Optimality conditions for spatial search with multiple marked vertices

We contribute to fulfill the long-lasting gap in understanding of spatial search with multiple marked vertices. The theoretical framework is that discrete-time quantum walks (QW), i.e., local unitary matrices drive evolution a single particle on lattice. QW based algorithms are well understood when they have tackle fundamental problem finding only one element $d\text{\ensuremath{-}}\text{dimensional}$ grid and it has been proven provide quadratic advantage over classical searching protocols. However, once we consider more than element, behavior algorithm may be affected by configuration elements even no longer guaranteed. Here our main contribution threefold: (i) sufficient conditions for optimality multi-items QWSearch algorithm; (ii) analytical evidence almost, but not all configurations optimal; (iii) numerically show computational respect counterpart always certain does depend proportion searched total number points.