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Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments
"... Background. Triangle finding is a graphtheoretic problem whose complexity is deeply connected to the complexity of several other computational tasks in theoretical computer science, such as solving path or matrix problems [3, 8, 9, 13, 18, 17, 19]. In its standard version (sometimes called unweight ..."
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Background. Triangle finding is a graphtheoretic problem whose complexity is deeply connected to the complexity of several other computational tasks in theoretical computer science, such as solving path or matrix problems [3, 8, 9, 13, 18, 17, 19]. In its standard version (sometimes called unweighted triangle finding), it asks to find, given an undirected and unweighted graph G = (V,E), three vertices v1, v2, v3 ∈ V such that {v1, v2}, {v1, v3} and {v2, v3} are edges of the graph. Problems like triangle finding can be studied in the query complexity setting. In the usual model used to describe the query complexity of such problems, the set of edges E of the graph is unknown but can be accessed through an oracle: given two vertices u and v in V, one query to the oracle outputs one if {u, v} ∈ E and zero if {u, v} / ∈ E. In the quantum query complexity setting, one further assume that the oracle can be queried in superposition. Besides its intrinsic interest, the triangle finding problem has been one of the main problems that stimulated the development of new techniques in quantum query complexity, and the history of improvement of upper bounds on the query complexity of triangle finding parallels the development of general techniques in the quantum complexity setting, as we explain below. Grover search immediately gives, when applied to triangle finding as a search over the space of triples of vertices of the graph, a quantum algorithm with query complexity O(n3/2). Using amplitude
4 Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments
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Optimal parallel quantum query algorithms
"... We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number o ..."
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We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Θ((n/p)2/3) pparallel queries for element distinctness and Θ((n/p)k/(k+1)) for ksum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical pparallel complexity are polynomially related for all total functions when p is not too large.