Algorithmica

, Volume 74, Issue 2, pp 851–907 | Cite as

Quantum Walks Can Find a Marked Element on Any Graph

  • Hari Krovi
  • Frédéric Magniez
  • Maris Ozols
  • Jérémie Roland
Article

Abstract

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set \(M\) consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time \({{\mathrm{HT}}}(P,M)\) of any reversible random walk \(P\) on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity \({\hbox {HT}}^{+}(P,M)\) which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk \(P\) and the absorbing walk \(P'\), whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk \(P\) is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters \(p_M\) (the probability of picking a marked vertex from the stationary distribution) and \({\hbox {HT}}^{+}(P,M)\) are known.

Keywords

Quantum algorithms Quantum walks Markov chains  Interpolated quantum walks 

Notes

Acknowledgments

MO would like to acknowledge Andrew Childs for many helpful discussions. The authors would also like to thank Andris Ambainis for useful comments. Part of this work was done while HK, MO, and JR were at NEC Laboratories America in Princeton. MO also was affiliated with University of Waterloo and Institute for Quantum Computing (supported by QuantumWorks) and IBM TJ Watson Research Center (supported by DARPA QUEST program under Contract No. HR0011-09-C-0047) during this project. Presently FM, MO and JR are supported by the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 600700 (QALGO). FM is also supported by the French ANR Blanc project ANR-12-BS02-005 (RDAM). Last, JR acknowledges support from the Belgian ARC project COPHYMA.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Hari Krovi
    • 1
  • Frédéric Magniez
    • 2
  • Maris Ozols
    • 3
  • Jérémie Roland
    • 4
  1. 1.Quantum Information Processing GroupRaytheon BBN TechnologiesCambridgeUSA
  2. 2.CNRS, LIAFAUniversity Paris Diderot, Sorbonne Paris-CitéParisFrance
  3. 3.DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  4. 4.QuIC, Ecole Polytechnique de BruxellesUniversité Libre de Bruxelles (ULB)BrusselsBelgium

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