Mean field approximation for solving QUBO problems.

Area of Science:

• Computational physics
• Optimization algorithms
• Quantum computing

Background:

• The Quadratic Unconstrained Binary Optimization (QUBO) problem is computationally challenging (NP-hard).
• Existing solutions include exact methods (e.g., Branch-and-Bound) for small instances and approximations (e.g., simulated annealing, mean-field annealing) for larger ones.
• Quantum computing, particularly quantum adiabatic annealing, offers a potential avenue for solving complex optimization problems.

Purpose of the Study:

• To investigate the mean-field approximation of quantum adiabatic annealing.
• To compare its behavior with traditional thermal mean-field annealing.
• To introduce a novel mean-field quantum adiabatic annealing approach for optimization.

Main Methods:

• Applying mean-field approximation to quantum adiabatic annealing.
• Developing a modified annealing process utilizing a new sigmoid function.
• Testing the method on benchmark Maximum Cut (Max-Cut) problems.

Main Results:

• The mean-field approximation of quantum adiabatic annealing yields equations analogous to thermal mean-field annealing.
• A novel sigmoid function replaces the standard thermal sigmoid function.
• The proposed mean-field quantum adiabatic annealing method successfully replicates state-of-the-art cut values on benchmark Max-Cut instances.

Conclusions:

• Mean-field quantum adiabatic annealing provides a viable computational approach for QUBO problems.
• The novel sigmoid function is key to the method's effectiveness.
• This technique demonstrates strong performance on challenging Max-Cut optimization tasks.