Ising-machine-like Driving Condition Restrictions in SSBM and Additional Pseudo-annealing- Driving Method for SSBM that Achieves Best Cut Value Search -

The gist

By adding an effect similar to physical annealing to Spontaneous Symmetry Breaking Machines (SSBM), researchers can make them search for the best known cuts in K2000

SSBM as a Dissipative Connection System

SSBM is a solver for combinatorial optimization problems that utilizes the SSB phenomenon, consisting of basic elements equal to the number of elements N in the problem Each basic element is a dissipative system, not a Hamiltonian system. This system involves an electrical pulse derived from an optical clock pulse driving the input gate to control transmittance while simultaneously flowing out of the system completely. Under specific conditions, where pulse widths and repetition periods satisfy σpw ≪ τpw ≪ ∆t, the basic element exhibits dissipative causality. When parameters are set to γ = π and θB = 0, a pseudo-potential featuring a single unstable fixed point (ϕ = 1/2), an axis of symmetry, and two additional attractors (ϕ = 0 and ϕ = 1) emerges. At the start of the optical clock pulse input, a state fixed at the attractor ϕ = 0 (a state with broken symmetry) appears, known as the hidden state (HS).

Pseudo-Spin Interaction and Dynamics

The multiple dissipative systems in SSBM can be realized through parallelization or time-division multiplexing. Using a delayed optical interferometer, a pseudospin interaction (pSI) is introduced among basic elements, described by Eq. (2) The interaction term involves terms like Q FM i and Q AFM i which depend on ferromagnetic and anti-ferromagnetic couplings. To counteract the tendency of pseudo-spins to accumulate in the zero state due to ferromagnetic interactions, averaging using the conjugate term is employed. To enhance the effectiveness of DDA when pSI competes with DDA, a modified scheme involving Eq. (8) and Eq. (9) is introduced. This pseudo-spin interaction differs from Ising interactions because it can produce effects consistent with the exchange interaction energy across the entire range of values of the continuous variable ϕ.

Driving Conditions for Best Cut Search

To find a way to solve the problem of SSBM not finding the best solution, two differences from improved SBM are focused on Point 1 involves performing operations in accordance with quasi-static operations under constraints that maintain a state approximating the steady state to avoid issues inherited from Ising interaction. Point 2 is an improvement adopted in HbSBM to enable best-cut search, introducing an effect that acts as noise in the SBM—a model free of noise—and realizing a behavior where this effect “diminishes” over time. This method incorporates pseudo-annealing based on the essence of that method, even though SSBM is based on discrete dynamics.

Pseudo-Annealing Effect and Results

Numerical simulations under new driving conditions confirmed that the pseudo-spins are strongly confined to the vicinity of the up- or down states when only the restriction related to Point 1 is imposed. When a pseudo-annealing effect is added, it can be confirmed that transitions between up and down occur with a frequency corresponding to the intensity of the applied noise. The specific scheduling of noise intensity was adjusted so that it fell between a sufficiently large value where pSI effect can be neglected and a value small enough to be negligible compared to the effect of pSI. The distribution of cut values obtained from 1,000 trials shows a shift toward larger cut values compared to the case without pseudo-annealing. This shift is understood as the stabilizing effect of pSI being more effectively elicited as noise intensity decreases through scheduling, competing with entropic effects originating from noise. The similarity of this distribution shape to dSBM, HdSBM, and HbSBM suggests that both points act as dominant conditions in search behavior.

Conclusion on Search Behavior

The paper reports that under the imposed constraints and pseudo-annealing effect, the system becomes capable of searching for the best cut even though the convergence effect of pseudo-spin patterns is lost. It is noted that while adjusting SSBM driving conditions to resemble a continuous-variable IM is possible, eliciting behavior similar to that of a SSBM from a continuous-variable IM is difficult due to the difference between Ising interaction and the pseudo-spin interaction introduced in SSBM. The pseudo-spin interaction allows utilizing the flexibility of pattern transitions inherent in the continuous-variable nature of the system and eliciting entropic effects not derived from noise, enabling convergence. In cases involving pseudo-annealing operations, entropic effects originating from noise play a significant role in determining the statistical distribution of obtained cut values. The quantitative discussions regarding solution-search characteristics accounting for these different entropic effects remain a topic for future research.

ACKNOWLEDGMENTS

The authors thank T. Hashimoto and H. Takenouchi for their support

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1. This system can search for high-quality solutions in combinatorial optimization problems by inducing convergence effect of solutions that does not appear in continuous-variable IM, as described when introducing a modified pseudo-spin interaction (pSI) in SSBM.

2. The improved system can achieve the best cut on benchmark problems like K2000 by employing driving conditions that combine constraints mimicking quasi-static operations and a noise effect, similar to the improvement adopted in HbSBM.

3. The AI system can utilize pseudo-annealing through intensity-scheduled noise, which, when balanced correctly between pSI and noise effects, leads to a statistical distribution of cut values that shifts toward larger values compared to cases without pseudo-annealing.