Abstract
We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarse-grained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.
References
[1] Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. In STOC’11-Proceedings of the 43rd ACM Symposium on Theory of Computing, 333-342, ACM, New York, 2011.10.1145/1993636.1993682Search in Google Scholar
[2] Dorit Aharonov, Itai Arad, Elad Eban, and Zeph Landau. Polynomial quantum algorithms for additive approximations of the Potts model and other points of the Tutte plane. arXiv preprint quant-ph/0702008, 2007.Search in Google Scholar
[3] László Babai. Graph isomorphism in quasipolynomial time. In STOC’16-Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, 684-697, ACM, New York, 2016.10.1145/2897518.2897542Search in Google Scholar
[4] L. Babai, D.Yu. Grigoryev and D.M. Mount. Isomorphism of graphs with bounded eigenvalue multiplicities. In STOC’82-Proceedings of the 14th ACM symposium on Theory of Computing, 310–324, ACM, New York,1982.10.1145/800070.802206Search in Google Scholar
[5] George A Baker Jr. Drum shapes and isospectral graphs. J. Mathematical Phys., 7 (1966), no 12, 2238–2242.10.1063/1.1704911Search in Google Scholar
[6] Alexander Barvinok. Combinatorics and Complexity of Partition Functions, Algorithms and Combinatorics, 30. Springer, Cham, 2016. vi+303 pp.10.1007/978-3-319-51829-9Search in Google Scholar
[7] Scott D Berry and Jingbo B Wang. Two-particle quantum walks: Entanglement and graph isomorphism testing. Phys. Rev. A, 83 (2011), no. 4, 042317, 12 pp.Search in Google Scholar
[8] Andreas Björklund, Brajesh Gupt, and Nicolás Quesada. A faster hafnian formula for complex matrices and its benchmarking on a supercomputer. ACM J. Exp. Algorithmics, 24 (2019), Art. 1.11, 17 pp.10.1145/3325111Search in Google Scholar
[9] Kamil Brádler, Pierre-Luc Dallaire-Demers, Patrick Rebentrost, Daiqin Su, and Christian Weedbrook. Gaussian boson sampling for perfect matchings of arbitrary graphs. Physical Review A, 98 (2018), no. 3, 032310, 15 pp. 2018.Search in Google Scholar
[10] Kamil Brádler, Robert Israel, Maria Schuld, Daiqin Su. A duality at the heart of Gaussian boson sampling, arXiv preprint arXiv:1910.04022, 2019.Search in Google Scholar
[11] Fernando GSL Brandão and Krysta M Svore. Quantum speed-ups for solving semidefinite programs. 58th Annual IEEE Symposium on Foundations of Computer Science-FOCS 2017, 415-426, IEEE Computer Soc., Los Alamitos, CA, 201710.1109/FOCS.2017.45Search in Google Scholar
[12] Fernando GSL Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M Svore, and Xiaodi Wu. Exponential quantum speed-ups for semidefinite programming with applications to quantum learning. arXiv preprint arXiv:1710.02581, 2017.Search in Google Scholar
[13] Andries E. Brouwer. Parameters of strongly regular graphs. https://www.win.tue.nl/~aeb/graphs/srg/srgtab.html, June 2017.Search in Google Scholar
[14] Eduardo R Caianiello. On quantum field theory I: explicit solution of Dyson’s equation in electrodynamics without use of Feynman graphs. Nuovo Cimento (9) 10 (1953), 1634-1652.10.1007/BF02781659Search in Google Scholar
[15] Cristian S Calude, Michael J Dinneen, and Richard Hua. QUBO formulations for the graph isomorphism problem and related problems. Theoret. Comput. Sci. 701 (2017), 54-69.Search in Google Scholar
[16] Andrew M Childs, Robin Kothari, and Rolando D Somma. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM J. Comput. 46 (2017), no. 6, 1920-1950.Search in Google Scholar
[17] Brendan L Douglas and Jingbo B Wang. A classical approach to the graph isomorphism problem using quantum walks. J. Phys. A 41 (2008), no. 7, 075303, 15 pp.10.1088/1751-8113/41/7/075303Search in Google Scholar
[18] David Emms, Simone Severini, Richard C. Wilson, and Edwin R. Hancock. Coined quantum walks lift the cospectrality of graphs and trees. In: Rangarajan A., Vemuri B., Yuille A.L. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2005. Lecture Notes in Computer Science, vol 3757. Springer, Berlin, Heidelberg.10.1007/11585978_22Search in Google Scholar
[19] Michael H Freedman, Alexei Kitaev, and Zhenghan Wang. Simulation of topological field theories by quantum computers. Comm. Math. Phys. 227 (2002), no. 3, 587-603.Search in Google Scholar
[20] Michael H Freedman, Michael Larsen, and Zhenghan Wang. A modular functor which is universal for quantum computation. Comm. Math. Phys. 227 (2002), no. 3, 605–622.Search in Google Scholar
[21] Frank Gaitan and Lane Clark. Graph isomorphism and adiabatic quantum computing. Phys. Rev. A, 89 (2014), no. 2, 022342, 20 pp.10.1103/PhysRevA.89.022342Search in Google Scholar
[22] John King Gamble, Mark Friesen, Dong Zhou, Robert Joynt, and S. N. Coppersmith. Two-particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A, 81 (2010), no. 5, 052313, 11 pp.10.1103/PhysRevA.81.052313Search in Google Scholar
[23] Joseph Geraci and Daniel A Lidar. On the exact evaluation of certain instances of the Potts partition function by quantum computers. Comm. Math. Phys. 279 (2008), no. 3, 735-768.Search in Google Scholar
[24] Chris Godsil and Gordon Royle. Strongly regular graphs. In Algebraic graph theory, pages 217–247. Springer, 2001.10.1007/978-1-4613-0163-9_10Search in Google Scholar
[25] C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex. Gaussian boson sampling. Phys. Rev. Lett., 119 (2017), no. 17, 170501, 5 pp.10.1103/PhysRevLett.119.170501Search in Google Scholar PubMed
[26] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 103 (2009), no. 15, 150502, 4 pp.10.1103/PhysRevLett.103.150502Search in Google Scholar PubMed
[27] Harald Andrés Helfgott, Jitendra Bajpai, and Daniele Dona. Graph isomorphisms in quasi-polynomial time. arXiv preprint arXiv:1710.04574, 2017.Search in Google Scholar
[28] Christian Kleiber and Jordan Stoyanov. Multivariate distributions and the moment problem. Journal of Multivariate Analysis, 113 (2013), 7-18.10.1016/j.jmva.2011.06.001Search in Google Scholar
[29] Regina Kruse, Craig S Hamilton, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, and Igor Jex. Detailed study of gaussian boson sampling. Phys. Rev. A, 100 (2019), no. 3, 032326, 15 pp.10.1103/PhysRevA.100.032326Search in Google Scholar
[30] C.H.C. Little. Combinatorial Mathematics V.: Proceedings of the Fifth Australian Conference, Held at the Royal Melbourne Institute of Technology, August 24 - 26, 1976. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2006.Search in Google Scholar
[31] Anuradha Mahasinghe, Josh A Izaac, Jingbo B Wang, and Jagath K Wijerathna. Phase-modified CTQW unable to distinguish strongly regular graphs efficiently. J. Phys. A, 48 (2015), no. 26, 265301, 13 pp.10.1088/1751-8113/48/26/265301Search in Google Scholar
[32] PW Mills, RP Rundle, VM Dwyer, Todd Tilma, Simon J Devitt, JH Samson, and Mark J Everitt. A proposal for an efficient quantum algorithm solving the graph isomorphism problem. arXiv preprint arXiv:1711.09842, 2017.Search in Google Scholar
[33] Kenneth Rudinger, John Gamble, Mark Wellons, Eric Bach, Mark Friesen, Robert Joynt, and S. Coppersmith. Noninteracting multiparticle quantum random walks applied to the graph isomorphism problem for strongly regular graphs. J. Phys. A, 86 (2012), no. 2, 022334, 10 pp.10.1103/PhysRevA.86.022334Search in Google Scholar
[34] Kenneth Rudinger, John King Gamble, Eric Bach, Mark Friesen, Robert Joynt, and S. N. Coppersmith. Comparing algorithms for graph isomorphism using discrete- and Continuous-Time quantum random walks. Journal of Computational and Theoretical Nanoscience, 10 (2013), no. 7, 1653–1661.Search in Google Scholar
[35] Stefan Scheel. Permanents in linear optical networks. arXiv preprint quant-ph/0406127, 2004.Search in Google Scholar
[36] Peter W Shor. Algorithms for quantum computation: Discrete logarithms and factoring. In 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM, 1994), 124-134, IEEE Comput. Soc. Press, Los Alamitos, CA, 1994.Search in Google Scholar
[37] Peter W Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM review, 41 (1999), no. 2, 303–332.10.1137/S0036144598347011Search in Google Scholar
[38] Jamie Smith. Algebraic Aspects of Multi-Particle Quantum Walks. PhD, University of Waterloo, December 2012.Search in Google Scholar
[39] Ted Spence. Strongly regular graphs. http://www.maths.gla.ac.uk/~es/srgraphs.php, August 2018.Search in Google Scholar
[40] Maarten Van den Nest, Wolfgang Dür, Robert Raussendorf, and Hans J Briegel. Quantum algorithms for spin models and simulable gate sets for quantum computation. Phys. Rev. A, 80 (2009), no. 5, 052334, 5 pp.10.1103/PhysRevA.80.052334Search in Google Scholar
[41] Huiquan Wang, Junjie Wu, Xuejun Yang, and Xun Yi. A graph isomorphism algorithm using signatures computed via quantum walk search model. J. Phys. A, 48 (2015), no. 11, 115302, 23 pp.10.1088/1751-8113/48/11/115302Search in Google Scholar
[42] Paweł Wocjan and Jon Yard. The Jones polynomial: quantum algorithms and applications in quantum complexity theory. Quantum Inf. Comput. 8 (2008), no. 1-2, 147–180.Search in Google Scholar
[43] Han-Sen Zhong et al. Quantum computational advantage using photons. Science 370 (2020), 1460-1463.10.1126/science.abe8770Search in Google Scholar PubMed
[44] Kenneth M Zick, Omar Shehab, and Matthew French. Experimental quantum annealing: case study involving the graph isomorphism problem. Scientific Reports, 5 (2015), 11168, 11 pp.Search in Google Scholar
© 2021 Kamil Brádler et al., published by De Gruyter
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