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Solving Set Cover with Pairs Problem using Quantum Annealing.

Yudong Cao1, Shuxian Jiang1, Debbie Perouli2

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Quantum annealing offers a novel approach to solving the Set Cover with Pairs (SCP) problem, an important NP-hard optimization challenge. This study details Hamiltonian constructions and simulations, comparing quantum annealing performance against simulated annealing for SCP instances.

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Area of Science:

  • Quantum Computing
  • Combinatorial Optimization
  • Computational Complexity

Background:

  • The Set Cover with Pairs (SCP) problem is an NP-hard combinatorial optimization problem with significant applications in networking, computational biology, and biochemistry.
  • Solving NP-hard problems efficiently remains a major challenge in computer science and various scientific domains.

Purpose of the Study:

  • To explore the application of quantum annealing for solving the Set Cover with Pairs (SCP) problem.
  • To develop and analyze Ising Hamiltonians encoding SCP solutions.
  • To compare the performance of quantum annealing with simulated annealing for SCP instances.

Main Methods:

  • Explicit construction of Ising Hamiltonians for SCP instances.
  • Numerical simulation of the time-dependent Schrödinger equation to model quantum annealing.
  • Comparison of quantum annealing results with simulated annealing.
  • Development of embedding strategies for D-Wave type architectures (Chimera graphs).

Main Results:

  • Demonstrated an explicit construction of Ising Hamiltonians whose ground states represent solutions to SCP.
  • Numerical simulations showed the performance of quantum annealing for random SCP instances.
  • Comparison indicated the potential advantages of quantum annealing over simulated annealing for SCP.
  • Developed embedding strategies for Chimera graphs that preserve SCP instance structure.

Conclusions:

  • Quantum annealing is a viable approach for tackling the Set Cover with Pairs (SCP) problem.
  • The developed Hamiltonian construction and embedding strategies are effective for quantum hardware.
  • The embedding techniques may have broader applicability beyond SCP, including for complete bipartite graphs and logical disjunctions.