Continuous-time quantum walks (CTQWs) play a crucial role in quantum computing, especially for designing quantum algorithms. However, how to efficiently implement CTQWs is a challenging issue. In this paper, we study the implementation of CTQWs on sparse graphs, i.e., constructing efficient quantum circuits for implementing the unitary operator ${e}^{\ensuremath{-}iHt}$, where $H=\ensuremath{\gamma}A$ ($\ensuremath{\gamma}$ is a constant and
A corresponds to the adjacency matrix of a graph). Our result is, for a
d-sparse graph with
N vertices and evolution time
t, we can approximate ${e}^{\ensuremath{-}iHt}$ by a quantum circuit with gate complexity $({d}^{3}\ensuremath{\parallel}H\ensuremath{\parallel}tN{log}_{2}{(N))}^{1+o(1)}$, compared to the general Pauli decomposition, which scales like $(\ensuremath{\parallel}H\ensuremath{\parallel}t{N}^{4}{log}_{2}{(N))}^{1+o(1)}$. For sparse graphs, for instance,
d=O(1), we obtain a noticeable improvement. Interestingly, our technique is related to graph decomposition. More specifically, we decompose the graph into a union of star graphs, and correspondingly, the Hamiltonian
H can be represented as the sum of some Hamiltonians
Hj, where each ${e}^{\ensuremath{-}i{H}_{j}t}$ is a CTQW on a star graph which can be implemented efficiently.
In this paper, we study coherent exciton transport of continuous-time quantum walks on star graphs.Exact analytical results of the transition probabilities are obtained by means of the Gram-Schmidt orthonormalization of the eigenstates.Our results show that the coherent exciton transport displays perfect revivals and strong localization on the initial node.When the initial excitation starts at the central node, the transport on a star graph is equivalent to the transport on a complete graph of the same size.