Discrete-time quantum walks with the Dirac equation as continuum limit localize on topological defects in O(√N) steps with probability O(1/log N), numerically matching Grover search scaling.
The Dirac equation as a quantum walk over the honeycomb and triangular lattices
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abstract
A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to well-known physics partial differential equations, such as the Dirac equation. We show that these simulation results need not rely on the grid: the Dirac equation in $(2+1)$--dimensions can also be simulated, through local unitaries, on the honeycomb or the triangular lattice. The former is of interest in the study of graphene-like materials. The latter, we argue, opens the door for a generalization of the Dirac equation to arbitrary discrete surfaces.
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The Grover search as a naturally occurring phenomenon
Discrete-time quantum walks with the Dirac equation as continuum limit localize on topological defects in O(√N) steps with probability O(1/log N), numerically matching Grover search scaling.