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126
Exponential Algorithmic Speedup by a Quantum Walk
"... We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a di#erent technique from previous quantum algorithms based on quantum Fouri ..."
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Cited by 158 (10 self)
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We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a di#erent technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk e#ciently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.
OneDimensional Quantum Walks
 STOC'01
, 2001
"... We define and analyze quantum computational variants of random walks on onedimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, ..."
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Cited by 134 (10 self)
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We define and analyze quantum computational variants of random walks on onedimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range [\Gamma t= p
Quantum algorithms for the triangle problem
 PROCEEDINGS OF SODA’05
, 2005
"... We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is b ..."
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Cited by 94 (10 self)
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We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.
Coins make quantum walks faster
 Proceedings of SODA’05. Also quantph/0402107
"... We show how to search N items arranged on a √ N × √ N grid in time O ( √ N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently ..."
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Cited by 61 (8 self)
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We show how to search N items arranged on a √ N × √ N grid in time O ( √ N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time Ω(N) to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of O ( √ N) and give several extensions of quantum walk search algorithms for general graphs. The coinflip operation needs to be chosen judiciously: we show that another “natural ” choice of coin gives a walk that takes Ω(N) steps. We also show that in 2 dimensions it is sufficient to have a twodimensional coinspace to achieve the time O ( √ N log N). 1
Quantum random walks in one dimension
 Quantum Information Processing
"... Abstract. This paper treats the quantum random walk on the line determined by a 2 × 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P,Q,R and S given by U. The dependence of the mth moment on U and initial qubit state ϕ i ..."
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Cited by 58 (27 self)
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Abstract. This paper treats the quantum random walk on the line determined by a 2 × 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P,Q,R and S given by U. The dependence of the mth moment on U and initial qubit state ϕ is clarified. Furthermore a new type of limit theorems for the Hadamard walk is given. It shows that the behavior of quantum random walk is striking different from that of the classical ramdom walk. 1
Quantum query complexity of some graph problems
 Proceedings of the 31st International Colloquium on Automata, Lanaguages, and Programming
, 2004
"... Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency listlike array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Sourc ..."
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Cited by 58 (3 self)
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Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency listlike array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(n 3/2) in the matrix model and in Θ ( √ nm) in the array model, while the complexity of Connectivity is also in Θ(n 3/2) in the matrix model, but in Θ(n) in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.
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.
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
Onedimensional quantum walks with absorbing boundaries. arXiv.org ePrint quantph/0207008
, 2002
"... In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the onedimensional case. We compute these probabilities both by employing generating functions and by use ..."
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Cited by 31 (2 self)
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In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the onedimensional case. We compute these probabilities both by employing generating functions and by use of an eigenfunction approach. The generating function method is used to determine some simple properties of the walks we consider, but appears to have limitations. The eigenfunction approach works by relating the problem of absorption to a unitary problem that has identical dynamics inside a certain domain, and can be used to compute several additional interesting properties, such as the time dependence of absorption. The eigenfunction method has the distinct advantage that it can be extended to arbitrary dimensionality. We outline the solution of the absorption probability problem of a (D − 1)dimensional wall in a Ddimensional space. 1