Abstract: A series of quantum search algorithms have been proposed recently providing a quadratic speedup compared to classical search algorithms. In particular, devising searches on regular lattices has become popular extending Grover’s original algorithm to spatial searching. However, the construction of continuous-time quantum search algorithms in two-dimensional lattices has proved difficult, requiring additional degrees of freedom. In this talk, I will demonstrate that a continuous-time quantum walk search is possible in two dimensions by changing the search topology to a graphene lattice, utilizing the Dirac point in the energy spectrum. The idea is based on bringing localized search states into resonance with an extended lattice state in an energy region of low spectral density—namely, at or near the Dirac point. I will discuss particular set-ups for a successful search algorithm and various other possible applications such as sensitive switching devices and using quantum searches for directed communication. I will also present a proof of principle experiment implementing wave search algorithms and directed wave transport. The experiment uses classical waves in a microwave setup containing weakly coupled dielectric resonators placed in a honeycomb arrangement, i.e., artificial graphene.

Quantum Search on Graphene Lattices, Iain Foulger, Sven Gnutzmann, and Gregor Tanner, PRL 112, 070504 (2014).
Quantum walks and quantum search on graphene lattices, Iain Foulger, Sven Gnutzmann, and Gregor Tanner, PHYSICAL REVIEW A 91, 062323 (2015).
Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene, Julian Böhm, Matthieu Bellec, Fabrice Mortessagne, Ulrich Kuhl,,* Sonja Barkhofen, Stefan Gehler, Hans-Jürgen Stöckmann, Iain Foulger, Sven Gnutzmann, and Gregor Tanner, PRL 114, 110501 (2015).