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arXiv:2402.09180 · Advanced Quantum Technologies

Approximating Maximum Independent Set on Rydberg Atom Arrays Using Local Detunings

Hyeonjun Yeo, Ha Eum Kim, Kabgyun Jeong

Published Sep 26, 2024• 4 citations
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Abstract

Abstract Rydberg atom arrays operated by a quantum adiabatic principle are among the most promising quantum simulating platforms due to their scalability and long coherence time. From the perspective of combinatorial optimization, they offer an efficient solution for an intrinsic maximum independent set problem because of the resemblance between the Rydberg Hamiltonian and the cost function of the maximum independent set problem. In this study, a strategy is suggested to approximate maximum independent sets by adjusting local detunings on the Rydberg Hamiltonian according to each vertex's vertex support, which is a quantity that represents connectivity between vertices. By doing so, the strategy successfully reduces the error rate three times for the checkerboard graphs with defects when the adiabaticity is sufficient. In addition, the strategy decreases the error rate for random graphs even when the adiabaticity is relatively insufficient. Moreover, it is shown that the strategy helps to prepare a quantum many‐body ground state by raising the fidelity between the evolved quantum state and a 2D cat state on a square lattice. Finally, the strategy is combined with the non‐abelian adiabatic mixing and this approach is highly successful in finding maximum independent sets compared to the conventional adiabatic evolution with local detunings.

DOI
10.1002/qute.202400291

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Abstract Rydberg atom arrays operated by a quantum adiabatic principle are among the most promising quantum simulating platforms due to their scalability and long coherence time. From the perspective of combinatorial optimization, they offer an efficient solution for an intrinsic maximum independent set problem because of the resemblance between the Rydberg Hamiltonian and the cost function of the maximum independent set problem. In this study, a strategy is suggested to approximate maximum independent sets by adjusting local detunings on the Rydberg Hamiltonian according to each vertex's vertex support, which is a quantity that represents connectivity between vertices. By doing so, the strategy successfully reduces the error rate three times for the checkerboard graphs with defects when the adiabaticity is sufficient. In addition, the strategy decreases the error rate for random graphs even when the adiabaticity is relatively insufficient. Moreover, it is shown that the strategy helps to prepare a quantum many‐body ground state by raising the fidelity between the evolved quantum state and a 2D cat state on a square lattice. Finally, the strategy is combined with the non‐abelian adiabatic mixing and this approach is highly successful in finding maximum independent sets compared to the conventional adiabatic evolution with local detunings.
Adiabatic quantum computation with Rydberg atoms provides a natural route for solving combinatorial optimization problems such as the maximum independent set (MIS). However, its performance is fundamentally limited by the reduction of the spectral gap with increasing system size and connectivity, which induces population leakage from the ground state during finite-time evolution. Here we introduce the Adjusted Detuning for Ground-Energy Leakage Blockade (ADGLB), a spectral-gap-guided schedule engineering method that modifies the laser detuning profile to suppress leakage without introducing additional Hamiltonian terms or iterative optimization loops. We experimentally benchmark ADGLB on a quasi-one-dimensional chain of N=10N=10N=10 atoms, and the MIS preparation probability increases substantially compared with the standard adiabatic schedule. Furthermore, we show that the schedule optimized for smaller instances can be directly applied to larger two-dimensional triangular lattices with N=25N=25N=25 and N=37N=37N=37. With a small heuristic offset, the method also remains effective for instances with higher hardness parameters. These findings demonstrate that spectral-gap-guided schedule engineering offers a scalable and hardware-efficient strategy for enhancing adiabatic quantum optimization on neutral-atom platforms.
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.
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.
In the past years, many quantum algorithms have been proposed to tackle hard combinatorial problems. These algorithms, which have been studied in depth in complexity theory, are at the heart of many industrial applications. 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 node-to-atom on such devices are limited to Unit-Disk graphs. In this setting, the inherent locality of the graphs can be leveraged by classical polynomial-time approximation schemes (PTAS) that guarantee an ε-approximate solution. In this work, we present a deterministic and polynomial-time construction to embed a large family of non-local graphs in 3D atomic arrays. This construction is a first crucial step towards tackling combinatorial tasks on quantum computers for which no classical efficient ε-approximation scheme exists.
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.
Rydberg atom arrays are a powerful platform for solving combinatorial optimization problems, owing to the Rydberg blockade mechanism, which imposes effective constraints on simultaneous atomic excitations. These constraints have enabled the encoding of the Maximum Independent Set (MIS) problem on unit disk graphs, where atoms interact within a fixed, globally defined blockade radius. However, this restriction limits the class of addressable problems. A natural extension is to consider disk graphs, which generalize unit disk graphs by allowing arbitrary disk radii and correspond to the intersection graphs of disks in the plane. Embedding such graphs in Rydberg systems requires moving beyond the standard, globally uniform blockade model. In this work, we introduce a local Rydberg blockade regime, which emerges when local drives are applied to different pairs of atoms involved in a potential interaction. We develop a general theoretical framework for this regime and propose two novel metrics, the correlation matrix and the maximum independence violation, to quantify the quality of the embedding. Using these metrics, we demonstrate that disk graphs can be meaningfully embedded into Rydberg atom arrays under local drive schemes, thereby expanding the landscape of quantum-addressable optimization problems. Finally, when evaluating approximate solutions of the MIS problem, characterized by near-optimal independent sets, local drive approaches exhibit significantly improved performance over global ones. These results highlight the practical advantage of local blockade engineering for approximate combinatorial optimization and open a path toward leveraging the analog capabilities of Rydberg platforms beyond conventional geometric constraints.