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.
From quantum cellular automata to quantum lattice gases
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abstract
A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas.
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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.