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arXiv:2209.03965 · PRX Quantum

Quantum Optimization with Arbitrary Connectivity Using Rydberg Atom Arrays

Minh-Thi Nguyen, Jin-Guo Liu, Jonathan Wurtz, Mikhail D. Lukin, Sheng-Tao Wang, Hannes Pichler

Published Feb 14, 2023• 119 citations
arXivHTMLPDF

Abstract

Programmable quantum systems based on Rydberg atom arrays have recently been used for hardwareefficient tests of quantum optimization algorithms [Ebadi et al., Science, 376, 1209 (2022)] 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.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.

DOI
10.1103/prxquantum.4.010316

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Programmable quantum systems based on Rydberg atom arrays have recently been used for hardwareefficient tests of quantum optimization algorithms [Ebadi et al., Science, 376, 1209 (2022)] 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.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 based on Rydberg atom arrays have recently emerged as a promising test bed for combinatorial optimization. Indeed, the maximum weighted independent set problem on unit-disk graphs can be efficiently mapped to such systems due to their geometric constraints. However, extending this capability to arbitrary graph instances typically necessitates the use of reduction gadgets, which introduce additional experimental overhead and complexity. Here, we analyze the complexity\penalty1000-\hskip0pttheoretic limits of polynomial reductions from arbitrary graphs to unit\penalty1000-\hskip0ptdisk instances. We prove any such reduction incurs a quadratic blow up in vertex count and degrades solution approximation guarantees. As a practical alternative, we propose a divide\penalty1000-\hskip0ptand\penalty1000-\hskip0ptconquer heuristic with only linear overhead, which leverages precalibrated atomic layouts. We benchmark it on Erd\"os-R\'enyi graphs, and demonstrate feasibility on the Orion Alpha processor.
Rydberg atom arrays are among the leading contenders for the demonstration of quantum speedups. Motivated by recent experiments with up to 289 qubits [Ebadi et al., Science 376, 1209 (2022)], we study the maximum-independent-set problem on unit-disk graphs with a broader range of classical solvers beyond the scope of the original paper. We carry out extensive numerical studies and assess problem hardness, using both exact and heuristic algorithms. We find that quasiplanar instances with Union-Jack-like connectivity can be solved to optimality for up to thousands of nodes within minutes, with both custom and generic commercial solvers on commodity hardware, without any instance-specific fine-tuning. We also perform a scaling analysis, showing that by relaxing the constraints on the classical simulated annealing algorithms considered in Ebadi et al., our implementation is competitive with the quantum algorithms. Conversely, instances with larger connectivity or less structure are shown to display a time-to-solution potentially orders of magnitudes larger. Based on these results, we propose protocols to systematically tune problem hardness, motivating experiments with Rydberg atom arrays on instances orders of magnitude harder (for established classical solvers) than previously studied.
Realizing quantum speedup for practically relevant, computationally hard problems is a central challenge in quantum information science. Using Rydberg atom arrays with up to 289 qubits in two spatial dimensions, we experimentally investigate quantum algorithms for solving the maximum independent set problem. We use a hardware-efficient encoding associated with Rydberg blockade, realize closed-loop optimization to test several variational algorithms, and subsequently apply them to systematically explore a class of graphs with programmable connectivity. We find that the problem hardness is controlled by the solution degeneracy and number of local minima, and we experimentally benchmark the quantum algorithm's performance against classical simulated annealing. On the hardest graphs, we observe a superlinear quantum speedup in finding exact solutions in the deep circuit regime and analyze its origins.
Programmable quantum systems utilizing Rydberg atoms have recently been shown to efficiently encode combinatorial optimization problems. The relationships among the variables in these optimization problems can be transformed into a maximum weighted independent set problem within the Rydberg atom system, requiring at most a quadratic overhead in terms of the number of qubits. Although experiments with current Rydberg quantum systems have demonstrated linear speedup compared to classical algorithms, the overhead in both the number of qubits and time remains substantial. To further reduce the overhead of encoding these optimization problems in Rydberg atom systems, we propose a simplification strategy and design a new heuristic algorithm that incorporates a topological sort approach to optimize the encoding process. We conduct extensive experiments on small graphs with dozens of qubits. The numerical results demonstrate the effectiveness of our algorithm in both runtime and optimization efficiency. Specifically, our algorithm reduces the number of required encoding qubits by 12.0% compared to existing methods. Additionally, our approach reduces the time complexity to a quadratic overhead, enabling experiments on larger graphs. The results further indicate that our algorithm provides significant time advantages over existing solutions.
Neutral atom arrays provide a versatile platform to implement coherent quantum annealing as an approach to solving hard combinatorial optimization problems. Here we present and experimentally demonstrate an efficient encoding scheme based on chains of Rydberg-blockaded atoms, which we call quantum wires, to natively embed maximum weighted independent set (MWIS) and quadratic unconstrained binary optimization (QUBO) problems on a neutral atom architecture. For graphs with quasi-unit-disk connectivity, in which only a few long-range edges are required, our approach requires a significantly lower overhead in the number of ancilla qubits than previous proposals, facilitating the implementation on currently available hardware. To demonstrate the approach, we perform annealing of weighted graphs on a programmable atom array using local light shifts to encode problem-specific weights across graphs of varying sizes. This approach successfully identifies the solutions to the original MWIS and QUBO graph instances. Our work expands the operational toolkit of near-term neutral atom arrays, enhancing their potential for scalable quantum optimization.
Recent progress in quantum computing and quantum simulation of many-body systems with arrays of neutral atoms using Rydberg excitation has provided unforeseen opportunities towards computational advantage in solving various optimization problems. The problem of a maximum-weight independent set of unit-disk graphs is an example of an NP-hard optimization problem. It involves finding the largest set of vertices with the maximum sum of their weights for a graph which has edges connecting all pairs of vertices within a unit distance. This problem can be solved using quantum annealing with an array of interacting Rydberg atoms. For a particular graph, a spatial arrangement of atoms represents vertices of the graph, while the detuning from resonance at Rydberg excitation defines the weights of these vertices. The edges of the graph can be drawn according to the unit-disk criterion. Maximum-weight independent sets can be obtained by applying a variational quantum adiabatic algorithm. We consider driving the quantum system of interacting atoms to the many-body ground state using a nonlinear quasiadiabatic profile for sweeping the Rydberg detuning. We also propose using a quantum wire, which is a set of auxiliary atoms of a different chemical element, to mediate strong coupling between the remote vertices of the graph. We investigate this effect for different lengths of the quantum wire. We also investigate the quantum phases of matter realizing commensurate and incommensurate phases in one- and two-dimensional spatial arrangements of the atomic array.