A quantum algorithm for solving eigenproblem of the Laplacian matrix of a fully connected weighted graph

TL;DR

This paper presents a quantum algorithm that efficiently solves the eigenproblem of the Laplacian matrix of fully connected weighted graphs, offering significant speedups over classical methods in data science and machine learning applications.

Contribution

The paper introduces a novel quantum algorithm utilizing block-encoding and quantum phase estimation to solve Laplacian eigenproblems with polynomial and exponential speedups.

Findings

Achieves polynomial speedup on the number of vertices

Provides exponential speedup on the dimension of each vertex

Extends to normalized Laplacian matrices

Abstract

Solving eigenproblem of the Laplacian matrix of a fully connected weighted graph has wide applications in data science, machine learning, and image processing, etc. However, this is very challenging because it involves expensive matrix operations. Here, we propose an efficient quantum algorithm to solve it based on a assumption that the element of each vertex and its norms can be effectively accessed via a quantum random access memory data structure. Specifically, we adopt the optimal Hamiltonian simulation technique based on the block-encoding framework to implement the quantum simulation of the Laplacian matrix. Then, the eigenvalues and eigenvectors of the Laplacian matrix are extracted by the quantum phase estimation algorithm. The core of our entire algorithm is to construct the block-encoding of the Laplacian matrix. To achieve this, we propose in detail how to construct the block-encodings of operators containing the information of the weight matrix and the degree matrix respectively, and further obtain the block-encoding of the Laplacian matrix. Compared with its classical counterpart, our algorithm has a polynomial speedup on the number of vertices and an exponential speedup on the dimension of each vertex. We also show that our algorithm can be extended to solve the eigenproblem of symmetric (non-symmetric) normalized Laplacian matrix.