Quantum walks and quantum search on graphene lattices

Quantum walks have been very useful in developing search algorithms in quantum information, in particular for devising of spatial search algorithms. However, the construction of continuous-time quantum search algorithms in two-dimensional lattices has proved difficult, requiring additional degrees of freedom. Here, we 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. This is made possible by making a change to standard methods of marking a particular site in the lattice. Various ways of marking a site are shown to result in successful search protocols. We further establish that the search can be adapted to transfer probability amplitude across the lattice between specific lattice sites thus establishing a line of communication between these sites.

Figure 1

Left: Graphene with lattice vectors ${\underline{a}}_{1/2}$, translation vectors ${\underline{\delta }}_i$, and unit cell (dashed lines). Right: Reciprocal lattice with basis vectors ${\underline{b}}_{1/2}$, symmetry points $\underline{\mathrm{\Gamma }}$, $\underline{K}$, ${\underline{K}}^{\prime }$, $\underline{M}$, and first Brillouin zone (hexagon).

Figure 2

Dispersion relation for infinite graphene sheet (${\epsilon }_D=0$ and $v=-1$).

Figure 3

Spectrum of ${\mathbf{H}}_{\gamma }$ in Eq. (13) as a function of $\gamma$ for a $12\times 12$ cell torus ($N=288$). The spectrum is symmetric around ${\epsilon }_D=0$. Inset: Scaling of the gap $\mathrm{\Delta }={\tilde{E}}_{+}-{\tilde{E}}_{-}$ (dots) and curves $c_1/\sqrt{N}$ (solid blue), $c_2/\sqrt{NlogN}$ (dashed red) for comparison.

Figure 4

Search on $12\times 12$ cell graphene lattice with a triple-bond perturbation, using starting state $\left|s\right.〉$ from Eq. (17). For tori with $m=n$ the dynamics at each neighboring site is the same so only one is shown.

Figure 5

Numerically calculated signal transfer on a $12\times 12$ cell graphene lattice between equivalent vertices, using the communication Hamiltonian in Eq. (45). The system is initialized in $\left|{\ell }_s\right.〉$ and localizes on $\left|{\ell }_t\right.〉$. Only the sum of probabilities to be found on the neighbor vertices is shown.

Figure 6

Upper: Numerically calculated signal transfer on a $12\times 12$ cell graphene lattice between nonequivalent vertices, using the communication Hamiltonian in Eq. (45). The system is initialized in $\left|{\ell }_s\right.〉$ and localizes on $\left|{\ell }_t\right.〉$. Only the sum of probabilities to be found on the neighbor vertices is shown. Lower: Analytically calculated behavior for the same system, using the reduced Hamiltonian method and Eq. (57).

Figure 7

Numerically calculated signal transfer on a $12\times 12$ cell graphene lattice between vertices on different sublattices, using the communication Hamiltonian in Eq. (45). The system is initialized in $\left|{\ell }_s\right.〉$ and localizes on $\left|{\ell }_t\right.〉$. Only the sum of probabilities to be found on the neighbor vertices is shown. The two figures have the same source position but different positions of the target site.

Figure 8

Spectrum of single-bond search Hamiltonian in Eq. (13) using perturbation from Eq. (58) as a function of $\gamma$ for a $12\times 12$ cell torus ($N=288$). The spectrum is symmetric around ${\epsilon }_D=0$. Inset: States nearest the Dirac energy have been colored depending on their parity with respect to the $C_2$ operator: even (solid blue), odd (dashed red), undefined (dot-dashed black).

Figure 9

Search on $12\times 12$ cell graphene lattice with single-bond perturbation, using an optimal starting state. The behavior at both marked vertices is the same, as is the behavior at each of their neighboring sites.

Figure 10

Spectrum of the Hamiltonian in Eq. (62) as a function of $\gamma$ for a $12\times 12$ cell torus ($N=288$). The symmetry of the graphene spectrum is broken by the choice of the perturbation in terms of an additional site. Inset: Search on $12\times 12$ cell graphene lattice with a single additional site perturbation, using starting state $\left|s\right.〉$ from Eq. (17).

Figure 11

Example of a armchair nanotube cell. The nanotube is periodic along the vertical axis and finite in the horizontal direction with a width of $N_x$ sites.

Figure 12

Searches on an armchair nanotube ($N=320$) with triple-bond perturbation located in the bulk (upper figure) and placed near the edge of the nanotube (lower figure), using numerically calculated optimal starting state: $\left|m_o-1,B,l_o\right.〉$ (dotted blue), $\left|m_o+1,B,l_o\right.〉$ (dashed red), $\left|m_o,B,l_o\right.〉$ (dot-dashed green), sum of neighbor probabilities (solid black).

Figure 13

Examples of two finite graphene sheets, with dimensions in terms of primitive cells $\left(N_x,N_y\right)=\left(4,4\right)$. Along the vertical axis of both sheets are armchair edges. The horizontal boundaries are formed by bearded edges (left) and zigzag edges (right).

Figure 14

Spectrum of triple-bond perturbation search Hamiltonian in Eq. (13) as a function of $\gamma$ for a $10\times 10$ cell bearded graphene sheet ($N=200$). We focus only on the relevant section of the spectrum.

Figure 15

Searching on a bearded edge graphene sheet with dimensions $\left(N_x,N_y\right)=\left(10,10\right)$ using the Hamiltonian, Eq. (13). All the searches are initialized in the first unperturbed, non-edge eigenstate above the Dirac energy. The locations of the marked vertices are as follows: upper, halfway along the left armchair edge; middle, center of the sheet; lower, at the midpoint of the lower bearded edge and a third of the way along the vertical axis. (Colors and line styles as in Fig. 12.)