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Area of Science:

  • Quantum computing
  • Combinatorial optimization
  • Quantum algorithms

Background:

  • The quantum approximate optimization algorithm (QAOA) is a prominent quantum algorithm for combinatorial optimization.
  • QAOA commonly employs a transverse field mixer, which can lead to nonuniform sampling of degenerate ground states.
  • Fair sampling is crucial for ensuring that all optimal solutions have an equal chance of being found.

Purpose of the Study:

  • To numerically examine and compare the fair sampling properties of the transverse field mixer QAOA and the Grover mixer QAOA (GM-QAOA).
  • To quantify fair sampling using Shannon entropy of ground-state amplitudes.
  • To investigate these properties on various quantum signature Hamiltonians and spin glass instances.

Main Methods:

  • Numerical simulations using the JuliQAOA software.
  • Comparison of transverse field mixer QAOA and GM-QAOA performance.
  • Quantification of fair sampling via Shannon entropy.
  • Analysis of QAOA angles and approximation ratios for increasing parameter p.

Main Results:

  • GM-QAOA provides theoretical guarantees for fair sampling of degenerate optimal solutions.
  • Transverse field mixer QAOA demonstrates nonuniform sampling, with some instances showing exponential suppression of degenerate ground states.
  • Some problem instances with transverse field mixer QAOA saturate Shannon entropy at 0 (maximally biased distribution) as approximation ratio approaches 1.
  • Other instances maintain maximum Shannon entropy (uniform distribution) regardless of p.

Conclusions:

  • GM-QAOA exhibits superior fair sampling properties compared to the transverse field mixer QAOA.
  • The choice of mixer significantly impacts the sampling distribution of degenerate ground states in QAOA.
  • Understanding fair sampling is critical for developing effective quantum optimization algorithms.