Implementation of Continuous-Time Quantum Walk on Sparse Graph

TL;DR

This paper presents an efficient method for implementing continuous-time quantum walks on sparse graphs by decomposing the graph into star graphs, significantly reducing the quantum circuit complexity compared to general methods.

Contribution

The authors introduce a graph decomposition technique that enables efficient quantum circuit implementation of CTQWs on sparse graphs, improving scalability over previous approaches.

Findings

Quantum circuit complexity scales as (d^3 ||H|| t N log N)^{1+o(1)} for sparse graphs.

Decomposition into star graphs allows efficient implementation of CTQWs.

Significant complexity reduction for graphs with bounded degree d=O(1).

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 implementation of CTQWs on sparse graphs, i.e., constructing efficient quantum circuits for implementing the unitary operator $e^{-iHt}$, where $H=\gamma A$ ($\gamma$ 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 $e^{-iHt}$ by a quantum circuit with gate complexity $(d^3 \|H\| t N \log N)^{1+o(1)}$, compared to the general Pauli decomposition, which scales like $(\|H\| t N^4 \log N)^{1+o(1)}$. 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 $e^{-iH_jt}$ is a CTQW on a star graph which can be implemented efficiently.