Search via Quantum Walk

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...

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Vydáno v:SIAM journal on computing Ročník 40; číslo 1; s. 142 - 164
Hlavní autoři: Magniez, Frédéric, Nayak, Ashwin, Roland, Jérémie, Santha, Miklos
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Témata:
Algorithmics. Computability. Computer arithmetics
Algorithms
Amplification
Applied sciences
Approximation
Computer science
Computer science; control theory; systems
Exact sciences and technology
Foundations
Markov analysis
Markov chains
Mathematics
Miscellaneous
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in probability and statistics
Operators
Presses
Quantum theory
Reflection
Sciences and techniques of general use
Studies
Theoretical computing
amplitude amplification
Costs
68Q12
Estimation
quantum walk
State space
05C81
State space method
hitting time
Implementation
Search algorithm
Markov chain
Computer science
phase estimation
search
81P68
Algorithm analysis
ISSN:0097-5397, 1095-7111
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Popis
Shrnutí: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. [PUBLICATION ABSTRACT]
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ISSN:0097-5397
1095-7111
DOI:10.1137/090745854