Entanglement in quantum graph states is intrinsically linked to rank-width, a graph complexity measure introduced by Oum and Seymour. In this work, we enable the preparation of maximally entangled deterministic graph states in constant depth by developing a general method to derive lower bounds on the rank-width of regular graphs from their edge expansion. By bridging edge-isoperimetric inequalities with the strong chromatic index and Jelínek's approach for lower bounding cut-rank, we systematically establish lower bounds for the rank-width of Cartesian products, including hypercubes, Hamming graphs, and grids. Extending this framework via Boolean function analysis, using a generalization of the Kahn-Kalai-Linial's Theorem, we strengthen the bounds for all Cartesian products by a non-trivial logarithmic factor. These methods result in the discovery of deterministic families of graphs on n vertices with a provably maximum rank-width Θ(n). Our results fill the previous gap in the literature for deterministic graph families of rank-width greater than Θ(n).
We present heuristic algorithms for minor embedding planar graphs in grids. Our work is motivated by the development of quantum computing hardware that performs quantum annealing. This hardware can be used to solve hard combinatorial problems, but requires the graph of each problem instance to be minor embedded in a graph that models the hardware. Hence, there is a need for practical minor embedding algorithms. We restrict our attention to planar graphs, and thus are able to make use of existing graph drawing methods. We present two algorithms for minor embedding planar graphs in grid graphs, and provide an experimental evaluation of one.