A proof-of-principle experiment on the spontaneous symmetry breaking machine and numerical estimation of its performance on the $K_{2000}$ benchmark problem

TL;DR

This work introduces the spontaneous symmetry breaking machine (SSBM) as a photonic, dissipative simulator for combinatorial optimization, and validates its operation experimentally on a 16-node MaxCut3 instance. It then develops an evolved, nested-action version of SSBM and performs extensive numerical simulations on the large-scale $K_{2000}$ benchmark, demonstrating convergence to a single stable state with a cut value reaching 99.7% of the best-known result. The findings highlight a novel duality between SSBM states and Ising-model stable states, identify dynamic asymmetries and competition between dynamics and pseudo-spin interactions, and propose practical paths to scale and stabilize the approach. Overall, SSBM shows promise as a high-performance, low-variability COP solver with unique dynamics distinct from existing Ising machines, while indicating clear avenues for further refinement.

Abstract

In a previous paper, we proposed a unique physically implemented type simulator for combinatorial optimization problems, called the spontaneous symmetry breaking machine (SSBM). In this paper, we first report the results of experimental verification of SSBM using a small-scale benchmark system, and then describe numerical simulations using the benchmark problems (K2000) conducted to confirm its usefulness for large-scale problems. From 1000 samples with different initial fluctuations, it became clear that SSBM can explore a single extremely stable state. This is based on the principle of a phenomenon used in SSBM, and could be a notable advantage over other simulators.

Figures (14)

• Figure 1: Schematic diagram of the circuit configuration generating dissipative causality. An optical coherent clock pulses (OCPs) $P_{C}(t)$ are supplied to the circuit via the input port ${\bf C}_{\rm in}$, and a 1$\times$2 MZM functions as a partial inflow gate to the system. The electrical pulse originating from the $i$-th OCP is timed by a tunable optical delay line (TODL) for use in intensity modulation of the ($i\!+\!m$)-th OCP, and its width is expanded by a Bessel filter to be sufficiently wider than the OCP width.
• Figure 2: Conceptual diagram of a full-dissipative system. This system possesses appropriate boundaries where the inflow of the ($i\!+\!m$)-th clock pulse is regulated by a partial inflow gate that utilizes the previously inflowed $i$-th clock pulse as an intensity-modulated signal. Simultaneously with the inflow of the ($i\!+\!m$)-th clock pulse, the $i$-th clock pulse completely flows out of the system.
• Figure 3: A 3rd-order graph of a target COP ($N\!=\!16$), which is addressed in the proof-of-principle experiment of the SSBM with ${\cal J}_{i:k}^{\rm AF}\!=\!1/30\, (k\!=\!i\!+\!8, i\!+\!1, i\!-\!1)$ and ${\cal J}_{i:k}^{\rm F}\!=\!0$. This graph represents NP-hard instances, equivalent to MaxCut3 problem.
• Figure 4: Schematic diagram of the dedicated MaxCut3 ($N\!=\!16$) problem system for the SSBM proof-of-principle experiment. (a) Optical delay interference circuit (ODIC) for physically implementing the PSI in the target problem (see Fig. 3), (b) Configuration diagram of SSBM physically implementing the PSI using the ODIC. The output from the 1$\times$2 MZM's output port ${\bf A}$ and its complementary output port $\overline{\bf A}$ are input to the corresponding input ports of the ODIC. Physical implementation of the PSIs are achieved through delay interference between optical pulses within the ODIC.
• Figure 5: Observed waveforms from the SSBM principle verification experiment. (a): Operational verification waveform with the PSI off (16 independent SSB phenomena observed), (b): Overall observed waveform of one trial with the PSI on, (b-1)-(b-3): Time-domain enlarged waveforms of (b), (c): Example of the most stable state pattern in the target problem. The electrical signal data monitored during 800 repeated trials was acquired in a single batch using a real-time oscilloscope. It can be confirmed that the state created by SSBM corresponds to one of the most stable state patterns of the target problem (see (b-3)). NMS: Normalized Monitoring Signal