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.

Physical annealing systems provide heuristic approaches to solving combinatorial optimization problems. Here, we benchmark two types of annealing machines—a quantum annealer built by D-Wave Systems and measurement-feedback coherent Ising machines (CIMs) based on optical parametric oscillators—on two problem classes, the Sherrington-Kirkpatrick (SK) model and MAX-CUT. The D-Wave quantum annealer outperforms the CIMs on MAX-CUT on cubic graphs. On denser problems, however, we observe an exponential penalty for the quantum annealer [exp(–αDWN2)] relative to CIMs [exp(–αCIMN)] for fixed anneal times, both on the SK model and on 50% edge density MAX-CUT. This leads to a several orders of magnitude time-to-solution difference for instances with over 50 vertices. An optimal–annealing time analysis is also consistent with a substantial projected performance difference. The difference in performance between the sparsely connected D-Wave machine and the fully-connected CIMs provides strong experimental support for efforts to increase the connectivity of quantum annealers.

Computers based on physical systems are increasingly anticipated to overcome the impending limitations on digital computer performance. One such computer is a coherent Ising machine (CIM) for solving combinatorial optimization problems. Here, we report a CIM with 100,512 degenerate optical parametric oscillator pulses working as the Ising spins. We show that the CIM delivers fine solutions to maximum cut problems of 100,000-node graphs drastically faster than standard simulated annealing. Moreover, the CIM, when operated near the phase transition point, provides some extremely good solutions and a very broad distribution. This characteristic will be useful for applications that require fast random sampling such as machine learning.

Terahertz (THz)-bandwidth continuous-wave (CW) squeezed light is essential for integrating quantum processors with time-domain multiplexing (TDM) by using optical delay line interferometers. Here, we utilize a single-pass optical parametric amplifier (OPA) based on a single-spatial-mode periodically poled ZnO:LiNbO3 waveguide, which is directly bonded onto a LiTaO3 substrate. The single-pass OPA allows THz bandwidth, and the absence of higher-order spatial modes in the single-spatial-mode structure helps to avoid degradation of squeezing. In addition, the directly bonded ZnO-doped waveguide has durability for highpower pump and shows small photorefractive damage. Using this waveguide, we observe CW 6.3-dB squeezing at 20-MHz sideband by balanced homodyne detection. This is the first realization of CW squeezing with a single-pass OPA at a level exceeding 4.5 dB, which is required for the generation of a two-dimensional cluster state. Furthermore, the squeezed light shows 2.5-THz spectral bandwidth. The squeezed light will lead to the development of a high-speed on-chip quantum processor using TDM with a centimeter-order optical delay line.

Many problems in mathematics, statistical mechanics, and computer science are computationally hard but can often be mapped onto a ground-state-search problem of the Ising model and approximately solved by artificial spin-networks of coupled degenerate optical parametric oscillators (DOPOs) in coherent Ising machines. To better understand their working principle and optimize their performance, we analyze the dynamics during the ground state search of 2D Ising models with up to 1936 mutually coupled DOPOs. For regular as well as frustrated and disordered 2D lattices, the machine finds the correct solution within just a few milliseconds. We determine that calculation performance is limited by freeze-out effects and can be improved by controlling the DOPO dynamics, which allows to optimize performance of coherent Ising machines in various tasks. Comparisons with Monte Carlo simulations reveal that coherent Ising machines behave like low temperature spin systems, thus making them suitable for optimization tasks.