Search via Quantum Walk

• Published in: SIAM journal on computing Vol. 40; no. 1; pp. 142 - 164
• Main Authors: Magniez, Frédéric, Nayak, Ashwin, Roland, Jérémie, Santha, Miklos
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: 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
• DOI: 10.1137/090745854

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]