A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Strengths and Weaknesses of Quantum Computing
SIAM Journal on Computing
Quantum simulations of classical random walks and undirected graph connectivity
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Classical and Quantum Computation
Classical and Quantum Computation
Quantum Walk on the Line
Quantum lower bounds for the collision and the element distinctness problems
Journal of the ACM (JACM)
Quantum Walk Algorithm for Element Distinctness
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Quantum Speed-Up of Markov Chain Based Algorithms
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Coins make quantum walks faster
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Quantum Algorithms for Element Distinctness
SIAM Journal on Computing
Quantum verification of matrix products
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Quantum Complexity of Testing Group Commutativity
Algorithmica
Quantum Algorithms for the Triangle Problem
SIAM Journal on Computing
On the hitting times of quantum versus random walks
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Quantum search on bounded-error inputs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Finding is as easy as detecting for quantum walks
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
The quantum query complexity of algebraic properties
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
Eigenpath traversal by phase randomization
Quantum Information & Computation
Improved output-sensitive quantum algorithms for Boolean matrix multiplication
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Exact Quantum Algorithms for the Leader Election Problem
ACM Transactions on Computation Theory (TOCT)
Time-Efficient quantum walks for 3-distinctness
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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 à 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.