We contribute to fulfil the long-lasting gap in the understanding of the spatial search with multiple marked vertices. The theoretical framework is that of discrete-time quantum walks (QW), i.e. local unitary matrices that drive the evolution of a single particle on the lattice. QW based search algorithms are well understood when they have to tackle the fundamental problem of finding only one marked element in a $d-$dimensional grid and it has been proven they provide a quadratic advantage over classical searching protocols. However, once we consider to search more than one element, the behaviour of the algorithm may be affected by the spatial configuration of the marked elements, due to the quantum interference among themselves and even the quantum advantage is no longer granted. Here our main contribution is twofold : (i) we provide strong numerical evidence that spatial configurations are almost all optimal; and (ii) we analytically prove that the quantum advantage with respect to the classical counterpart is not always granted and it does depend on the proportion of searched elements over the total number of grid points $\tau$. We finally providing a clear phase diagram for the QW search advantage with respect to the classical random algorithm.
翻译:理论框架是离散时间量子漫游(QW)的理论框架,即地方单一矩阵,它驱动着一个粒子在悬浮层上的演进。基于QW的搜索算法在它们必须解决在美元-美元网格中只找到一个标记元素这一根本问题,而且事实证明它们为古典搜索协议提供了四级优势。然而,当我们考虑搜索不止一个元素时,算法的行为可能受到标记元素的空间配置的影响,因为它们之间的量子干扰,甚至量子优势也不再被授予。 我们在这方面的主要贡献是双重的:(一) 我们提供强有力的数字证据,表明空间配置几乎都是最佳的;(二) 我们分析证明,古典对应方的量子优势并非总能得到,而且它取决于检索元素在总电网点上的比例 $\tau$。我们最后提供了一份清晰的QW搜索优势与古典随机算法有关的阶段图表。