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48
Quantum walk algorithms for element distinctness
 In: 45th Annual IEEE Symposium on Foundations of Computer Science, OCT 1719, 2004. IEEE Computer Society Press, Los Alamitos, CA
, 2004
"... We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrm ..."
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Cited by 174 (13 self)
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We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrman et al. [11] and matches the lower bound by [1]. We also give an O(N k/(k+1) ) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items. 1
Quantum and Classical Strong Direct Product Theorems and Optimal TimeSpace Tradeoffs
 SIAM Journal on Computing
, 2004
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum ..."
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Cited by 65 (13 self)
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...
Search via quantum walk
 LOGIC PROGRAMMING, PROC. OF THE 1994 INT. SYMP
, 2007
"... 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 [24] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimat ..."
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Cited by 56 (8 self)
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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 [24] 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 [6] and Szegedy [24]. 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 chain. In addition, it is conceptually simple, avoids several technical difficulties in the previous analyses, and leads to improvements in various aspects of several algorithms based on quantum walk.
All quantum adversary methods are equivalent
 THEORY OF COMPUTING
, 2006
"... The quantum adversary method is one of the most versatile lowerbound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), an ..."
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Cited by 50 (4 self)
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The quantum adversary method is one of the most versatile lowerbound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), and the Kolmogorov complexity adversary (Laplante and Magniez, 2004). We also present a few new equivalent formulations of the method. This shows that there is essentially one quantum adversary method. From our approach, all known limitations of these versions of the quantum adversary method easily follow.
Subcubic Equivalences Between Path, Matrix, and Triangle Problems
"... We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvab ..."
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Cited by 42 (11 self)
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We say an algorithm on n × n matrices with entries in [−M,M] (or nnode graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n 3) time are equivalent under subcubic reductions. Namely, the following weighted problems either all have truly subcubic algorithms, or none of them do: • The allpairs shortest paths problem on weighted digraphs (APSP). • Detecting if a weighted graph has a triangle of negative total edge weight. • Listing up to n 2.99 negative triangles in an edgeweighted graph. • Finding a minimum weight cycle in a graph of nonnegative edge weights. • The replacement paths problem on weighted digraphs. • Finding the second shortest simple path between two nodes in a weighted digraph. • Checking whether a given matrix defines a metric. • Verifying the correctness of a matrix product over the (min,+)semiring. Therefore, if APSP cannot be solved in n 3−ε time for any ε> 0, then many other problems also
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
Quantum complexity of testing group commutativity
 Proceedings of ICALP’05
, 2005
"... Abstract. We consider the problem of testing the commutativity of a blackbox group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorit ..."
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Cited by 30 (5 self)
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Abstract. We consider the problem of testing the commutativity of a blackbox group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in Õ(k2/3). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Ω(k 2/3), we introduce a new technique of reduction for quantum query complexity. Along the way, we prove the optimality of the algorithm of Pak for the randomized model. 1
Quantum Search Algorithms
, 2005
"... We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1 ..."
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Cited by 27 (1 self)
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We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1
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.