A coherent Ising machine for 2000-node optimization problems

Abstract

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.

Coherent Ising machines—optical neural networks operating at the quantum limit

“…25 Major changes from the Stanford CIM 24 are the fiber length increased from 300 m to 1 km and the pulse repetition frequency increased from 100 MHz to 1 GHz. Figure 9b shows the observed Ising energy vs. computation time for the MAX-CUT problems in N = 2000 complete graphs with all-to-all connections.…”

“…The complicated task of the synchronous computation of the vector-vector multiplication between J ij andX j , which must be completed in the pulse interval of 100 ps -1 ns, is achieved by a single measurement-feedback circuit consisting of an analog-to-digital converter, a field programmable gate array (FPGA), a digital-to-analog converter and optical amplitude/phase modulators. 24,25 The feedback pulse used as an input to the optical modulator and the local oscillator pulse used for optical homodyne detection are both provided by a part of the pump laser output, as shown in Fig. 6b.…”

“…40 Here we consider a CIM consisting of synchronously pumped DOPO pulses circulating in a fiber ring cavity. 24,25 The density matrix elements Xjρ X 0 j h iin the (X+X′, X−X′) phase space for two out-of-phase (anti-ferromagnetically) coupled DOPOs are plotted, where the pump rate is just above the threshold. In Fig.…”

“…6b. 24,25 Instead of directly connecting the DOPO pulses with optical delay lines, we can measure approximately the in-phase amplitude of the internal DOPO pulse by the optical balanced homodyne detectors. If the inferred in-phase amplitude of the jth DOPO pulse is represented byX j , the feedback pulse to the ith DOPO pulse should have an in-phase amplitude proportional to P j J ijXj .…”

“…15,22,23 Our measurement-feedback-based DOPO Ising machine allows connections between any spin and any other spin, and is fully programmable. 24,25 Fully connected superconducting Ising machines have been also proposed recently. 26,27 CLASSICAL VS. QUANTUM NEURAL NETWORKS Nonlinear devices that have a characteristic input-output relation shown in Fig.…”