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A phase transition is the process in which a substance changes from one state of matter to another, like from a solid to a liquid, liquid to gas, or vice versa, at a specific temperature and under given pressure conditions. This change is spontaneous and is affected by alterations in temperature and pressure. These parameters impact the strength of the forces between molecules (intermolecular forces) in the substance.During a phase transition, both the initial and final phases of the substance...
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Combinatorial optimization using dynamical phase transitions in driven-dissipative systems.

Timothée Leleu1, Yoshihisa Yamamoto2,3, Shoko Utsunomiya4

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This study shows driven-dissipative systems can efficiently solve combinatorial optimization problems by minimizing Ising Hamiltonians. A novel hybrid analog-digital approach improves solution quality near dynamic phase transitions.

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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.