Large-scale Ising spin network based on degenerate optical parametric oscillators

Abstract

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...

Nondegenerate parametric oscillations in a tunable superconducting resonator

“…The JPO regime is established at parametric pump amplitudes above a critical value (threshold), where an instability of the resonator ground state is developed, and it is stabilized by the Kerr nonlinearity. The JPO can be used for vacuum squeezing and photonic entanglement [13,14], photonic qubit operation [15], and catstate engineering [16][17][18]. The JPO has also been employed for high-fidelity readout of superconducting qubits [19,20].…”

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

“…Figure 6a shows one experimental scheme to implement the Ising coupling J ij in the DOPO network, where a part of each DOPO pulse circulating in a fiber ring resonator is picked-off at every round trip by the output coupler, amplified by an external PSA, split into multiple optical delay lines including intensity and phase modulators and then injected back to the target DOPO pulse at appropriate timing. 15,22,23 The external PSA is provided by another DOPA and boosts the in-phase amplitudeX of a picked-off pulse before it is attenuated by the beam splitter and modulator loss. Using this technique, we can implement a relatively large Ising coupling constant (J ij ≲ 1) even if the system size is fairly large (N ≳ 10 4 ).…”

“…Such an all-optical coupling scheme has been experimentally demonstrated for N = 4 and 16 pulses using free-space optics with a 4.8 m ring cavity and 250 MHz and 1 GHz repetition frequencies, respectively 15,22 and for N = 10 4 pulses using a planar lightwave circuit and fiber optics with 1 km fiber ring cavity and 2 GHz repetition frequency. 23 In the latter system, for example, the optical fiber length is 1 km so that a round trip time is 5 μs while the pulse interval is 0.5 ns, which leads to N = 10 4 DOPO pulses inside a single ring cavity.…”

“…The feedback pulses, which are produced by combining the outputs from N−1 intensity and phase modulators, are injected back to the target DOPO pulse by the injection coupler. 15,22,23 b A CIM with a measurement-feedback circuit. A small portion of each DOPO signal pulse is out-coupled through the output coupler, and its in-phase amplitude is measured by optical balanced homodyne detectors, where LO pulse is directly obtained from the pulsed pump laser.…”

“…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.…”