(b) MWIS representation of n 1 n 2 = 0, with the third, unlabeled vertex being an ancillary vertex. The MWISs correspond to the satisfying assignments to the corresponding constraint-satisfaction problem. For each example, the degenerate MWIS configurations are shown by identifying vertices in a MWIS with a red boundary. The weight of the vertices is indicated by its interior color on a gray scale. Each bit is represented by a corresponding vertex in the MWIS problem graph. MWIS representation of some example constraints. 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. 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. In this work we significantly expand the class of problems that can be addressed with Rydberg atom arrays, overcoming the limitations to geometric graphs. For instance, Rydberg atom arrays naturally allow encoding maximum independent set problems, but native encodings are restricted to so-called unit disk graphs. In particular, the native connectivity of the qubits for a given platform typically restricts the class of problems that can addressed. Some of the main practical limitations in this context are often set by specific hardware restrictions. Programmable quantum systems offer unique possibilities to test the performance of various quantum optimization algorithms.
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. 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. 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. 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. In particular, the maximum independent set problem on so-called unit-disk graphs, was shown to be efficiently encodable in such a quantum system. Programmable quantum systems based on Rydberg atom arrays have recently been used for hardware-efficient tests of quantum optimization algorithms with hundreds of qubits.
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