Searching via Nonlinear Quantum Walk on the 2D-Grid

We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage with respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge The numerical simulations showed that the walker finds the marked vertex in O(N1/4log3/4N) steps, with probability O(1/logN), for an overall complexity of O(N1/4log5/4N), using amplitude amplification. We also proved that there exists an optimal choice of the walker parameters to avoid the time measurement precision affecting the complexity searching time of the algorithm.

Analysis of lackadaisical quantum walks

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any regular locally arc-transitive graph.

Impact of global and local interaction on quantum spatial search on chimera graph

In this paper, we investigated the influence of local and global interaction on the efficiency of continuous-time quantum spatial search. To do so, we analyzed numerically the chimera graph, which is defined as 2D grid with each node replaced by complete bipartite graph. Our investigation provides a numerical evidence that with a large number of local interactions the quantum spatial search is optimal, contrary to the case with limited number of such interactions. The result suggests that relatively large number of local interactions with the marked vertex is necessary for optimal search, which in turn would imply that poorly connected vertices are hard to be found.

Defective 3-Paintability of Planar Graphs

A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.

Bikei invariants and gauss diagrams for virtual knotted surfaces

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in [Formula: see text]; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.

Correcting for potential barriers in quantum walk search

A randomly walking quantum particle searches in Grover’s Θ(√ N) iterations for a marked vertex on the complete graph of N vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a “coin” flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with N. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(√ N) runtime.