Spatial search via an interpolated memoryless walk
Peter Friedrich Hoyer, Janet Leahy · Physical Review A · 2022
The defining feature of memoryless quantum walks is that they operate on the vertex space of a graph and therefore can be used to produce search algorithms with minimal memory. We present a memoryless walk that can find a unique marked vertex on a two-dimensional lattice. Our walk is based on the construction proposed by Falk, which tessellates the lattice with squares of size $2\ifmmode\times\else\texttimes\fi{}2$. Our walk uses minimal memory, $O(\sqrt{NlogN})$ applications of the walk operator, and outputs the marked vertex with vanishing error probability. To accomplish this, we apply a self-loop to the marked vertex---a technique we adapt from interpolated walks. We prove that with our explicit choice of self-loop weight, this forces the action of the walk asymptotically into a single rotational space. We characterize this space and as a result show that our memoryless walk produces the marked vertex with a success probability asymptotically approaching one.