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Network of Time-Multiplexed Optical Parametric Oscillators as a Coherent Ising Machine
Alireza Marandi, Zhe Wang, Kenta Takata, Robert L. Byer, Yoshihisa Yamamoto
TL;DR
The paper addresses the challenge of finding ground states for NP-hard Ising problems, for which no efficient classical or quantum algorithm is known. It introduces a coherent network of time-multiplexed degenerate optical parametric oscillators and demonstrates its operation on MAX-CUT, with no computational error detected in 1000 runs.
Problem
No efficient classical or quantum algorithm is known for finding ground states of Ising Hamiltonians underlying many important combinatorial optimization problems.
Method
The paper implements Ising spins as binary above-threshold OPO phases and realizes couplings through mutual injections in a time-multiplexed OPO ring-cavity network.
Results
1000 trials of the NP-hard MAX-CUT problem detected no excited phase state or computational error, while numerical benchmarks produced outputs about 2–6% better than the 0.878-performance guarantee of Goemans-Williamson SDP.
Takeaways & Limitations
The demonstrated OPO network is capable of solving NP-hard MAX-CUT for N=4 and is presented as a route toward approximate solutions for larger Ising problems with reasonable accuracy and speed.
Takeaways & Limitations
The machine is stochastic, so its success probability for obtaining a ground state is always smaller than one, and practical scalability is constrained by stabilizing the phases of long fiber links.
Abstract
from arXiv · showhide
Finding the ground states of the Ising Hamiltonian [1] 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 Hamiltonian [2-6]. Here we report a novel 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 injections [7]. 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 programed the smallest non-deterministic polynomial time (NP)- hard Ising problem on the machine, and in 1000 runs of the machine no computational error was detected.
2 National Institute of Informatics, Tokyo 101-8403, Japan,
The paper develops a time-division-multiplexed network of degenerate OPOs that maps binary phase states and mutual injections onto an Ising machine, demonstrating NP-hard MAX-CUT for four spins. Experiments and modeling show ground-state selection, while scalability and universal performance remain bounded by phase stabilization and stochastic success probability.
- OPO Ising mapping: Above threshold, each degenerate OPO oscillates in one of two phase states, |0⟩ or |π⟩, representing an Ising spin; mutual injections realize the couplings.Different phase configurations produce different network photon losses corresponding to the Ising energy landscape.
- Experimental architecture: Time-division multiplexing places multiple independent OPO spins in one ring resonator, while delayed out-coupled light supplies configurable pairwise interactions.The four-pulse cavity uses three delay-line coupling paths, and the scheme avoids mismatch and phase-decoherence noise.
- Experimental architecture: The interferometer measures differential phase between adjacent OPOs by comparing consecutive temporal slots with a one-bit delay.Fast and slow detectors provide complementary measurements of the phase-state patterns.
- Coupling validation: In-phase coupling produces aligned OPO phases, whereas out-of-phase coupling produces alternating phases and tolerates coupling-phase deviations of at least ±30°.These observations identify ferromagnetic and antiferromagnetic interactions in the corresponding Ising spin ring.
- MAX-CUT demonstration: NP-hard MAX-CUT is implemented by setting all mutual OPO couplings out of phase, making answer states the configurations with lowest total photon decay.For the four-vertex cubic graph, any subset containing two vertices is an answer.
- Experimental result: Across 1000 MAX-CUT trials, no excited phase state was detected and the computation error rate was less than 10^-3.The reported error rate is limited by the length of the measurements.
- Scalability and scope: The proposed approach is intended to scale through longer ring cavities and additional delay lines, but phase stabilization remains a main challenge and stochastic success probability is below one.The authors report no mathematical proof of superior performance for all Ising problems and acknowledge possible failures on some instances.
Contributions
The authors divide the work across conceptual design, experiment, simulation, guidance, and manuscript preparation.
- A.M. and Y.Y. conceived the idea and designed the experiment.
- A.M. and K.T. carried out the experiment, while Z.W. performed numerical simulations.
- Y.Y. and R.L.B. guided the work, and A.M. wrote the manuscript with input from all authors.
1 Principle of Operation
The OPO-based machine searches for low-loss Ising states by increasing parametric gain, unlike classical thermal hopping or quantum tunneling through metastable states.
- 1 Principle of Operation: The OPO machine searches upward in network loss by gradually increasing parametric gain toward the first-touch ground state.The ground state is represented by minimum network loss.
- 1 Principle of Operation: Classical simulated annealing searches vertically through repeated temperature decreases and increases, whereas quantum annealing searches horizontally via tunneling.
2 Theoretical Modeling and Numerical Test of OPO Network
The study models mutually coupled degenerate OPO networks with c-number Langevin equations and tests their performance on MAX-CUT instances ranging from small cubic graphs to large G-set graphs.
- Theoretical Modeling: The network model uses c-number Langevin equations derived from the quantum mechanical Fokker–Planck description of degenerate OPOs.The model represents each OPO through normalized in-phase and quadrature amplitudes and includes Gaussian vacuum-noise processes.
- Theoretical Modeling: The C-LGE and Q-FPE descriptions produce completely agreeing squeezing and anti-squeezing characteristics across the oscillation threshold.This agreement is reported at pump rate p = 1.
- Numerical Test: The numerical test solves MAX-CUT instances on cubic graphs with N = 4 to N = 20 and random graphs with N = 800 to N = 20000.For a graph with N vertices, the simulation solves 2N coupled Langevin equations using the Dormand–Prince method with adaptive integration.
- Numerical Test: 99% of tested graphs have normalized build-up time t ≃100, largely independent of graph order N.Only a slight increase appears for the worst-case graphs, so computational time is mainly determined by ground-state success probability.
- Numerical Test: The worst-instance success probability at the optimal pump rate is independent of graph order and ranges from 0.7 ∼1.0.At fixed p = 1.1 and ξ = −0.1, qmin denotes worst-case success probability, while qopt is achieved at the optimal pump rate.
- Numerical Test: On 71 G-set benchmark instances, the network's best and average outputs are about 2 - 6% better than the 0.878-performance guarantee of the Goemans-Williamson algorithm.The best and average values differ by less than 1% for most instances, while average computational time scales as O(N0.2).
3 Measurement of Phase States
The 4-OPO phase-state measurement uses a one-bit delay interferometer, with detector choice determining how many distinct pulse patterns can be resolved.
- 3 Measurement of Phase States: A one-bit delay interferometer measures the phase states of the 4-OPO system and maps them to output pulse trains.Complementary phase states produce the same output, so 16 phase states yield 8 different pulse trains.
- 3 Measurement of Phase States: Fast detection resolves four pulse patterns without a time reference, whereas slow detection produces three patterns proportional to the number of ones.The detector bandwidth therefore changes the distinguishable output patterns.
4 Details of Experimental Setup
The experiment uses a ring-cavity OPO with stabilized cavity and delay phases, while exploiting intrinsic pump–signal phase locking so pump stabilization is unnecessary.
- Optical system: The OPO ring has a 16-ns round-trip time and uses a MgO:PPLN crystal for degenerate operation near 2 µm at room temperature.The cavity includes pellicle couplers, and degeneracy is achieved by cavity-length tuning despite nonoptimal pump phase matching.
- Pump source: A free-running mode-locked Yb-fiber laser supplies approximately 80-fs pulses at 1045 nm and 250 MHz repetition rate.Its maximum average power exceeds 1 W, and the network is pumped approximately 2.2 times above threshold.
- Stabilization: Five dither-and-lock servo controllers stabilize the cavity length, delay-line phases, and interferometer arm-length difference.Each controller applies sub-10-nm modulation at 5–20 kHz and has a 10-Hz 3-dB bandwidth.
- Free-running pump: The experiment shows that the servo controllers suffice for Ising-machine operation without stabilizing the free-running pump.Smooth repetition-rate changes transfer intrinsically from pump to signal, while carrier-envelope-offset changes are accommodated by intrinsic phase locking and servo control.
- Free-running pump: Degenerate OPO phase-sensitive gain locks signal-pulse phase slips to pump-pulse slips with a factor of one-half.The servo loop follows this phase slip to maximize output power, locking the pump and signal carrier-envelope-offset frequencies.
5 OPO Characterization
The four-OPO system operates at degeneracy with 15 mW output centered at 2090 nm, producing short pulses and a characterized spatial beam profile.
- Output characterization: 15 mW of average output power is obtained at degeneracy from 290 mW of pump power, with the spectrum centered at 2090 nm.The system threshold reaches 135 mW when all input and output couplers are installed.
- Output characterization: The output spectrum has a 3-dB bandwidth of 91 nm centered at 2090 nm.This spectrum is reported for the four-OPO system.
- Pulse characterization: The interferometric autocorrelation has a full width at half maximum of approximately 120 fs, suggesting approximately 85-fs Gaussian pulses.The spatial beam profile is also measured at 2090 nm.
6 Extended Slow-Detector Results
Slow-detector measurements show that coupling phase controls aligned, alternating, paired, and MAX-CUT-related OPO phase configurations, with regenerative behavior across broad phase ranges.
- Adjacent couplings: In-phase adjacent coupling produces identical OPO phase states and high interferometer output, whereas out-of-phase coupling produces alternating states and low output.These behaviors are observed when scanning delays 1 and 3 individually.
- Next-nearest coupling: Delay-2 in-phase coupling groups OPOs into two same-phase pairs, while out-of-phase coupling yields constant output Im/2.The paired groups may share either the same or different phase states under in-phase coupling.
- Regeneration: The coupling network exhibits regenerative behavior and insensitivity to a wide range of coupling-phase changes.This behavior is observed in the three delay-scan plots.
- MAX-CUT configuration: For the MAX-CUT configuration, scanned delay phases produce the expected outcomes, including the anti-ferromagnetic case at phase π.The center of each scan corresponds to the all-antiferromagnetic MAX-CUT problem.
7 Scalability
Time-division multiplexing scales the OPO network by placing many identical pulses in one ring cavity while requiring only N−1 delay lines for N spins.
- Scalable architecture: For an Ising problem with N sites, time-division multiplexing uses N−1 delay lines instead of N^2−N possible couplings, so physical size scales linearly with N.The architecture benefits from intrinsically identical network nodes.
- Scalable architecture: A single ring resonator with Tcavity = NTR and N−1 delay lines can realize a network of N OPOs.A 10-GHz pump and 200-m fiber resonator would produce 10,000 temporally separated OPOs.
- Practical limits: A fiber-based network for N = 10,000 has an expected photon lifetime of approximately 6 × 10^-6 s, promising reasonably fast MAX-CUT computation.The main challenge is stabilizing the phases of long fiber links, although regenerative OPO behavior may tolerate relatively large coupling-phase noise.
- Programmable couplings: Each delay line can implement multiple coupling terms by using electrooptic modulators to switch individual time-slot couplings on or off.A delay of mTR realizes couplings of the form J(i)(i+m).
- Implementation examples: The scalable implementation is illustrated schematically as a fiber-based OPO network with configurable delay phases and phase-state configurations.The supplementary configuration table identifies the MAX-CUT phase arrangement among four tested configurations.