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.

Simulating a network of Ising spins with physical systems is now emerging as a promising approach for solving mathematically intractable problems [1][2][3][4][5]. Here we report a large-scale network of artificial spins based on degenerate optical parametric oscillators (DOPO), paving the way towards a photonic Ising machine capable of solving difficult combinatorial optimization problems. We generated >10,000 time-divisionmultiplexed DOPOs using dual-pump four-wave mixing (FWM) [6,7] in a highly nonlinear fibre (HNLF) placed in a fibre cavity. Using those DOPOs, a one-dimensional (1D) Ising model was simulated by introducing nearest-neighbour optical coupling. We observed the formation of spin domains and found that the domain size diverged near the DOPO threshold, which suggests that the DOPO network can simulate the behaviour of low-temperature Ising spins. Combinatorial optimization problems are becoming increasingly important in our society, for example in applications such as artificial intelligence, drug discovery, optimization of cognitive wireless networks, and analysis of social networks. Many such problems are classified as non-deterministic polynomial time (NP)-hard or NP-complete problems, which are considered to be hard to solve efficiently with modern computers [8]. It is well known that many combinatorial optimization problems can be mapped onto the ground-state-search problems of the Ising Hamiltonian [9] demonstrated a CIM using DOPOs [13]. A DOPO can be utilized as a stable artificial spin because it takes only the 0 or π phase relative to the pump phase [14]. The spinspin interaction can be simply implemented with mutual injections of DOPO lights using delay interferometers. In [13], a spin system composed of four DOPOs was employed for a proof-of-principle CIM experiment. However, to simulate a more complex Ising Hamiltonian to verify the advantages of the CIM over existing methods, we need to implement a CIM with a much larger number of spins. Here we report a large scale network of artificial spins realized with as many as 10,000 time-divisionmultiplexed DOPOs generated via dual-pump FWM in an HNLF placed in a fibre cavity. We successfully simulated the ferro-and anti-ferromagnetic-like behaviour of a 1D Ising spin chain by introducing uni-directional nearest-neighbour coupling between DOPOs. In addition, we observed a formation of domain walls with which we could obtain information on how much the state of the spin network was excited from the ground state. We believe the present result will provide a promising platform on which to realize an efficient machine for solving the Ising model based on the CIM concept. A dimensionless Hamiltonian of an N -spin Ising model without an external magnetic field (Fig. 1 a) is given bywhere J ij is the coupling coefficient between the ith and jth spins, and σ ℓ (ℓ ∈ {i, j}) denotes the z projection of the ℓth spin, which can take ±1 values. The purpose of an Ising machine is to find the ground state of the above Hamiltonian with a given set of J ij us...

The structure of the thin film phase of pentacene was investigated using x-ray diffraction reciprocal space mapping (RSM). The crystal structure was found to be triclinic with the following lattice parameters: a=0.593nm, b=0.756nm, c=1.565nm, α=98.6°, β=93.3°, and γ=89.8°. Atomic positions were determined by comparing the observed RSM diffraction intensities with theoretical calculations.

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