Optimality conditions for spatial search with multiple marked vertices

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), \textit{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 and even the quantum advantage is no longer guaranteed. Here our main contribution is threefold~: (i)~we provide \textit{sufficient conditions for optimality} for a multi-items QWSearch algorithm~; (ii)~we provide analytical evidences that \textit{almost, but not all} spatial configurations with multiple marked elements are optimal; and (iii)~we numerically show that the computational advantage with respect to the classical counterpart is not always certain and it does depend on the proportion of searched elements over the total number of grid points.