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

2021 ◽  
Vol 2 (1) ◽  
pp. 1-30
Author(s):  
Joseph X. Lin ◽  
Eric R. Anschuetz ◽  
Aram W. Harrow
Keyword(s):  
Graph Theory ◽  
Graph Drawing ◽  
Undirected Graph ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Graph Laplacian ◽  
Spectral Graph Theory ◽  
One Dimensional ◽  
Spectral Graph ◽  
Logical Qubits

We propose an efficient heuristic for mapping the logical qubits of quantum algorithms to the physical qubits of connectivity-limited devices, adding a minimal number of connectivity-compliant SWAP gates. In particular, given a quantum circuit, we construct an undirected graph with edge weights a function of the two-qubit gates of the quantum circuit. Taking inspiration from spectral graph drawing, we use an eigenvector of the graph Laplacian to place logical qubits at coordinate locations. These placements are then mapped to physical qubits for a given connectivity. We primarily focus on one-dimensional connectivities and sketch how the general principles of our heuristic can be extended for use in more general connectivities.

scholarly journals Coherent structure colouring: identification of coherent structures from sparse data using graph theory

2016 ◽  
Vol 811 ◽  
pp. 468-486 ◽  
Author(s):  
Kristy L. Schlueter-Kuck ◽  
John O. Dabiri
Keyword(s):  
Graph Theory ◽  
Coherent Structures ◽  
Graph Drawing ◽  
Practical Interest ◽  
Spectral Graph Theory ◽  
Graph Colouring ◽  
Time Interval ◽  
Trajectory Data ◽  
Spectral Graph ◽  
Lagrangian Flow

We present a frame-invariant method for detecting coherent structures from Lagrangian flow trajectories that can be sparse in number, as is the case in many fluid mechanics applications of practical interest. The method, based on principles used in graph colouring and spectral graph drawing algorithms, examines a measure of the kinematic dissimilarity of all pairs of fluid trajectories, measured either experimentally, e.g. using particle tracking velocimetry, or numerically, by advecting fluid particles in the Eulerian velocity field. Coherence is assigned to groups of particles whose kinematics remain similar throughout the time interval for which trajectory data are available, regardless of their physical proximity to one another. Through the use of several analytical and experimental validation cases, this algorithm is shown to robustly detect coherent structures using significantly less flow data than are required by existing spectral graph theory methods.

scholarly journals Geary’s c and Spectral Graph Theory

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2465
Author(s):  
Hiroshi Yamada
Keyword(s):  
Fourier Transform ◽  
Graph Theory ◽  
Spatial Autocorrelation ◽  
Correlation Coefficient ◽  
Graph Laplacian ◽  
Spectral Graph Theory ◽  
Algebraic Graph Theory ◽  
Spectral Graph ◽  
New Perspective ◽  
Geary’S C

Spatial autocorrelation, of which Geary’s c has traditionally been a popular measure, is fundamental to spatial science. This paper provides a new perspective on Geary’s c. We discuss this using concepts from spectral graph theory/linear algebraic graph theory. More precisely, we provide three types of representations for it: (a) graph Laplacian representation, (b) graph Fourier transform representation, and (c) Pearson’s correlation coefficient representation. Subsequently, we illustrate that the spatial autocorrelation measured by Geary’s c is positive (resp. negative) if spatially smoother (resp. less smooth) graph Laplacian eigenvectors are dominant. Finally, based on our analysis, we provide a recommendation for applied studies.

Sign in / Sign up

Export Citation Format

Share Document