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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.
Generalized Flow and Determinism in Measurementbased Quantum Computation
 New J. Physics
, 2007
"... Abstract. We extend the notion of quantum information flow defined by Danos and Kashefi [1] for the oneway model [2] and present a necessary and sufficient condition for the deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the (X, ..."
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Cited by 26 (13 self)
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Abstract. We extend the notion of quantum information flow defined by Danos and Kashefi [1] for the oneway model [2] and present a necessary and sufficient condition for the deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the (X, Y), (X, Z) and (Y, Z) planes. We apply both measurement calculus and the stabiliser formalism to derive our main theorem which for the first time gives a full characterization of the deterministic computation in the oneway model. We present several examples to show how our result improves over the traditional notion of flow, such as geometries (entanglement graph with input and output) with no flow but having generalized flow and we discuss how they lead to an optimal implementation of the unitaries. More importantly one can also obtain a better quantum computation depth with the generalized flow rather than with flow. We believe our characterization result is particularly essential for the study of the algorithms and complexity in the oneway model. Generalized Flow and Determinism 2 1.
Quantum Period Query Proves NP⊆BQP
, 2001
"... Quantum period query (QPQ) is a quantum algorithm for decision problems. A decision problem is viewed as a funtion f A: {0,1} n fi {0,1} where A specifies an instance of the problem and f A(x)=1 iff there is a solution to that instance of that problem. The function f A is transformed into a periodic ..."
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Quantum period query (QPQ) is a quantum algorithm for decision problems. A decision problem is viewed as a funtion f A: {0,1} n fi {0,1} where A specifies an instance of the problem and f A(x)=1 iff there is a solution to that instance of that problem. The function f A is transformed into a periodic function h A: {0,1} 2n fi {0,1} such that the period of h A is 1 iff instance A of the problem has no solution, and is 2 n+1 otherwise. QPQ is a straightforward polynomial quantum implementation of h A followed by quantum period finding, which is known to be also polynomial. We present QPQ through its application to number partitioning, thus showing that number partitioning is in BQP, hence that every NP problem lies within BQP. 1 Number partitioning Number partitioning is the NPcomplete problem of deciding whether there exists a bipartition A 0˙A 1 of a set of strictly positive integers A={a 0, a 2, …, a n1}, such that: A
Finding optimal flows efficiently
 In Proceeding of 35th International Colloquium on Automata, Languages and Programming
, 2008
"... ..."
Quantum query complexity of graph connectivity
 IN QUANTPH/0303169
, 2003
"... Harry Buhrman et al gave an Ω (√n) lower bound for Graph Connectivity in the adjacency matrix query model. Their proof is based on the polynomial method. We give an Ω(n 3/2) bound using Andris Ambainis’ method, and an O(n 3/2 log n) upper bound based on Grover’s search algorithm. In addition we stud ..."
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Cited by 4 (0 self)
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Harry Buhrman et al gave an Ω (√n) lower bound for Graph Connectivity in the adjacency matrix query model. Their proof is based on the polynomial method. We give an Ω(n 3/2) bound using Andris Ambainis’ method, and an O(n 3/2 log n) upper bound based on Grover’s search algorithm. In addition we study the adjacency list query model, where we have almost matching lower and upper bounds for Strong Connectivity of directed graphs.
Creative Commons Attribution License. Information Flow in Secret Sharing Protocols
"... The entangled graph states [9] have emerged as an elegant and powerful quantum resource, indeed almost all multiparty protocols can be written in terms of graph states including measurement based quantum computation (MBQC), error correction and secret sharing amongst others. In addition they are at ..."
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The entangled graph states [9] have emerged as an elegant and powerful quantum resource, indeed almost all multiparty protocols can be written in terms of graph states including measurement based quantum computation (MBQC), error correction and secret sharing amongst others. In addition they are at the forefront in terms of implementations. As such they represent an excellent opportunity to move towards integrated protocols involving many of these elements. In this paper we look at expressing and extending graph state secret sharing [3] and MBQC in a common framework and graphical language related to flow [5, 4]. We do so with two main contributions. First we express in entirely graphical terms which set of players can access which information in graph state secret sharing protocols. These succinct graphical descriptions of access allow us to take known results from graph theory to make statements on the generalisation of the previous schemes to present new secret sharing protocols. Second, we give a set of necessary conditions as to when a graph with flow, i.e. capable of performing a class of unitary operations, can be extended to include vertices which can be ignored, pointless measurements, and hence considered as unauthorised players in terms of secret sharing, or error qubits in terms of fault tolerance. This offers a way to extend existing MBQC patterns to secret sharing protocols. Our characterisation of pointless measurements is believed also to be a useful tool for further integrated measurement based schemes, for example in constructing fault tolerant MBQC schemes. 1
The Multiplicative Quantum Adversary
, 2007
"... We present a new variant of the quantum adversary method. All adversary methods give lower bounds on the quantum query complexity of a function by bounding the change of a progress function caused by one query. All previous variants upperbound the difference of the progress function, whereas our ne ..."
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Cited by 9 (1 self)
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We present a new variant of the quantum adversary method. All adversary methods give lower bounds on the quantum query complexity of a function by bounding the change of a progress function caused by one query. All previous variants upperbound the difference of the progress function, whereas our new variant upperbounds the ratio and that is why we coin it the multiplicative adversary. The new method generalizes to all functions the new quantum lowerbound method by Ambainis [Amb05, A ˇ SW06] based on the analysis of eigenspaces of the density matrix. We prove a strong direct product theorem for all functions that have a multiplicative adversary lower bound. 1
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