1 Quantum query complexity of some graph problems C. DürrUniv. Paris-Sud M. HeiligmanNational...
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1 Quantum query complexity of some graph problems C. Dürr Univ. Paris-Sud M. Heiligman National Security Agency P. Høyer Univ. of Calgary M. Mhalla Institut IMAG, Grenoble
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Transcript of 1 Quantum query complexity of some graph problems C. DürrUniv. Paris-Sud M. HeiligmanNational...
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Quantum query complexity of some graph problems
C. Dürr Univ. Paris-SudM. Heiligman National Security AgencyP. Høyer Univ. of Calgary M. Mhalla Institut IMAG, Grenoble
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Single Source Shortest Paths
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Given a directed graph G(V,E), with non-negative edge weights and a source vertex v0
find the shortest paths to all vertices v
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How many queries of the type ''what is the weight of the edge (u,v)?'' are necessary to solve the problem with bounded error?
Classical (n2)Quantum(n3/2), O(n3/2log3/2n)
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Single Source Shortest Paths
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Given a directed graph G(V,E), with non-negative edge weights and a source vertex v0
find the shortest paths to all vertices v
v01 1 4 3 12
How many queries of the type ''what is the weight of the edge (u,v)?'' are necessary to solve the problem with bounded error?
Classical (n2)Quantum(n3/2), O(n3/2log3/2n)
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General algorithm
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Tree T={v0} covering vertices S={v0}while |S|<n
add cheapest border edge (u,v)∈E∩Sx(V\S) to A
add v to S
Definition cost of edge (u,v) =shortest path weight(v0,u) + edge
weight(u,v)
v0
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Quantum procedure for finding cheapest border edgeConsider the decomposition of |S| into powers of 2Decompose S into P1∪…∪Pk s.t.
●|P1|>…>|Pk| ●and each |Pi| is a power of 2
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Quantum procedure for finding cheapest border edgeConsider the decomposition of |S| into powers of 2Decompose S into P1∪…∪Pk s.t.
●|P1|>…>|Pk| ●and each |Pi| is a power of 2●Suppose for every Pi we computed Ai : the |Pi| cheapest border edges of Pi with distinct targets(for edges with source∈Pi and target∉P1∪…∪Pi)
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Observations●Ai∩Sx(V\S) (restricted to targets∉S) is non empty for every i ●The cheapest border edge of S (u,v) has its source u∈Pi for some i, and therefore v∈Ai●Thus (A1∪…∪Ak)∩Sx(V\S) contains the cheapest border edge of S
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A1 A2
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Computing Ak usinga minimum search procedure
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Inputmatrix ℕa×b
Outputa column disjoint minimal entriesBounded error quantum query complexity(ab)
8 5 ∞∞
∞ 2 9∞
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Single source (n2) (n3/2), O(n3/2log2n) (m) ((nm)), O((nm)log2n)shortest pathsMinimum weight (n2) (n3/2) (m) ((nm))spanning treeConnectivity (n2) (n3/2) (m) (n)(undirected graph)Strong Connectivity (n2) (n3/2)(m) ((nm)), O((nmlogn))(directed graph)
Bounded error quantum query complexity
Adjacency matrix model
1: 2: 3: 4:1: 0 1 1 02: 1 0 0 03: 1 1 0 14: 0 0 0 0
Bounded error (classical) quantum query complexity
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Adjacency array model1: 2 32: 13: 1 4 24:
