Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene

Abstract

A series of quantum search algorithms have been proposed recently providing an algebraic speedup compared to classical search algorithms from N to ffiffiffiffi N p , where N is the number of items in the search space. In particular, devising searches on regular lattices has become popular in extending Grover's original algorithm to spatial searching. Working in a tight-binding setup, it could be demonstrated, theoretically, that a search is possible in the physically relevant dimensions 2 and 3 if the lattice spectrum possesses Dirac points. We present here a proof of principle experiment implementing wave search algorithms and directed wave transport in a graphene lattice arrangement. 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. The experiment is implemented using classical waves in a microwave setup containing weakly coupled dielectric resonators placed in a honeycomb arrangement, i.e., artificial graphene. Furthermore, we investigate the scaling behavior experimentally using linear chains. Introduction.-Currently, one of the most fruitful branches of quantum information is the field of quantum search algorithms. It started with Grover's work [1] describing a search algorithm for unstructured databases, which has been implemented experimentally in NMR [2,3] and in optical experiments [4]. More recently, spatial quantum search algorithms have been proposed based on the quantum walk mechanism [5,6]. All of these algorithms can achieve up to quadratic speedup compared to the corresponding classical search. For quantum searches on generic d-dimensional lattices, certain restrictions have been observed, however, depending on whether the underlying quantum walk is discrete [7] or continuous [8]. While effective search algorithms for discrete walks on square lattices have been reported for d ≥ 2 [9,10], continuoustime quantum search algorithms on the same lattice show speedup compared to the classical search only for d ≥ 4 [11]. Experimental implementations of discrete quantum walks need time stepping mechanisms such as laser pulses [12][13][14][15][16][17]. By switching to a continuous-time evolution based, for example, on tight-binding coupling between sites, one can avoid time discretization in an experiment. It has been shown in Ref.[18] that continuous-time quantum search in 2D is indeed possible when performed near the Dirac point in graphene or, more generally, for lattices with a cone structure in the dispersion relation [19], i.e., a linear growth of the density of states (DOS). This effect adds a

Experimental observation of wave localization at the Dirac frequency in a two-dimensional photonic crystal microcavity

“…The electric current imported from the side of the PC excites the localized modes at the Dirac frequency. This structure is different from that of Refs [21][22][23]26], where the PCs are thin slabs with non-negligible thickness effects.…”

“…Edge modes were found in these studies. Meanwhile, J. Böhm et al [26] demonstrated the implementations of the continuous quantum wave search algorithms and directed wave transport in artificial graphene lattice. In this ingenious experiment, localized search states were brought into resonance with an extended lattice state near the Dirac point.…”

Staggered quantum walk on hexagonal lattices

“…All these lattices have attracted the attention of researchers. For instance, coined quantum walks were analyzed on the two-dimensional square lattice [10][11][12][13][14][15], hexagonal lattice [16][17][18][19], and triangular lattice [19][20][21]. Since the seminal paper by Shenvi et al [22], coined quantum walks are being used for search algorithms [23].…”

Driven discrete time quantum walks

“…The advantages of this scheme is that it requires no extended initial state and also the walkers do not oscillate in-and-out of the marked vertex, thus we can measure at any time, after the initial waiting period. However, like another experimental scheme [13], we have no reduction in the number of vertices needed so there is no resource advantage over the Grover search.…”

“…They have also been used to model transport processes such as quantum heat transport [7] and percolation [8]. Various experimental implementations have also been explored in optical delay loops [9][10][11], trapped atoms [12], microwave scattering [13] and waveguide arrays [14][15][16][17].…”