# Laplacian versus adjacency matrix in quantum walk search

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## Abstract

A quantum particle evolving by Schrödinger’s equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace’s operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.

## Keywords

Quantum walk Continuous time Spatial search Laplacian Adjacency matrix## Notes

### Acknowledgments

TW and NN were supported by the European Union Seventh Framework Programme (FP7/2007-2013) under the QALGO (Grant Agreement No. 600700) project, and the ERC Advanced Grant MQC. LT was supported by CNPq CSF/BJT grant reference 301181/2014-4.

## References

- 1.Schrödinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev.
**28**, 1049–1070 (1926)ADSCrossRefMATHGoogle Scholar - 2.Griffiths, D.J.: Introduction to Quantum Mechanics. Prentice Hall, New Jersey (2005)Google Scholar
- 3.Bloch, I.: Ultracold quantum gases in optical lattices. Nat. Phys.
**1**, 23–30 (2005)CrossRefGoogle Scholar - 4.Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A
**58**, 915–928 (1998)ADSMathSciNetCrossRefGoogle Scholar - 5.Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A
**70**, 022314 (2004)ADSMathSciNetCrossRefGoogle Scholar - 6.Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing. STOC ’03, pp. 59–68. ACM, New York, NY, USA (2003)Google Scholar
- 7.Farhi, E., Goldstone, J., Gutmann, S.: A quantum algorithm for the Hamiltonian NAND tree. Theory Comput.
**4**(8), 169–190 (2008)MathSciNetCrossRefMATHGoogle Scholar - 8.Bose, S., Casaccino, A., Mancini, S., Severini, S.: Communication in XYZ all-to-all quantum networks with a missing link. Int. J. Quantum Inf.
**07**(04), 713–723 (2009)CrossRefMATHGoogle Scholar - 9.Alvir, R., Dever, S., Lovitz, B., Myer, J., Tamon, C., Xu, Y., Zhan, H.: Perfect state transfer in Laplacian quantum walk. J. Algebraic Combin.
**43**, 801–826 (2016)Google Scholar - 10.Ackelsberg, E., Brehm, Z., Chan, A., Mundinger, J., Tamon, C.: Laplacian state transfer in coronas. Linear Algebra Appl.
**506**, 154–167 (2016)Google Scholar - 11.Wong, T.G.: Grover search with lackadaisical quantum walks. J. Phys. A: Math. Theor.
**48**(43), 435304 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar - 12.Janmark, J., Meyer, D.A., Wong, T.G.: Global symmetry is unnecessary for fast quantum search. Phys. Rev. Lett.
**112**, 210502 (2014)ADSCrossRefGoogle Scholar - 13.Meyer, D.A., Wong, T.G.: Connectivity is a poor indicator of fast quantum search. Phys. Rev. Lett.
**114**, 110503 (2015)ADSCrossRefGoogle Scholar - 14.Wong, T.G.: Spatial search by continuous-time quantum walk with multiple marked vertices. Quantum Inf. Process.
**15**(4), 1411–1443 (2016)Google Scholar - 15.Wong, T.G., Ambainis, A.: Quantum search with multiple walk steps per oracle query. Phys. Rev. A
**92**, 022338 (2015)ADSCrossRefGoogle Scholar - 16.Wong, T.G.: Faster quantum walk search on a weighted graph. Phys. Rev. A
**92**, 032320 (2016) arXiv:1508.01327v3 - 17.Novo, L., Chakraborty, S., Mohseni, M., Neven, H., Omar, Y.: Systematic dimensionality reduction for quantum walks: optimal spatial search and transport on non-regular graphs. Sci. Rep.
**5**, 13304 (2015)ADSCrossRefGoogle Scholar - 18.Chakraborty, S., Novo, L., Ambainis, A., Omar, Y.: Spatial search by quantum walk is optimal for almost all graphs. Phys. Rev. Lett.
**116**, 100501 (2016)Google Scholar - 19.Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortsch. Phys.
**46**(4–5), 493–505 (1998)ADSCrossRefGoogle Scholar - 20.Motwani, R., Raghavan, P.: Randomized algorithms. Cambridge University Press, New York (1995)CrossRefMATHGoogle Scholar
- 21.von Schelling, H.: Auf der spur des zufalls. Deutsches Statistisches Zentralblatt
**26**, 137–146 (1934)MATHGoogle Scholar - 22.von Schelling, H.: Coupon collecting for unequal probabilities. Amer. Math. Mon.
**61**(5), 306–311 (1954)MathSciNetCrossRefMATHGoogle Scholar - 23.Flajolet, P., Gardy, D., Thimonier, L.: Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Appl. Math.
**39**, 207–229 (1992)MathSciNetCrossRefMATHGoogle Scholar - 24.Wong, T.G.: Diagrammatic approach to quantum search. Quantum Inf. Process.
**14**(6), 1767–1775 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar