Implementation of Boolean Functions

Abstract

The main task of classical computations is the computation of functions defined in analytical or tabular form. Let us start with Boolean functions that depend on several arguments. The implementation of Boolean functions with a quantum computer is based on the construction of a quantum circuit that depends on the kind \(f(\mathbf {x})\). We represent the given function using only operations of conjunctions and addition modulo two. This representation is called algebraic normal form (ANF), or Reed–Muller expansion, or Zhegalkin polynomial. The algebraic normal form of the function \(f(\mathbf {x})\) is the sum modulo two of several elementary conjunctions of the form $$G=K_1\oplus K_2\oplus \ldots \oplus K_s,$$ where \(K_i\), \(i=1,2,\dots ,s\) are pairwise different monotone elementary conjunctions over some set of variables \(\{x_1\), ..., \(x_n\}\), where \(n=1,2,\dots \). One of the \(K_i\) can be a constant one. The greatest of ranks of elementary conjunctions included in the polynomial G is called a degree of the function. It is known that any Boolean function is uniquely represented as a Reed–Muller decomposition accurate to the order of summands in the sum and the order of cofactors in the conjunctions [1].