和算法可以推广到具有定义的多位域和值的多位集合的任意布尔函数的情况，以及当\（k时）具有多值（\（k \）- value）逻辑的情况。 = p \）是素数。所开发的方法对于实现量子计算机技术非常重要，并且是从经典计算机逻辑向量子硬件过渡的基础。