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The defining feature of memoryless quantum walks is that they operate on thevertex space of a graph, and therefore can be used to produce search algorithmswith minimal memory. We present a memoryless walk that can find a unique markedvertex on a two-dimensional grid. Our walk is based on the constructionproposed by Falk, which tessellates the grid with squares of size $2 \times 2$.Our walk uses minimal memory, $O(\sqrt{N \log N})$ applications of the walkoperator, and outputs the marked vertex with vanishing error probability. Toaccomplish this, we apply a selfloop to the marked vertex - a technique weadapt from interpolated walks. We prove that with our explicit choice ofselfloop weight, this forces the action of the walk asymptotically into asingle rotational space. We characterize this space and as a result, show thatour memoryless walk produces the marked vertex with a success probabilityasymptotically approaching one.

Spatial Search via Memoryless Walk with Selfloop