Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
The anatomy of a large-scale hypertextual Web search engine
WWW7 Proceedings of the seventh international conference on World Wide Web 7
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
An Example of the Difference Between Quantum and Classical Random Walks
Quantum Information Processing
On the Digraph of a Unitary Matrix
SIAM Journal on Matrix Analysis and Applications
Edit Distance From Graph Spectra
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
Graph matching using Random Walks
ICPR '04 Proceedings of the Pattern Recognition, 17th International Conference on (ICPR'04) Volume 3 - Volume 03
Quantum Walk Algorithm for Element Distinctness
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A Schrödinger Wave Equation Approach to the Eikonal Equation: Application to Image Analysis
EMMCVPR '09 Proceedings of the 7th International Conference on Energy Minimization Methods in Computer Vision and Pattern Recognition
A correspondence measure for graph matching using the discrete quantum walk
GbRPR'07 Proceedings of the 6th IAPR-TC-15 international conference on Graph-based representations in pattern recognition
Graph embedding using quantum commute times
GbRPR'07 Proceedings of the 6th IAPR-TC-15 international conference on Graph-based representations in pattern recognition
Graph similarity using interfering quantum walks
CAIP'07 Proceedings of the 12th international conference on Computer analysis of images and patterns
Approximate axial symmetries from continuous time quantum walks
SSPR'12/SPR'12 Proceedings of the 2012 Joint IAPR international conference on Structural, Syntactic, and Statistical Pattern Recognition
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In this paper we consider the problem of distinguishing graphs that are cospectral with respect to the standard adjacency and Laplacian matrix representations. Borrowing ideas from the field of quantum computing, we define a new matrix based on paths of the coined quantum walk. Quantum walks exhibit interference effects and their behaviour is markedly different to that of classical random walks. We show that the spectrum of this new matrix is able to distinguish many graphs which cannot be distinguished by standard spectral methods. We pay particular attention to strongly regular graphs; if a pair of strongly regular graphs share the same parameter set then there is no efficient algorithm that is proven to be able distinguish them. We have tested the method on large families of co-parametric strongly regular graphs and found it to be successful in every case. We have also tested the spectra's performance when used to give a distance measure for inexact graph matching tasks.