The analysis and optimization of complex systems can be reduced to mathematical problems collectively known as combinatorial optimization. Many such problems can be mapped onto ground-state search problems of the Ising model, and various artificial spin systems are now emerging as promising approaches. However, physical Ising machines have suffered from limited numbers of spin-spin couplings because of implementations based on localized spins, resulting in severe scalability problems. We report a 2000-spin network with all-to-all spin-spin couplings. Using a measurement and feedback scheme, we coupled time-multiplexed degenerate optical parametric oscillators to implement maximum cut problems on arbitrary graph topologies with up to 2000 nodes. Our coherent Ising machine outperformed simulated annealing in terms of accuracy and computation time for a 2000-node complete graph.

We demonstrate unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using a digital coherent receiver, where the decision-directed carrier-phase estimation is employed. The phase fluctuation is effectively eliminated in the 16-QAM system with such a phase-estimation method, when the linewidth of semiconductor lasers for the transmitter and the local oscillator is 150 kHz. Finite-impulse-response (FIR) filters at the receiver compensate for 4,000-ps/nm group-velocity dispersion (GVD) of the 200-km-long single-mode fiber and a part of self-phase modulation (SPM) in the digital domain. In spite of the launched power limitation due to SPM, the acceptable bit-error rate performance is obtained owing to high sensitivity of the digital coherent receiver.

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Ultra-dense spatial-division multiplexing (SDM) is achieved by mode multiplexed technique with multiple cores in a single fiber, namely few-mode multi-core fiber. Using a 9.8-km six-mode nineteen-core fiber, we demonstrate an ultra-dense SDM transmission of 16-channels wavelength-division-multiplexed (WDM) dual-polarization quadrature phase shift keying signals, achieving a record spatial multiplicity of 114. With the help of ultra-dense Super-Nyquist WDM techniques in the 4.5-THz bandwidth of the full C-band, we demonstrate 2.05 Pbit/s transmission over 9.8-km six-mode nineteen-core fibers. In this experiment, the highest aggregate spectral efficiency of 456 bit/s/Hz is achieved.

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