The Grover search as a naturally occurring phenomenon

TL;DR

This paper presents evidence that 1/2-spin fermions can naturally perform Grover-like searches for topological defects via quantum walks, potentially enabling new applications in quantum computing without explicit oracle steps.

Contribution

It demonstrates that certain quantum walks on Dirac fermions on different grids can naturally localize around topological defects, suggesting a new paradigm for quantum search.

Findings

Fermions can localize around defects in O(√N) steps

Localization probability is about 1/ log N

Quantum walks can naturally perform search without explicit oracle steps

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 (QW), i.e. local unitary matrices that drive the evolution of a single particle on the lattice. Some QW 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 QW, one on the square grid and the other on a triangular grid reminiscent of graphene-like materials. The numerical simulations show that the walker localises around the defects in $O(\sqrt{N})$ steps with probability $O(1/\log{N})$, in line with previous QW search on the grid. The main advantage brought by those of this paper is that they could be implemented as `naturally occurring' freely propagating particles over a surface featuring topological---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.