Combinatorial optimization using dynamical phase transitions in driven-dissipative systems

Area of Science:

• Physics
• Computational Science
• Applied Mathematics

Background:

• Combinatorial optimization problems are computationally challenging.
• Ising Hamiltonians are a common framework for representing such problems.
• Driven-dissipative systems offer novel computational paradigms.

Purpose of the Study:

• To propose a method for efficient combinatorial optimization using driven-dissipative systems.
• To demonstrate the applicability to problems reducible to Ising Hamiltonian minimization.
• To explore physical implementations for solving optimization tasks.

Main Methods:

• Utilizing the normal form of the supercritical pitchfork bifurcation for generic dynamics.
• Employing a hybrid analog-digital representation of Ising spins.
• Minimizing a Lyapunov function composed of an Ising Hamiltonian and tunable well potentials.

Main Results:

• Stable steady states correspond to global minima of the Ising Hamiltonian under specific conditions.
• Amplitude heterogeneity in analog spins can lead to local minima, degrading solution quality.
• Tuning driving signal parameters near a dynamic phase transition improves solution accuracy by reducing amplitude heterogeneity.

Conclusions:

• Driven-dissipative systems provide a viable platform for efficient combinatorial optimization.
• The proposed hybrid method offers a pathway to enhanced computational performance.
• Physical implementation using degenerate optical parametric oscillators is a promising direction.