Using Spectral Graph Theory to Map Qubits onto Connectivity-limited Devices

Introduction to spectral graph theory

We write M ∈ Rn×n to denote that M is an n×n matrix with real elements, and v ∈ Rn to denote that v is a vector of length n. Vectors are usually taken to be column vectors unless otherwise specified. Recall that a real matrix M ∈ Rn×n represents a linear operator from Rn to Rn. In other words, given any vector v ∈ Rn, we think of M as a function which maps it to another vector w ∈ Rn, namely th...

Discovering genetic ancestry using spectral graph theory.

As one approach to uncovering the genetic underpinnings of complex disease, individuals are measured at a large number of genetic variants (usually SNPs) across the genome and these SNP genotypes are assessed for association with disease status. We propose a new statistical method called Spectral-GEM for the analysis of genome-wide association studies; the goal of Spectral-GEM is to quantify th...

Algorithm Design Using Spectral Graph Theory

Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. Laplace’s equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial optimization, computer vision, computer graphics, and machine le...

Spectral graph theory

With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices pro...

Spectral graph theory