Hardness of the Maximum Independent Set Problem on Unit-Disk Graphs and Prospects for Quantum Speedups

• URL: http://arxiv.org/abs/2307.09442v3
• Date: Tue, 9 Jan 2024 17:07:00 GMT
• Title: Hardness of the Maximum Independent Set Problem on Unit-Disk Graphs and Prospects for Quantum Speedups
• Authors: Ruben S. Andrist, Martin J. A. Schuetz, Pierre Minssen, Romina Yalovetzky, Shouvanik Chakrabarti, Dylan Herman, Niraj Kumar, Grant Salton, Ruslan Shaydulin, Yue Sun, Marco Pistoia, Helmut G. Katzgraber
• Abstract summary: Rydberg atom arrays are among the leading contenders for the demonstration of quantum speedups. We study the maximum independent set problem on unit-disk graphs with a broader range of classical solvers. We find that quasi-planar instances with Union-Jack-like connectivity can be solved to optimality for up to thousands of nodes within minutes.
• Score: 9.927706464212168
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 quasi-planar 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.

Related papers

• Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints [76.53316706600717]
  Recently proposed quantum algorithm arXiv:2206.14999 is based on semidefinite programming (SDP) We generalize the SDP-inspired quantum algorithm to sum-of-squares. Our results show that our algorithm is suitable for large problems and approximate the best known classicals.
  arXiv  Detail & Related papers  (2024-08-14T19:04:13Z)
• An Analysis of Quantum Annealing Algorithms for Solving the Maximum Clique Problem [49.1574468325115]
  We analyse the ability of quantum D-Wave annealers to find the maximum clique on a graph, expressed as a QUBO problem. We propose a decomposition algorithm for the complementary maximum independent set problem, and a graph generation algorithm to control the number of nodes, the number of cliques, the density, the connectivity indices and the ratio of the solution size to the number of other nodes.
  arXiv  Detail & Related papers  (2024-06-11T04:40:05Z)
• Quantum Optimization for the Maximum Cut Problem on a Superconducting Quantum Computer [0.3518016233072556]
  We investigate the performance of a hybrid quantum-classical algorithm inspired by semidefinite approaches for solving the maximum cut problem. We attain an average performance of 99% over a random ensemble of thousands of problem instances. A runtime analysis shows that the quantum solver on large-scale problems is competitive against Gurobi but short of others.
  arXiv  Detail & Related papers  (2024-04-26T17:59:22Z)
• Quantum speedup for combinatorial optimization with flat energy landscapes [0.0]
  We develop a theoretical framework to analyze the relative performance of the optimized quantum adiabatic algorithm and a broad class of classical Markov chain Monte Carlo algorithms.
  arXiv  Detail & Related papers  (2023-06-22T18:00:00Z)
• NISQ-compatible approximate quantum algorithm for unconstrained and constrained discrete optimization [0.0]
  We present an approximate gradient-based quantum algorithm for hardware-efficient circuits with amplitude encoding. We show how simple linear constraints can be directly incorporated into the circuit without additional modification of the objective function with penalty terms.
  arXiv  Detail & Related papers  (2023-05-23T16:17:57Z)
• Quantum Clustering with k-Means: a Hybrid Approach [117.4705494502186]
  We design, implement, and evaluate three hybrid quantum k-Means algorithms. We exploit quantum phenomena to speed up the computation of distances. We show that our hybrid quantum k-Means algorithms can be more efficient than the classical version.
  arXiv  Detail & Related papers  (2022-12-13T16:04:16Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Quantum optimization with arbitrary connectivity using Rydberg atom arrays [0.0]
  We extend the classes of problems that can be efficiently encoded in Rydberg arrays by constructing explicit mappings from 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.
  arXiv  Detail & Related papers  (2022-09-08T18:00:00Z)
• Entanglement and coherence in Bernstein-Vazirani algorithm [58.720142291102135]
  Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. We analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state.
  arXiv  Detail & Related papers  (2022-05-26T20:32:36Z)
• Quantum Optimization of Maximum Independent Set using Rydberg Atom Arrays [39.76254807200083]
  We experimentally investigate quantum algorithms for solving the Maximum Independent Set problem. We find the problem hardness is controlled by the solution degeneracy and number of local minima. On the hardest graphs, we observe a superlinear quantum speedup in finding exact solutions.
  arXiv  Detail & Related papers  (2022-02-18T19:00:01Z)
• Adiabatic Quantum Computing for Multi Object Tracking [170.8716555363907]
  Multi-Object Tracking (MOT) is most often approached in the tracking-by-detection paradigm, where object detections are associated through time. As these optimization problems are often NP-hard, they can only be solved exactly for small instances on current hardware. We show that our approach is competitive compared with state-of-the-art optimization-based approaches, even when using of-the-shelf integer programming solvers.
  arXiv  Detail & Related papers  (2022-02-17T18:59:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.