Quantum Search on Simplicial Complexes

Abstract

In this paper, we propose an extension of quantum searches on graphs driven by quantum walks to simplicial complexes. To this end, we newly define a quantum walk on simplicial complex which is an alternative of preceding studies by authors. We show that the quantum search on the specific simplicial complex corresponding to the triangulation of $n$-dimensional unit square driven by this new simplicial quantum walk works well, namely, a marked simplex can be found with probability $1+o(1)$ with in a time $O(\sqrt{N})$, where $N$ is the number of simplices with the dimension of marked simplex.