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Quantum walk based search algorithms
 In Proceedings of the 5th Conference on Theory and Applications of Models of Computation
, 2008
"... Abstract. In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We presen ..."
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Cited by 37 (1 self)
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Abstract. In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We present a rather detailed description of a somewhat simplified version of the MNRS algorithm. Finally, in the query complexity model, we show how quantum walks can be applied to the following
Any ANDOR formula of size N can be evaluated in time N1/2+o(1) on a quantum computer
 in Proceedings of the 48th IEEE FOCS
"... Abstract. Consider the problem of evaluating an ANDOR formula on an Nbit blackbox input. We present a boundederror quantum algorithm that solves this problem in time N1/2+o(1). In particular, approximately balanced formulas can be evaluated in O( N) queries, which is optimal. The idea of the alg ..."
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Cited by 35 (10 self)
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Abstract. Consider the problem of evaluating an ANDOR formula on an Nbit blackbox input. We present a boundederror quantum algorithm that solves this problem in time N1/2+o(1). In particular, approximately balanced formulas can be evaluated in O( N) queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discretetime quantum walk on a weighted tree whose spectrum encodes the value of the formula.
Spanprogrambased quantum algorithm for evaluating formulas
, 2008
"... We give a quantum algorithm for evaluating formulas over an extended gate set, including all two and threebit binary gates (e.g., NAND, 3majority). The algorithm is optimal on readonce formulas for which each gate’s inputs are balanced in a certain sense. The main new tool is a correspondence be ..."
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Cited by 34 (6 self)
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We give a quantum algorithm for evaluating formulas over an extended gate set, including all two and threebit binary gates (e.g., NAND, 3majority). The algorithm is optimal on readonce formulas for which each gate’s inputs are balanced in a certain sense. The main new tool is a correspondence between a classical linearalgebraic model of computation, “span programs,” and weighted bipartite graphs. A span program’s evaluation corresponds to an eigenvaluezero eigenvector of the associated graph. A quantum computer can therefore evaluate the span program by applying spectral estimation to the graph. For example, the classical complexity of evaluating the balanced ternary majority formula is unknown, and the natural generalization of randomized alphabeta pruning is known to be suboptimal. In contrast, our algorithm generalizes the optimal quantum ANDOR formula evaluation algorithm and is optimal for evaluating the balanced ternary majority formula.
Quantum walks: a comprehensive review
, 2012
"... Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting ..."
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Cited by 24 (0 self)
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Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete and continuoustime quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discretetime quantum walks. Furthermore, we have reviewed several algorithms based on both discrete and continuoustime quantum walks as well as a most important result: the computational universality of both continuous and discretetime quantum walks.
Quantum algorithms for algebraic problems
, 2008
"... Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational pro ..."
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Cited by 24 (2 self)
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Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum
Finding is as easy as detecting for quantum walks
, 2010
"... We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. The number of steps of the quantum walk is quadratically smaller than the classical hitting time of any reversible random walk P on the graph. Our approach is new, simpler and more g ..."
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Cited by 12 (2 self)
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We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. The number of steps of the quantum walk is quadratically smaller than the classical hitting time of any reversible random walk P on the graph. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the walk P and the absorbing walk P ′ , whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of the interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not statetransitive, and in the presence of multiple marked vertices. As a consequence we make a progress on an open problem related to the spatial search on the 2Dgrid.