Quantum walks as a probe of structural anomalies in graphs

We study how quantum walks can be used to find structural anomalies in graphs via several examples. Two of our examples are based on star graphs, graphs with a single central vertex to which the other vertices, which we call external vertices, are connected by edges. In the basic star graph, these are the only edges. If we now connect a subset of the external vertices to form a complete subgraph, a quantum walk can be used to find these vertices with a quantum speedup. Thus, under some circumstances, a quantum walk can be used to locate where the connectivity of a network changes. We also look at the case of two stars connected at one of their external vertices. A quantum walk can find the vertex shared by both graphs, again with a quantum speedup. This provides an example of using a quantum walk in order to find where two networks are connected. Finally, we use a quantum walk on a complete bipartite graph to find an extra edge that destroys the bipartite nature of the graph.Comment: 10 pages, 2 figure

Quantum searches on highly symmetric graphs

We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an $M$-partite graph. In all cases, the dimension of the Hilbert space in which the time evolution of the walk takes place is small (between three and six), so the walks can be completely analyzed analytically. Such dimensional reduction is due to the fact that these graphs have large automorphism groups. We find the usual quadratic quantum speedups in all cases considered.Comment: 11 pages, 6 figures; major revision

Searching via walking: How to find a marked subgraph of a graph using quantum walks

We show how a quantum walk can be used to find a marked edge or a marked complete subgraph of a complete graph. We employ a version of a quantum walk, the scattering walk, which lends itself to experimental implementation. The edges are marked by adding elements to them that impart a specific phase shift to the particle as it enters or leaves the edge. If the complete graph has N vertices and the subgraph has K vertices, the particle becomes localized on the subgraph in O(N/K) steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resource needed for a quantitative comparison of the efficiency of classical and quantum searches -- the number of oracle calls.Comment: 4 pages, 2 figure

Incompatible measurements on quantum causal networks

The existence of incompatible measurements, epitomized by Heisenberg's uncertainty principle, is one of the distinctive features of quantum theory. So far, quantum incompatibility has been studied for measurements that test the preparation of physical systems. Here we extend the notion to measurements that test dynamical processes, possibly consisting of multiple time steps. Such measurements are known as testers and are implemented by interacting with the tested process through a sequence of state preparations, interactions, and measurements. Our first result is a characterization of the incompatibility of quantum testers, for which we provide necessary and sufficient conditions. Then we propose a quantitative measure of incompatibility. We call this measure the robustness of incompatibility and define it as the minimum amount of noise that has to be added to a set of testers in order to make them compatible. We show that (i) the robustness is lower bounded by the distinguishability of the sequence of interactions used by the tester and (ii) maximum robustness is attained when the interactions are perfectly distinguishable. The general results are illustrated in the concrete example of binary testers probing the time evolution of a single-photon polarization