1 Quantum query complexity of some graph problems

1 Quantum query complexity of some graph problems C. Dürr M. Heiligman P. Høyer M. Mhalla Univ. Paris-Sud National Security Agency Univ. of Calgary Institut 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 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) • • • v 0 • • • •

1 General algorithm Tree T={v 0} covering vertices S={v 0} while |S|<n add cheapest border edge (u, v)∈E∩Sx(VS) to A add v to S Definition cost of edge (u, v) =shortest path weight(v 0, u) + edge weight(u, v) • • • v 0 • • • •

1 Quantum procedure for finding cheapest border edge Consider the decomposition of |S| into powers of 2 Decompose S into P 1∪…∪Pk s. t. ●|P |>…>|P | 1 k ●and each |P | is a power of 2 i • • • P 1 • • • P 2 • • • P 3 • •

1 Quantum procedure for finding cheapest border edge Consider the decomposition of |S| into powers of 2 Decompose S into P 1∪…∪Pk s. t. ●|P |>…>|P | 1 k ●and each |P | is a power of 2 i ●Suppose for every P we computed A : the |P | cheapest i i i border edges of Pi with distinct targets (for edges with source∈Pi and target∉P 1∪…∪Pi) • • • P 1 • • • P 2 • • A 1 • P 3 A 2 • A 3 • • •

1 Observations Ai∩Sx(VS) (restricted to targets∉S) is non empty for every i ●The cheapest border edge of S (u, v) has its source u∈P for i some i, and therefore v∈Ai ●Thus (A ∪…∪A )∩Sx(VS) 1 k contains the cheapest border edge of S ● • • u • P 1 • • • P 2 • • A 1 • P 3 A 2 v • A 3 • • •

1 Computing Ak using a minimum search procedure Input matrix ℕa×b Output a column disjoint minimal entries Bounded error quantum query complexity (a b) 8 ∞ • • • P 1 • • • P 2 • • P 3 • • • 5 ∞ 2 ∞ ∞ 9

1 Bounded error quantum query complexity Bounded error (classical) quantum query complexity Single source shortest paths Minimum weight spanning tree Connectivity (undirected graph) Strong Connectivity (directed graph) (n 2) (n 3/2), O(n 3/2 log 2 n) (m) ( (nm)), O( (nm)log 2 n) (n 2) (n 3/2) (m) ( (nm)) (n 2) (n 3/2) (m) (n) (n 2) (n 3/2) (m) ( (nm)), O( (nmlogn))