Efficiently Controllable Graphs

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that when a quantum network is described by such a graph, the network is efficiently controllable, in the sense that universal quantum computation can be performed using a control sequence polynomial in the size of the network while controlling a vanishingly small fraction of subsystems. We show that networks corresponding to finite-dimensional lattices are efficently controllable, and explore generalizations to percolation clusters and random graphs. We show that the classical computational complexity of estimating the ground state of Hamiltonians described by controllable graphs is polynomial in the number of subsystems/qubits

Hofer's metric on the space of diameters

The present paper considers Hofer's distance between diameters in the unit disk. We prove that this distance is unbounded and show its relation to Lagrangian intersections.Comment: 11 pages, 4 figure

On covering expander graphs by Hamilton cycles

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree $\Delta$ satisfies some basic expansion properties and contains a family of $(1-o(1))\Delta/2$ edge disjoint Hamilton cycles, then there also exists a covering of its edges by $(1+o(1))\Delta/2$ Hamilton cycles. This implies that for every $\alpha >0$ and every $p \geq n^{\alpha-1}$ there exists a covering of all edges of $G(n,p)$ by $(1+o(1))np/2$ Hamilton cycles asymptotically almost surely, which is nearly optimal.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some math/061275

On Temporal Graph Exploration

A temporal graph is a graph in which the edge set can change from step to step. The temporal graph exploration problem TEXP is the problem of computing a foremost exploration schedule for a temporal graph, i.e., a temporal walk that starts at a given start node, visits all nodes of the graph, and has the smallest arrival time. In the first part of the paper, we consider only temporal graphs that are connected at each step. For such temporal graphs with $n$ nodes, we show that it is NP-hard to approximate TEXP with ratio $O(n^{1-\epsilon})$ for any $\epsilon>0$. We also provide an explicit construction of temporal graphs that require $\Theta(n^2)$ steps to be explored. We then consider TEXP under the assumption that the underlying graph (i.e. the graph that contains all edges that are present in the temporal graph in at least one step) belongs to a specific class of graphs. Among other results, we show that temporal graphs can be explored in $O(n^{1.5} k^2 \log n)$ steps if the underlying graph has treewidth $k$ and in $O(n \log^3 n)$ steps if the underlying graph is a $2\times n$ grid. In the second part of the paper, we replace the connectedness assumption by a weaker assumption and show that $m$-edge temporal graphs with regularly present edges and with random edges can always be explored in $O(m)$ steps and $O(m \log n)$ steps with high probability, respectively. We finally show that the latter result can be used to obtain a distributed algorithm for the gossiping problem.Comment: This is an extended version of an ICALP 2015 pape

Universally Optimal Noisy Quantum Walks on Complex Networks

Transport properties play a crucial role in several fields of science, as biology, chemistry, sociology, information science, and physics. The behavior of many dynamical processes running over complex networks is known to be closely related to the geometry of the underlying topology, but this connection becomes even harder to understand when quantum effects come into play. Here, we exploit the Kossakoski-Lindblad formalism of quantum stochastic walks to investigate the capability to quickly and robustly transmit energy (or information) between two distant points in very large complex structures, remarkably assisted by external noise and quantum features as coherence. An optimal mixing of classical and quantum transport is, very surprisingly, quite universal for a large class of complex networks. This widespread behaviour turns out to be also extremely robust with respect to geometry changes. These results might pave the way for designing optimal bio-inspired geometries of efficient transport nanostructures that can be used for solar energy and also quantum information and communication technologies.Comment: 17 pages, 12 figure