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Quantum Hamiltonian algorithms for maximum independent sets.

Xianjue Zhao1, Peiyun Ge2, Hongye Yu3

  • 1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.

National Science Review
|September 11, 2025
PubMed
Summary
This summary is machine-generated.

We found two quantum Hamiltonian algorithms for the maximum independent set problem are equivalent. Despite this, simulations reveal performance differences, offering new insights into quantum dynamics and the PXP model.

Keywords:
Rydberg atom arraymaximum independent setnon-Abelian mixingquantum algorithm

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Area of Science:

  • Quantum Computing
  • Computational Complexity Theory
  • Condensed Matter Physics

Background:

  • The maximum independent set problem is a fundamental challenge in computer science.
  • Quantum Hamiltonian algorithms offer potential speedups for complex problems.
  • The PXP model is a significant area of research in quantum dynamics.

Purpose of the Study:

  • To compare two distinct quantum Hamiltonian algorithms for the maximum independent set problem.
  • To investigate the mathematical relationship and performance differences between these algorithms.
  • To reframe the PXP model within the context of quantum diffusion.

Main Methods:

  • Adiabatic evolution utilizing an emergent non-Abelian gauge matrix.
  • Designed application of single-qubit operations.
  • Mathematical analysis and numerical simulations to compare algorithm performance.

Main Results:

  • The two quantum Hamiltonian algorithms were proven to be mathematically equivalent (interaction pictures of each other).
  • Numerical simulations demonstrated significant performance disparities between the equivalent algorithms.
  • Analytical explanations were derived for the observed performance differences.

Conclusions:

  • The equivalence highlights a deep connection between different quantum algorithmic approaches.
  • Performance variations underscore the importance of implementation details in quantum algorithms.
  • The PXP model can be interpreted as quantum diffusion on a graph related to independent sets, governed by a non-Abelian gauge matrix.