Grover Search as a Naturally Occurring Phenomenon

Abstract

We provide first evidence that under certain conditions, 1/2-spin fermions may naturally behave like a Grover search, looking for topological defects in a material. The theoretical framework is that of discrete-time quantum walks (QWs), i.e., local unitary matrices that drive the evolution of a single particle on the lattice. Some QWs are well known to recover the (2+1)-dimensional Dirac equation in continuum limit, i.e., the free propagation of the 1/2-spin fermion. We study two such Dirac QWs, one on the square grid and the other on a triangular grid reminiscent of graphenelike materials. The numerical simulations show that the walker localizes around the defects in O(N) steps with probability O(1/logN), in line with previous QW search on the grid. The main advantage brought by those of this Letter is that they could be implemented as "naturally occurring" freely propagating particles over a surface featuring topological defects - without the need for a specific oracle step. From a quantum computing perspective, however, this hints at novel applications of QW search: instead of using them to look for "good" solutions within the configuration space of a problem, we could use them to look for topological properties of the entire configuration space.