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Spatial search and the Dirac equation

Andrew M. Childs·Jeffrey Goldstone
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Abstract

We consider the problem of searching a d-dimensional lattice of N sites for a single marked location. We present a Hamiltonian that solves this problem in time of order sqrt(N) for d>2 and of order sqrt(N) log(N) in the critical dimension d=2. This improves upon the performance of our previous quantum walk search algorithm (which has a critical dimension of d=4), and matches the performance of a corresponding discrete-time quantum walk algorithm. The improvement uses a lattice version of the Dirac Hamiltonian, and thus requires the introduction of spin (or coin) degrees of freedom.

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Andrew M. Childs, Jeffrey Goldstone. 2004-05-20. Spatial search and the Dirac equation. https://doi.org/10.1103/physreva.70.042312

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