# Spatial search by continuous-time quantum walk with multiple marked vertices

- 175 Downloads
- 7 Citations

## Abstract

In the typical spatial search problems solved by continuous-time quantum walk, changing the location of the marked vertices does not alter the search problem. In this paper, we consider search when this is no longer true. In particular, we analytically solve search on the “simplex of \(K_M\) complete graphs” with all configurations of two marked vertices, two configurations of \(M+1\) marked vertices, and two configurations of \(2(M+1)\) marked vertices, showing that the location of the marked vertices can dramatically influence the required jumping rate of the quantum walk, such that using the wrong configuration’s value can cause the search to fail. This sensitivity to the jumping rate is an issue unique to continuous-time quantum walks that does not affect discrete-time ones.

## Keywords

Grover’s algorithm Quantum search Spatial search Quantum random walk Multiple marked vertices## Notes

### Acknowledgments

Thanks to Andris Ambainis for useful discussions. This work was 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.

## References

- 1.Schrödinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev.
**28**, 1049–1070 (1926)ADSCrossRefMATHGoogle Scholar - 2.Sakurai, J.J.: Modern Quantum Mechanics, Revised edn. Addison Wesley, Boston (1993)Google Scholar
- 3.Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A
**58**, 915–928 (1998)ADSMathSciNetCrossRefGoogle Scholar - 4.Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A
**70**, 022314 (2004)ADSMathSciNetCrossRefGoogle Scholar - 5.Farhi, E., Goldstone, J., Gutmann, S.: A quantum algorithm for the Hamiltonian NAND tree. Theory Comput.
**4**(8), 169–190 (2008)MathSciNetCrossRefMATHGoogle Scholar - 6.Rudinger, K., Gamble, J.K., Wellons, M., Bach, E., Friesen, M., Joynt, R., Coppersmith, S.N.: Noninteracting multiparticle quantum random walks applied to the graph isomorphism problem for strongly regular graphs. Phys. Rev. A
**86**, 022334 (2012)ADSCrossRefGoogle Scholar - 7.Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett.
**102**, 180501 (2009)ADSMathSciNetCrossRefGoogle Scholar - 8.Mochon, C.: Hamiltonian oracles. Phys. Rev. A
**75**, 042313 (2007)ADSCrossRefGoogle Scholar - 9.Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing. STOC ’96, pp. 212–219. ACM, New York (1996)Google Scholar
- 10.Farhi, E., Gutmann, S.: Analog analogue of a digital quantum computation. Phys. Rev. A
**57**(4), 2403–2406 (1998)ADSMathSciNetCrossRefGoogle Scholar - 11.Wong, T.G.: Nonlinear quantum search. PhD dissertation (2014)Google Scholar
- 12.Wong, T.G.: Grover search with lackadaisical quantum walks. J. Phys. A: Math. Theor.
**48**, 435304 (2015). doi: 10.1088/1751-8113/48/43/435304 ADSMathSciNetCrossRefMATHGoogle Scholar - 13.Wong, T.G.: Quantum walk search through potential barriers. arXiv:1503.06605 [quant-ph] (2015)
- 14.Janmark, J., Meyer, D.A., Wong, T.G.: Global symmetry is unnecessary for fast quantum search. Phys. Rev. Lett.
**112**, 210502 (2014)ADSCrossRefGoogle Scholar - 15.Meyer, D.A., Wong, T.G.: Connectivity is a poor indicator of fast quantum search. Phys. Rev. Lett.
**114**, 110503 (2015)ADSCrossRefGoogle Scholar - 16.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 - 17.Wong, T.G., Ambainis, A.: Quantum search with multiple walk steps per oracle query. Phys. Rev. A
**92**, 022338 (2015)ADSCrossRefGoogle Scholar - 18.Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys.
**44**(4), 307–327 (2003)ADSMathSciNetCrossRefGoogle Scholar - 19.Meyer, D.A.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys.
**85**(5–6), 551–574 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar - 20.Meyer, D.A.: On the absence of homogeneous scalar unitary cellular automata. Phys. Lett. A
**223**(5), 337–340 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar - 21.Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’05, pp. 1099–1108 (2005)Google Scholar
- 22.Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A
**67**, 052307 (2003)ADSCrossRefGoogle Scholar - 23.Nahimovs, N., Rivosh, A.: Quantum walks on two-dimensional grids with multiple marked locations. In: Proceedings of the 42nd International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM ’16. Harrachov (2016). To appear arXiv:1507.03788
- 24.Ambainis, A., Rivosh, A.: Quantum walks with multiple or moving marked locations. In: V. Geffert, J. Karhumöki, A. Bertoni, B. Preneel, P. Návrat, M. Bieliková (eds.) Proceedings of the 34th Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2008, pp. 485–496 (2008)Google Scholar
- 25.Nahimovs, N., Rivosh, A.: Exceptional congurations of quantum walks with Grover’s coin. In: Proceedings of the 10th Doctoral Workshop on Mathematical and Engineering Methods in Computer Science, MEMICS ’15. Telc̆ (2015). To appear arXiv:1509.06862
- 26.Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’04, pp. 32–41 (2004)Google Scholar
- 27.Krovi, H., Magniez, F., Ozols, M., Roland, J.: Quantum walks can find a marked element on any graph. Algorithmica (2015). doi: 10.1007/s00453-015-9979-8
- 28.Wong, T.G.: Faster quantum walk search on a weighted graph. Phys. Rev. A
**92**, 032320 (2015)ADSCrossRefGoogle Scholar - 29.Wong, T.G.: Diagrammatic approach to quantum search. Quantum Inf. Process.
**14**(6), 1767–1775 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar