The problem of finding a marked node in a graph can be solved by the spatial search algorithm based on continuous-time quantum walks (CTQW). However, this algorithm is known to run in optimal time only for a handful of graphs. In this work, we prove that for Erdös-Renyi random graphs, i.e. graphs of n vertices where each edge exists with probability p, search by CTQW is almost surely optimal as long as p ≥ log 3/2 (n)/n. Consequently, we show that quantum spatial search is in fact optimal for almost all graphs, meaning that the fraction of graphs of n vertices for which this optimality holds tends to one in the asymptotic limit. We obtain this result by proving that search is optimal on graphs where the ratio between the second largest and the largest eigenvalue is bounded by a constant smaller than 1. Finally, we show that we can extend our results on search to establish high fidelity quantum communication between two arbitrary nodes of a random network of interacting qubits, namely to perform quantum state transfer, as well as entanglement generation. Our work shows that quantum information tasks typically designed for structured systems retain performance in very disordered structures.PACS numbers: 03.67. Ac, 03.67.Lx, 03.67.Hk Quantum walks provide a natural framework for tackling the spatial search problem of finding a marked node in a graph of n vertices. In the original work of Childs and Goldstone [1], it was shown that continuous-time quantum walks can search on complete graphs, hypercubes and lattices of dimension larger than four in O( √ n) time, which is optimal. More recently, new instances of graphs have been found where spatial search works optimally. These examples show that global symmetry, regularity and high connectivity are not necessary for the optimality of the algorithm [2][3][4]. However, it is not known how general is the class of graphs for which spatial search by quantum walk is optimal. Here we address the following question: If one picks at random a graph from the set of all graphs of n nodes, can one find a marked node in optimal time using quantum walks? We show that the answer is almost surely yes. Moreover, we adapt the spatial search algorithm to protocols, for state transfer and entanglement generation between arbitrary nodes of a network of interacting qubits, that work with high fidelity for almost all graphs, for large n (nodes and vertices are used interchangeably throughout the paper). Thus, besides showing that spatial search by quantum walk is optimal in a very general scenario, we also show that other important quantum information tasks, typically designed for ordered systems, can be accomplished efficiently in very disordered structures. We obtain our results by studying the spatial search problem in Erdös-Renyi random graphs, i.e. graphs of n vertices where an edge between any two vertices exists with probability p independently of all other edges, typically denoted as G(n, p) [5,6]. Note that our approach is different from the quantum random networks of non-intera...

Continuous time quantum walks provide an important framework for designing new algorithms and modelling quantum transport and state transfer problems. Often, the graph representing the structure of a problem contains certain symmetries that confine the dynamics to a smaller subspace of the full Hilbert space. In this work, we use invariant subspace methods, that can be computed systematically using the Lanczos algorithm, to obtain the reduced set of states that encompass the dynamics of the problem at hand without the specific knowledge of underlying symmetries. First, we apply this method to obtain new instances of graphs where the spatial quantum search algorithm is optimal: complete graphs with broken links and complete bipartite graphs, in particular, the star graph. These examples show that regularity and high-connectivity are not needed to achieve optimal spatial search. We also show that this method considerably simplifies the calculation of quantum transport efficiencies. Furthermore, we observe improved efficiencies by removing a few links from highly symmetric graphs. Finally, we show that this reduction method also allows us to obtain an upper bound for the fidelity of a single qubit transfer on an XY spin network.

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