Network of time-multiplexed optical parametric oscillators as a coherent Ising machine

Journal name:
Nature Photonics
Year published:
Published online

Finding the ground states of the Ising Hamiltonian1 maps to various combinatorial optimization problems in biology, medicine, wireless communications, artificial intelligence and social network. So far, no efficient classical and quantum algorithm is known for these problems and intensive research is focused on creating physical systems—Ising machines—capable of finding the absolute or approximate ground states of the Ising Hamiltonian2, 3, 4, 5, 6. Here, we report an Ising machine using a network of degenerate optical parametric oscillators (OPOs). Spins are represented with above-threshold binary phases of the OPOs and the Ising couplings are realized by mutual injections7. The network is implemented in a single OPO ring cavity with multiple trains of femtosecond pulses and configurable mutual couplings, and operates at room temperature. We programmed a small non-deterministic polynomial time-hard problem on a 4-OPO Ising machine and in 1,000 runs no computational error was detected.

At a glance


  1. Principle of operation of OPO Ising machine.
    Figure 1: Principle of operation of OPO Ising machine.

    a, Simplified illustration of vacuum squeezing in a degenerate OPO below threshold and the binary phase states above threshold shown in in-phase and quadrature-phase coordinates. b, Gradual pumping of the OPO network to find the ground states. The parametric gain is gradually increased from below threshold to reach the minimum loss of the ground state. Because the lowest loss of the network corresponds to the phase-state configuration representing the answer to the Ising problem, it is expected that only the ground state oscillates. c, Comparison of the search mechanism in the OPO network, classical simulated annealing and quantum annealing.

  2. Experimental set-up.
    Figure 2: Experimental set-up.

    a, A 4-OPO system is pumped by a femtosecond fibre laser at 1,045 nm, three delay lines provide couplings between the OPOs, and an unequal-arm Michelson interferometer measures the relative phase states of adjacent OPOs. A chopper (CP) restarts the OPO system periodically with a clock rate of ∼1 kHz and a rise time (10–90% power) of ∼180 µs. Output couplers (OCs) and input couplers (ICs) provide ∼4% of power reflection (varying between 2 and 6%). The nonlinear crystal is periodically poled lithium niobate (PPLN). b, Illustration of the output pulse train and the time slots assigned to OPO1 to OPO4. The repetition period of the pump and the OPO output (TR) is 4 ns and the cavity roundtrip time (Tcavity) is 16 ns. c, Illustration of couplings provided by different delay lines. Each delay line provides four couplings, that is, two-body interactions, among the temporally separated OPOs. Delay 1 couples OPOn to OPOn+1 (J12, J23, J34, J41), delay 2 couples OPOn to OPOn+2 (J13, J24, J31, J42), and delay 3 couples OPOn to OPOn+3 (J14, J21, J32, J43).

  3. Slow detector results.
    Figure 3: Slow detector results.

    a, All delay lines are blocked. The output is randomly at one of the three discrete intensity levels 0, Im/2 or Im. The zero intensity level is slightly above the noise floor mainly because the arms of the interferometer have different lengths and so the interfering beams are not completely overlapping. b, Only delay 1 is present and its phase is scanned slowly. Around zero, where the coupling is in phase, the output locks to Im, verifying that all OPOs are in the same phase states. Around 180°, where the coupling is out of phase, the output locks at 0, verifying that adjacent OPOs have opposite phase states. The three-state output around 90° shows the inefficiency of coupling at those phases. Regenerative behaviour due to the phase-sensitive amplification of the OPO is observed, for example, for the coupling phases between −30 and 30° and between 150 and 210°. c, All the delays are present and their phases are locked to 180° to represent the MAX-CUT problem, where the output toggles between 0 and Im/2. a.u., arbitrary units.

  4. Fast detector results.
    Figure 4: Fast detector results.

    a, Samples of the pulse patterns at the output of the interferometer. Each time the system is pumped above threshold, depending on the network configuration, one of these patterns is detected with a fast detector at the output of the interferometer. a.u., arbitrary units. b, Histogram of the phase states when all couplings are blocked (that is, the OPOs are independent), for 1,000 runs. c, Histogram of the phase states when all delays are on and their phases are locked to π for 1,000 runs. The OPO network represents the MAX-CUT problem for a four-vertex graph, and it only oscillates in the phase states corresponding to the answers to this problem.


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  1. E.L. Ginzton Laboratory, Stanford University, California 94305, USA

    • Alireza Marandi,
    • Zhe Wang,
    • Robert L. Byer &
    • Yoshihisa Yamamoto
  2. National Institute of Informatics, Tokyo 101-8403, Japan

    • Alireza Marandi,
    • Kenta Takata &
    • Yoshihisa Yamamoto
  3. Department of Information and Communication Engineering, University of Tokyo, Tokyo 113-8656, Japan

    • Kenta Takata &
    • Yoshihisa Yamamoto


A.M. and Y.Y. conceived the idea and designed the experiment. A.M. and K.T. carried out the experiment. Z.W. performed the numerical simulations. Y.Y. and R.L.B. guided the work. A.M. wrote the manuscript, with input from all authors.

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