Coined Quantum Walks Lift the Cospectrality of Graphs and Trees

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.

[1]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[2]  Xavier L. Hubaut,et al.  Strongly regular graphs , 1975, Discret. Math..

[3]  J. Cirac,et al.  Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network , 1996, quant-ph/9611017.

[4]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.

[5]  Nayak Ashwin,et al.  Quantum Walk on the Line , 2000 .

[6]  Edward Farhi,et al.  An Example of the Difference Between Quantum and Classical Random Walks , 2002, Quantum Inf. Process..

[7]  Alistair Sinclair,et al.  Algorithms for Random Generation and Counting: A Markov Chain Approach , 1993, Progress in Theoretical Computer Science.

[8]  Russell Merris,et al.  Almost all trees are co-immanantal , 1991 .

[9]  Simone Severini.  On the Digraph of a Unitary Matrix , 2003, SIAM J. Matrix Anal. Appl..

[10]  Andris Ambainis,et al.  One-dimensional quantum walks , 2001, STOC '01.

[11]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[12]  Julia Kempe,et al.  Quantum Random Walks Hit Exponentially Faster , 2002, ArXiv.

[13]  K. Birgitta Whaley,et al.  Quantum random-walk search algorithm , 2002, quant-ph/0210064.

[14]  L. Beineke,et al.  Topics in algebraic graph theory , 2004 .

[15]  Braunstein.  Quantum teleportation without irreversible detection. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[16]  Andris Ambainis,et al.  QUANTUM WALKS AND THEIR ALGORITHMIC APPLICATIONS , 2003, quant-ph/0403120.

[17]  Andris Ambainis,et al.  Quantum walks on graphs , 2000, STOC '01.

[18]  Antonio Robles-Kelly,et al.  Edit distance from graph spectra , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[19]  Edwin R. Hancock,et al.  String Edit Distance, Random Walks And Graph Matching , 2004, Int. J. Pattern Recognit. Artif. Intell..

[20]  Marco Gori,et al.  Graph matching using random walks , 2004, ICPR 2004.

[21]  Andris Ambainis.  Quantum Walk Algorithm for Element Distinctness , 2004, FOCS.