1 Quantum query complexity of some graph problems C. DürrUniv. Paris-Sud M. HeiligmanNational Security Agency P. HøyerUniv. of Calgary M. MhallaInstitut.

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1 Quantum query complexity of some graph problems C. DürrUniv. Paris-Sud M. HeiligmanNational Security Agency P. HøyerUniv. of Calgary M. MhallaInstitut IMAG, Grenoble

1 Single Source Shortest Paths Given a directed graph G(V,E), with non-negative edge weights and a source vertex v 0 find the shortest paths to all vertices v v0v 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  (n 2 ) Quantum  (n 3/2 ), O(n 3/2 log 3/2 n)

1 Single Source Shortest Paths Given a directed graph G(V,E), with non-negative edge weights and a source vertex v 0 find the shortest paths to all vertices v v0v 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  (n 2 ) Quantum  (n 3/2 ), O(n 3/2 log 3/2 n)

1 General algorithm Tree T={v 0 } covering vertices S={v 0 } 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(v 0,u) + edge weight(u,v) v0v0

1 P3P3 Quantum procedure for finding cheapest border edge Consider the decomposition of |S| into powers of 2 Decompose S into P 1 ∪ … ∪ P k s.t. ● |P 1 |>…>|P k | ● and each |P i | is a power of 2 P1P1 P2P2

1 P3P3 Quantum procedure for finding cheapest border edge Consider the decomposition of |S| into powers of 2 Decompose S into P 1 ∪ … ∪ P k s.t. ● |P 1 |>…>|P k | ● and each |P i | is a power of 2 ● Suppose for every P i we computed A i : the |P i | cheapest border edges of P i with distinct targets (for edges with source ∈ P i and target ∉ P 1 ∪ … ∪ P i ) P1P1 P2P2 A1A1 A2A2 A3A3

1 P3P3 Observations ● A i ∩ 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 ∈ P i for some i, and therefore v ∈ A i ● Thus (A 1 ∪ … ∪ A k ) ∩ Sx(V\S) contains the cheapest border edge of S P1P1 P2P2 A1A1 A2A2 A3A3 u v

1 Computing A k using a minimum search procedure P3P3 P1P1 P2P Input matrix ℕ a × b Output a column disjoint minimal entries Bounded error quantum query complexity  (a  b) 85∞∞∞29∞85∞∞∞29∞

1 Single source  (n 2 )  (n 3/2 ), O(n 3/2 log 2 n)  (m)  (  (nm)), O(  (nm)log 2 n) shortest paths Minimum weight  (n 2 )  (n 3/2 )  (m)  (  (nm)) spanning tree Connectivity  (n 2 )  (n 3/2 )  (m)  (n) (undirected graph) Strong Connectivity  (n 2 )  (n 3/2 )  (m)  (  (nm)), O(  (nmlogn)) (directed graph) Bounded error quantum query complexity Adjacency matrix model 1: 2: 3: 4: 1: : : : Bounded error (classical) quantum query complexity Adjacency array model 1: 2 3 2: 1 3: :