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arXiv:2405.09803 · Physical Review A

Generation of quantum phases of matter and finding a maximum-weight independent set of unit-disk graphs using Rydberg atoms

Ahmed M. Farouk, Ilya I. Beterov, Peng Xu, I. I. Ryabtsev

Published Aug 27, 2024• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.110.022442

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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.
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.
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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.
We describe and analyze an architecture for quantum optimization to solve maximum independent set (MIS) problems using neutral atom arrays trapped in optical tweezers. Optimizing independent sets is one of the paradigmatic, NP-hard problems in computer science. Our approach is based on coherent manipulation of atom arrays via the excitation into Rydberg atomic states. Specifically, we show that solutions of MIS problems can be efficiently encoded in the ground state of interacting atoms in 2D arrays by utilizing the Rydberg blockade mechanism. By studying the performance of leading classical algorithms, we identify parameter regimes, where computationally hard instances can be tested using near-term experimental systems. Practical implementations of both quantum annealing and variational quantum optimization algorithms beyond the adiabatic principle are discussed.
Neutral-atom arrays have emerged as a versatile platform toward scalable quantum computation and optimization. In this paper, we present demonstrations of solving maximum-weighted independent-set problems on a Rydberg-atom array using annealing with local light shifts. We verify the ability to prepare weighted graphs in one-dimensional (1D) and two-dimensional (2D) arrays, including embedding a five-vertex nonunit-disk graph using nine physical qubits and demonstration of a simple crossing gadget. We find common annealing ramps leading to preparation of the target ground state robustly over a substantial range of different graph weightings. This work provides a route to exploring large-scale optimization of nonplanar weighted graphs relevant for solving relevant real-world problems.
In the past years, many quantum algorithms have been proposed to tackle hard combinatorial problems. In particular, the maximum independent set (MIS) is a known NP-hard problem that can be naturally encoded in Rydberg atom arrays. By representing a graph with an ensemble of neutral atoms one can leverage Rydberg dynamics to naturally encode the constraints and the solution to MIS. However, the classes of graphs that can be directly mapped ``vertex-to-atom'' on standard devices with two-dimensional capabilities are currently limited to Unit-Disk graphs. In this setting, the inherent spatial locality of the graphs can be leveraged by classical polynomial-time approximation schemes (PTAS) that guarantee an $\ensuremath{\epsilon}$-approximate solution. In this work, we build upon recent progress made for using three-dimensioanl arrangements of atoms to embed more complex classes of graphs. We report experimental and theoretical results which represent important steps towards tackling combinatorial tasks on quantum computers for which no classical efficient $\ensuremath{\varepsilon}$-approximation scheme exists.