Quantum algorithms for the analysis of Boolean functions have received a lot of attention over the last few years. The algebraic normal form (ANF) of a linear Boolean function can be recovered by using the Bernstein–Vazirani (BV) algorithm. No research has been carried out on quantum algorithms for learning the ANF of general Boolean functions. In this paper, quantum algorithms for learning the ANF of quadratic Boolean functions are studied. We draw a conclusion about the influences of variables on quadratic functions, so that the BV algorithm can be run on them. We study the functions obtained by inversion and zero-setting of some variables in the quadratic function and show the construction of their quantum oracle. We introduce the concept of “club” to group variables that appear in quadratic terms and study the properties of clubs. Furthermore, we propose a bunch of algorithms for learning the full ANF of quadratic Boolean functions. The most efficient algorithm, among those we propose, provides an O(n) speedup over the classical one, and the number of queries is independent of the degenerate variables.

中文翻译：

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用于学习二次布尔函数的代数范式的量子算法

在过去的几年中，用于布尔函数分析的量子算法受到了很多关注。线性布尔函数的代数范式（ANF）可以通过使用Bernstein–Vazirani（BV）算法来恢复。尚未研究用于学习通用布尔函数的ANF的量子算法。本文研究了用于学习二次布尔函数ANF的量子算法。我们得出关于变量对二次函数的影响的结论，以便可以对它们运行BV算法。我们研究了通过二次函数中的一些变量的求逆和零设置获得的函数，并说明了它们的量子预言的构造。我们将“俱乐部”的概念引入以二次方形式出现的变量分组，并研究俱乐部的属性。此外，我们提出了一系列算法来学习二次布尔函数的完整ANF。在我们提出的算法中，最有效的算法提供了O（n）比传统的要快，查询的数量与简并变量无关。