### Abstract

Graph comparison is an established NP-hard problem. In this paper, we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for any pair of unlabelled graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part, we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with such a vanishingly small amplitude that even quantum algorithms such as Grover’s search are not fast enough to distil it efficiently. In order to extract the information, we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires O(n^{3}log ^{3}(n) log log (n)) elementary quantum gates, and the nonlinear evolution under the Gross–Pitaevskii equation has a time scaling of O(1gn2log3(n)loglog(n)), where n is the number of vertices in each graph and g is the strength of the Gross–Pitaevskii nonlinearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.

Original language | English |
---|---|

Article number | 302 |

Journal | Quantum Information Processing |

Volume | 18 |

Issue number | 10 |

DOIs | |

Publication status | Published - 1 Oct 2019 |

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### Cite this

*Quantum Information Processing*,

*18*(10), [302]. https://doi.org/10.1007/s11128-019-2407-2

}

*Quantum Information Processing*, vol. 18, no. 10, 302. https://doi.org/10.1007/s11128-019-2407-2

**Graph comparison via nonlinear quantum search.** / Chiew, M.; de Lacy, K.; Yu, C. H.; Marsh, S.; Wang, J. B.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Graph comparison via nonlinear quantum search

AU - Chiew, M.

AU - de Lacy, K.

AU - Yu, C. H.

AU - Marsh, S.

AU - Wang, J. B.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - Graph comparison is an established NP-hard problem. In this paper, we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for any pair of unlabelled graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part, we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with such a vanishingly small amplitude that even quantum algorithms such as Grover’s search are not fast enough to distil it efficiently. In order to extract the information, we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires O(n3log 3(n) log log (n)) elementary quantum gates, and the nonlinear evolution under the Gross–Pitaevskii equation has a time scaling of O(1gn2log3(n)loglog(n)), where n is the number of vertices in each graph and g is the strength of the Gross–Pitaevskii nonlinearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.

AB - Graph comparison is an established NP-hard problem. In this paper, we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for any pair of unlabelled graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part, we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with such a vanishingly small amplitude that even quantum algorithms such as Grover’s search are not fast enough to distil it efficiently. In order to extract the information, we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires O(n3log 3(n) log log (n)) elementary quantum gates, and the nonlinear evolution under the Gross–Pitaevskii equation has a time scaling of O(1gn2log3(n)loglog(n)), where n is the number of vertices in each graph and g is the strength of the Gross–Pitaevskii nonlinearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.

KW - Graph comparison

KW - Nonlinear quantum search

KW - Permutations

KW - Quantum computing

UR - http://www.scopus.com/inward/record.url?scp=85070981659&partnerID=8YFLogxK

U2 - 10.1007/s11128-019-2407-2

DO - 10.1007/s11128-019-2407-2

M3 - Article

VL - 18

JO - Quantum Information Processing

JF - Quantum Information Processing

SN - 1570-0755

IS - 10

M1 - 302

ER -