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Biomolecular and quantum algorithms for the dominating set problem in arbitrary networks.

Renata Wong1, Weng-Long Chang2, Wen-Yu Chung3

  • 1Physics Division, National Center for Theoretical Sciences, National Taiwan University, Taipei, 10617, Taiwan. renata.wong@phys.ncts.ntu.edu.tw.

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Researchers developed a quantum algorithm to solve the graph dominating-set problem, offering a significant speedup over classical methods. This quantum approach is the most efficient known solution for this complex optimization challenge.

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Area of Science:

  • Graph Theory
  • Quantum Computing
  • Computational Complexity

Background:

  • The dominating-set problem seeks a minimum set of vertices in a graph such that all other vertices are either in the set or adjacent to it.
  • This problem is crucial for organizing large-scale wireless ad hoc and sensor networks, but is computationally intractable for classical computers.
  • Efficiently solving the dominating-set problem has significant implications for network optimization and communication.

Purpose of the Study:

  • To propose novel biomolecular and quantum algorithms for the dominating-set problem.
  • To demonstrate a quadratic speedup for the dominating-set problem using a quantum algorithm compared to classical approaches.
  • To validate the quantum algorithm's performance and correctness through practical implementation and comparison.

Main Methods:

  • Development of a quantum algorithm tailored for the dominating-set problem.
  • Analysis of the quantum algorithm's query complexity, showing a solution in O(n) queries.
  • Experimental verification of the algorithm on IBM Quantum's qasm simulator and a superconducting quantum device.

Main Results:

  • The proposed quantum algorithm solves the dominating-set problem in O(n) queries, achieving a quadratic speedup.
  • The quantum algorithm represents the best-known procedure for solving the dominating-set problem to date.
  • Successful execution and validation of the algorithm on quantum hardware confirmed its correctness.

Conclusions:

  • Quantum computation offers a powerful approach to tackle NP-hard problems like the dominating-set problem.
  • The developed quantum algorithm provides a significant advancement in solving complex graph optimization problems.
  • Molecular solutions to the dominating-set problem can be represented within a finite-dimensional Hilbert space framework.