The Grover search as a naturally occurring phenomenon
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We provide the first evidence that under certain conditions, electrons may naturally behave like a Grover search, looking for 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 of these are well-known to recover the (2+1)--dimensional Dirac equation in continuum limit, i.e. the free propagation of the electron. 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 a defect in O(√(N)) steps with probability O(1/N). This in line with previous QW formulations of the Grover search on the 2D grid. But these Dirac QW are `naturally occurring' and require no specific oracle step other than a hole defect in a material.
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