We analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set of m such ‘chosen’ elements (out of \(n>m)\). Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change \(\varphi \), all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers \(\varphi \) and \(\alpha \) which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on \(\alpha \). In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change \(\varphi ,\) and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm (\(m=1,\phi =\pi \)), which are of the form, the found previously, unitary operators.

我们分析了经典Grover算法的数学结构，并将其置于复数上的线性代数的框架内。从某种意义上讲，我们也将其概括为：我们不是在寻找数据集中的一个“选择”元素（有时称为“解决方案”），而是在寻找一组m个这样的“选择”元素（在\（n> m）中\）。此外，我们不假定所谓的初始叠加是统一的。我们还假设我们有一个预言，它通过适当的相变\（\ varphi \）对所有这些“选择”元素进行“标记”。在本文的第一部分，我们构造了一个唯一的unit运算符，该运算符可以一步选择所有“选择的”元素。构造的运算符由数字\（\ varphi \）唯一定义和\（\ alpha \），它是初始叠加系数的特定函数。而且，它是由两个所谓的反射组成的形式。该结果纯粹是理论上的，因为达到此目标所需的相变很大程度上取决于\（\ alpha \）。在第二部分中，我们构造了of元算子，该given元算子在给定正交versors集合的情况下具有两个或多个反射的组合形式（概括构造的算子）。我们发现这些操作的属性，尤其是它们的组成。此外，通过考虑固定的“便捷”相变\（\ varphi，\）然后通过顺序应用如此构造的运算符，我们找到了几率来找到这些“选择”的元素。我们应用此知识来研究Grover算法的一般化（\（m = 1，\ phi = \ pi \）），其形式为先前发现的unit算子。