Year: 2024
Type: article
Abstract: 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 ${H}_{j}$, where each ${e}^{\ensuremath{-}i{H}_{j}t}$ is a CTQW on a star graph which can be implemented efficiently.
Source: Physical Review A
Authors: Zhaoyang Chen, Guanzhong Li, Lvzhou Li
Institution: Sun Yat-sen University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G4905797547, G5675855966, G5718002867
OpenAlex ID: W4404414276