Comparing Algorithms for Graph Isomorphism Using Discrete and ContinuousTime Quantum Random Walks
Abstract
Berry and Wang [Phys. Rev. A 83, 042317 (2011)] show numerically that a discretetime quan tum random walk of two noninteracting particles is able to distinguish some nonisomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuoustime quantum walks of two noninteracting parti cles cannot. We show analytically and numerically that even singleparticle discretetime quantum random walks can distinguish some strongly regular graphs, though not as many as twoparticle noninteracting discretetime walks. Additionally, we demonstrate how, given the same quantum random walk, subtle di erences in the graph certi cate construction algorithm can nontrivially im pact the walk's distinguishing power. We also show that no continuoustime walk of a xed number of particles can distinguish all strongly regular graphs when used in conjunction with any of the graph certi cates we consider. We extend this constraint to discretetime walks of xed numbers of noninteracting particles for one kind of graph certi cate; it remains an open question as to whether or not this constraint applies to the other graph certi cates we consider.
 Authors:

 Univ. of Wisconsin, Madison, WI (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1259509
 Report Number(s):
 SAND20156076J
Journal ID: ISSN 15461955; SICI 15461955(20130701)10:7L.1653;1
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational and Theoretical Nanoscience
 Additional Journal Information:
 Journal Volume: 10; Journal Issue: 7; Journal ID: ISSN 15461955
 Publisher:
 American Scientific Publishers
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
Citation Formats
Rudinger, Kenneth, Gamble, John King, Bach, Eric, Friesen, Mark, Joynt, Robert, and Coppersmith, S. N. Comparing Algorithms for Graph Isomorphism Using Discrete and ContinuousTime Quantum Random Walks. United States: N. p., 2013.
Web. doi:10.1166/jctn.2013.3105.
Rudinger, Kenneth, Gamble, John King, Bach, Eric, Friesen, Mark, Joynt, Robert, & Coppersmith, S. N. Comparing Algorithms for Graph Isomorphism Using Discrete and ContinuousTime Quantum Random Walks. United States. doi:10.1166/jctn.2013.3105.
Rudinger, Kenneth, Gamble, John King, Bach, Eric, Friesen, Mark, Joynt, Robert, and Coppersmith, S. N. Mon .
"Comparing Algorithms for Graph Isomorphism Using Discrete and ContinuousTime Quantum Random Walks". United States. doi:10.1166/jctn.2013.3105. https://www.osti.gov/servlets/purl/1259509.
@article{osti_1259509,
title = {Comparing Algorithms for Graph Isomorphism Using Discrete and ContinuousTime Quantum Random Walks},
author = {Rudinger, Kenneth and Gamble, John King and Bach, Eric and Friesen, Mark and Joynt, Robert and Coppersmith, S. N.},
abstractNote = {Berry and Wang [Phys. Rev. A 83, 042317 (2011)] show numerically that a discretetime quan tum random walk of two noninteracting particles is able to distinguish some nonisomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuoustime quantum walks of two noninteracting parti cles cannot. We show analytically and numerically that even singleparticle discretetime quantum random walks can distinguish some strongly regular graphs, though not as many as twoparticle noninteracting discretetime walks. Additionally, we demonstrate how, given the same quantum random walk, subtle di erences in the graph certi cate construction algorithm can nontrivially im pact the walk's distinguishing power. We also show that no continuoustime walk of a xed number of particles can distinguish all strongly regular graphs when used in conjunction with any of the graph certi cates we consider. We extend this constraint to discretetime walks of xed numbers of noninteracting particles for one kind of graph certi cate; it remains an open question as to whether or not this constraint applies to the other graph certi cates we consider.},
doi = {10.1166/jctn.2013.3105},
journal = {Journal of Computational and Theoretical Nanoscience},
number = 7,
volume = 10,
place = {United States},
year = {2013},
month = {7}
}
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