Fig. 1. Concepts for constructing the ISPIS based on XY model. (a) Schematic of the proposed N-channel ISPIS, comprising a coherent input module, a spin encoding module, a coherent superposition module, and affiliated electronics. (b) The theory drawing of amplitude-phase modulation method. The weighted interaction Jlk between two spins θl and θk can be transformed into a unitary interaction J between those spins under a specific phase χlk of deflection. (c) Microscopy image for the fabricated eight-channel ISPIS. (d) Packaged ISPIS chip on a PCB.

Fig. 2. Optimized high-uniformity MMI for an ISPIS system. (a) Schematic diagram of the 1×8 MMI coupler with unequal-width tapers. (b) Optimized electric field profiles at 1550 nm simulated using the MODE solver. (c) Normalized output light intensity from eight output ports at 1550 nm in experiments and simulations. (d), (e) Insertion loss and uniformity of the initial MMI (with equal-width tapers) and optimized MMI (with unequal-width tapers) and cascaded 1×2 MMIs.

Fig. 3. Ground state search for an ISPIS system. (a) Optical losses and the corresponding optical phase differences of the tested thermal modulators. (b) Process of ground state search of the XY model with ISPIM. (c) Calibration results for the phase shifters in eight channels.

Fig. 4. Simulation of phase transition based on ISPIS. (a) Experimental and simulated optical loss values of the ISPIS with the corresponding theoretical Hamiltonian of the fully connected eight-spin Ising model. (b)–(d) Hamiltonian and magnetization evolve as a function of temperature, for the ferromagnetic phase, ferrimagnetic phase, and spin glass phase transitions, as observed in eight-spin Ising system with different Markov chain lengths (L=100,1000). (e) Ferromagnetic phase transition with external magnetic field, as observed in six-spin Ising system with different Markov chain lengths (L=100,1000). (f) Simulated and experimental optical power for a weighted full-rank Ising model.

Fig. 5. Optically solving the MaxCut problems with ISPIS. (a), (b) Normalized optical power evolution curves of 100 experiment for the unweighted MaxCut problem and weighted MaxCut problem, respectively. (c), (d) Cut value evolution curves of 100 experiment for the unweighted MaxCut problem and weighted MaxCut problem, respectively. (e), (f) Observed probability distribution for the cut value and experimentally found ground-state solutions probability density function for the unweighted and weighted MaxCut problem, respectively.