![]() ![]() Programmable quantum systems based on Rydberg atom arrays have recently been used for hardware-efficient tests of quantum optimization algorithms with hundreds of qubits. In particular, the maximum independent set problem on so-called unit-disk graphs, was shown to be efficiently encodable in such a quantum system. Here, we extend the classes of problems that can be efficiently encoded in Rydberg arrays by constructing explicit mappings from a wide class of problems to maximum-weighted independent set problems on unit-disk graphs, with at most a quadratic overhead in the number of qubits. begingroup A concrete example of a graph that cant be represented as a unit disk graph is a star with more than 7 vertices (including the centre): All of the leaf disks need to overlap the central vertexs disk and not touch each other, but the kissing number of a circle is 6. We analyze several examples, including maximum-weighted independent set on graphs with arbitrary connectivity, quadratic unconstrained binary optimization problems with arbitrary or restricted connectivity, and integer factorization. Numerical simulations on small system sizes indicate that the adiabatic time scale for solving the mapped problems is strongly correlated with that of the original problems. Our work provides a blueprint for using Rydberg atom arrays to solve a wide range of combinatorial optimization problems with arbitrary connectivity, beyond the restrictions imposed by the hardware geometry. Programmable quantum systems offer unique possibilities to test the performance of various quantum optimization algorithms. We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. Some of the main practical limitations in this context are often set by specific hardware restrictions. In particular, the native connectivity of the qubits for a given platform typically restricts the class of problems that can addressed. Our main contribution is an approach to design subexponential-time FPT algorithms for problems on disk graphs, which we apply to several well-studied graph problems. For instance, Rydberg atom arrays naturally allow encoding maximum independent set problems, but native encodings are restricted to so-called unit disk graphs. In this work we significantly expand the class of problems that can be addressed with Rydberg atom arrays, overcoming the limitations to geometric graphs. We develop a specific encoding scheme to map a variety of problems into arrangements of Rydberg atoms, including maximum weighted independent sets on graphs with arbitrary connectivity, quadratic unconstrained binary optimization problems with arbitrary or restricted connectivity, and integer factorization. ![]() Our work thus provides a blueprint for using Rydberg atom arrays to solve a wide range of combinatorial optimization problems, using technology already available in experiments. MWIS representation of some example constraints. Each bit is represented by a corresponding vertex in the MWIS problem graph. The weight of the vertices is indicated by its interior color on a gray scale. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit. ![]() For each example, the degenerate MWIS configurations are shown by identifying vertices in a MWIS with a red boundary. A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. The MWISs correspond to the satisfying assignments to the corresponding constraint-satisfaction problem. Is allowed), both of our algorithms are almost optimal.(b) MWIS representation of n 1 n 2 = 0, with the third, unlabeled vertex being an ancillary vertex. The $\Omega(n \log n)$-time lower bound of the problem (even when approximation Berkeley, uncovers the parameters of a single disk. Download a PDF of the paper titled Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs, by Haitao Wang and 1 other authors Download PDF Abstract: We revisit a classical graph-theoretic problem, the \textit)))$ time. The Skippy algorithm, from work by Nisha Talagala and colleagues at U.C. ![]()
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