Journal
SIAM JOURNAL ON COMPUTING
Volume 40, Issue 1, Pages 142-164Publisher
SIAM PUBLICATIONS
DOI: 10.1137/090745854
Keywords
search; Markov chain; hitting time; quantum walk; phase estimation; amplitude amplification
Funding
- European Commission [QAP 015848, QCS 25596]
- French ANR [QRAC 08-EMER-012]
- NSERC Canada
- CIFAR
- ERA (Ontario)
- QuantumWorks
- CFI
- OIT
- MITACS
- ARO/NSA (USA)
- Government of Canada through Industry Canada
- Province of Ontario through MRI
- Belgian FNRS
- NSF [CCF-0524837]
- ARO [DAAD 19-03-1-0082]
- Singapore Ministry of Education
- National Research Foundation
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We propose a new method for designing quantum search algorithms for finding a marked element in the state space of a classical Markov chain. The algorithm is based on a quantum walk a la Szegedy [Quantum speed-up of Markov chain based algorithms, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 2004, pp. 32-41] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [Quantum walk algorithm for Element Distinctness, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 2004, pp. 22-31] and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.
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