Current quantum computing devices have different strengths and weaknesses depending on their architectures. This means that flexible approaches to circuit design are necessary. We address this task by introducing a novel space-efficient quantum optimization algorithm for the graph coloring problem. Our circuits are deeper than the ones of the standard approach. However, the number of required qubits is exponentially reduced in the number of colors. We present extensive numerical simulations demonstrating the performance of our approach. Furthermore, to explore currently available alternatives, we also perform a study of random graph coloring on a quantum annealer to test the limiting factors of that approach, too.
In this paper, we show that all nodes can be found optimally for almost all random Erdős-Rényi G(n, p) graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires p = ω(log 8 (n)/n), while the second requires p ≥ (1+ε) log(n)/n, where ε > 0. The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the · ∞ norm. At the same time for p < (1 − ε) log(n)/n, the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.