# Quantum Walks Can Find a Marked Element on Any Graph

- 354 Downloads
- 13 Citations

## 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.

## References

- 1.Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput.
**37**(1), 210–239 (2007)CrossRefMathSciNetMATHGoogle Scholar - 2.Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM J. Comput.
**37**(2), 413–424 (2007)CrossRefMathSciNetMATHGoogle Scholar - 3.Buhrman, H., Špalek, R: Quantum verification of matrix products. In: Proceedings of the 17th ACM-SIAM symposium on discrete algorithms (SODA’06), pp. 880–889. ACM (2006)Google Scholar
- 4.Magniez, F., Nayak, A.: Quantum complexity of testing group commutativity. Algorithmica
**48**(3), 221–232 (2007)CrossRefMathSciNetMATHGoogle Scholar - 5.Aaronson, S., Ambainis, A.: Quantum search of spatial regions. Theory Comput.
**1**(4), 47–79 (2005)CrossRefMathSciNetGoogle Scholar - 6.Shenvi, N., Kempe, J., Whaley, B.K.: Quantum random-walk search algorithm. Phys. Rev. A
**67**(5), 052307 (2003)CrossRefGoogle Scholar - 7.Childs, A.M., Goldstone, J.: Spatial search and the Dirac equation. Phys. Rev. A
**70**(4), 042312 (2004)CrossRefMathSciNetGoogle Scholar - 8.Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the 16th ACM-SIAM symposium on discrete algorithms (SODA’05), pp. 1099–1108. SIAM (2005)Google Scholar
- 9.Kempe, J.: Discrete quantum walks hit exponentially faster. Probab. Theory Relat. Fields
**133**(2), 215–235 (2005)CrossRefMathSciNetMATHGoogle Scholar - 10.Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of the 45th IEEE symposium on foundations of computer science (FOCS’04), pp. 32–41. IEEE Computer Society Press (2004)Google Scholar
- 11.Krovi, H., Brun, T.A.: Hitting time for quantum walks on the hypercube. Phys. Rev. A
**73**(3), 032341 (2006)CrossRefMathSciNetGoogle Scholar - 12.Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proceedings of the 39th ACM symposium on theory of computing (STOC’07), pp. 575–584. ACM Press (2007)Google Scholar
- 13.Magniez, F., Nayak, A., Richter, P., Santha, M.: On the hitting times of quantum versus random walks. Algorithmica
**63**(1), 91–116 (2012)CrossRefMathSciNetMATHGoogle Scholar - 14.Varbanov, M., Krovi, H., Brun, T.A.: Hitting time for the continuous quantum walk. Phys. Rev. A
**78**(2), 022324 (2008)CrossRefMathSciNetGoogle Scholar - 15.Tulsi, A.: Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A
**78**(1), 012310 (2008)CrossRefGoogle Scholar - 16.Krovi, H., Magniez, F., Ozols, M., Roland, J.: Finding is as easy as detecting for quantum walks. In: Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 6198, pp. 540–551. Springer, Berlin-Heidelberg (2010)Google Scholar
- 17.Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A
**70**(2), 022314 (2004)CrossRefMathSciNetGoogle Scholar - 18.Ambainis, A., Bačkurs, A., Nahimovs, N., Ozols, R., Rivosh, A.: Lecture Notes in Computer Science, vol. 7582. Springer, Berlin (2013)Google Scholar
- 19.Krovi, H., Ozols, M., Roland, J.: Adiabatic condition and the quantum hitting time of Markov chains. Phys. Rev. A
**82**(2), 022333 (2010)CrossRefGoogle Scholar - 20.Grinstead, C.M., Snell, J.L.: Introduction to Probability, 2nd edn. American Mathematical Society, Providence (1997)MATHGoogle Scholar
- 21.Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Undergraduate Texts in Mathematics. Springer, Berlin (1960)MATHGoogle Scholar
- 22.Koralov, L.B., Sinai, Y.G.: Theory of Probability and Random Processes. Springer, Berlin (2007)CrossRefMATHGoogle Scholar
- 23.Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)MATHGoogle Scholar
- 24.Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
- 25.Meyer, C.D.: Matrix Analysis and Applied Linear Algebra, vol. 1. SIAM (Society for Industrial and Applied Mathematics), Philadelphia (2000)Google Scholar
- 26.Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. Proc. R. Soc. Lond.
**454**(1969), 339–354 (1998)CrossRefMathSciNetMATHGoogle Scholar - 27.Høyer, P., Mosca, M., de Wolf, R.: Quantum search on bounded-error inputs. In: Proceedings of the 30th international colloquium on automata, languages and programming (ICALP’03), volume 2719 of lecture notes in computer science, pp. 291–299. Springer (2003)Google Scholar
- 28.Feige, U., Raghavan, P., Peleg, D., Upfal, E.: Computing with noisy information. SIAM J. Comput.
**23**(5), 1001–1018 (1994)CrossRefMathSciNetMATHGoogle Scholar - 29.Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science
**292**(5516), 472–475 (2001)CrossRefMathSciNetMATHGoogle Scholar