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Analog Coupled Oscillator Based Weighted Ising Machine
Jeffrey Chou, Suraj Bramhavar, Siddhartha Ghosh, William Herzog
TL;DR
The paper targets difficult combinatorial optimization by mapping weighted Ising problems onto a coupled-oscillator system. It combines theoretical stochastic modeling with a fully connected LC circuit and experiments, achieving high MAX-CUT ground-state probabilities while indicating favorable scaling and a hardware limitation for very large networks.
Problem
Combinatorial optimization is difficult for standard von Neumann architectures, motivating alternative computing methodologies for complex energy landscapes.
Method
The paper develops and characterizes a fully connected 4-node differential LC oscillator circuit with cross-bar programmable coupling, modeled through stochastic oscillator dynamics and annealing.
Results
98% ground-state probability was achieved for randomized MAX-CUT problems with 1-bit weights, versus 84% with 5-bit weights.
Takeaways & Limitations
The proof-of-concept supports weighted-Ising optimization using low-cost hardware and simulations predict favorable scaling against existing optoelectronic and GPU-based implementations.
Takeaways & Limitations
Densely interconnecting very large oscillator networks above 1e3 nodes remains a major challenge for the interconnect architecture.
Abstract
from arXiv · showhide
We report on an analog computing system with coupled non-linear oscillators which is capable of solving complex combinatorial optimization problems using the weighted Ising model. The circuit is composed of a fully-connected 4-node LC oscillator network with low-cost electronic components and compatible with traditional integrated circuit technologies. We present the theoretical modeling, experimental characterization, and statistical analysis our system, demonstrating single-run ground state accuracies of 98% on randomized MAX-CUT problem sets with binary weights and 84% with 5-bit weight resolutions. Solutions are obtained within 5 oscillator cycles, and the time-to-solution has been demonstrated to scale directly with oscillator frequency. We present scaling analysis which suggests that large coupled oscillator networks may be used to solve computationally intensive problems faster and more efficiently than conventional algorithms. The proof-of-concept system presented here provides the foundation for realizing such larger scale systems using existing hardware technologies and could pave the way towards an entirely novel computing paradigm.
1. Introduction
The paper addresses difficult combinatorial optimization by implementing a weighted-Ising solver with a fully connected, four-node LC oscillator circuit. Its cross-bar coupling architecture supports larger weight bit-depth and potential CMOS scalability.
- Combinatorial optimization is difficult for standard von Neumann architectures and appears in applications including scheduling, routing, finance, and drug discovery.
- The paper demonstrates a fully connected 4-node differential LC oscillator circuit using standard electronic components that maps to the Ising model.
- The circuit uses gain ratios in a cross-bar architecture instead of direct resistor-applied weights.
- The cross-bar architecture enables larger weight bit-depth and straightforward CMOS scalability for large, fully connected systems.
- The system is evaluated on MAX-CUT problems with binary and multi-bit weights through detailed circuit analysis and performance characterization.
2. Theory
The theory maps weighted Ising optimization onto coupled nonlinear oscillator phases, using super-harmonic injection locking and annealing to obtain binary phase states. Simulations show rapid convergence to degenerate ground states across randomized initial conditions.
- The Ising Hamiltonian represents node interactions through weights J_ij, with spin variables s_i restricted to +1 or -1.
- MAX-CUT maps to the Ising Hamiltonian by selecting the spin state s that minimizes H for a graph G(V,E).
- Coupled nonlinear oscillator phases map to the Ising model when phases settle at 0° or 180°.
- A super-harmonic injection-locking signal at twice the natural frequency polarizes oscillator phases into binary values.
- Gaussian phase fluctuations produce stochastic differential equations, while Euler-Maruyama approximates their solutions iteratively.
- Annealing gradually increases the injection-locking amplitude A, with τ set to 5 oscillation cycles.
- The simulation always reached one of the correct degenerate ground states, including from an intentionally incorrect initial state.
- 2.3 oscillation cycles was the average settling time across 1000 randomized-initial-condition trials.
3. Results
The 4-node fully connected LC oscillator system implements weighted Ising couplings through digitally controlled analog gain ratios and settles to binary phase solutions. Experiments characterize solution speed, accuracy, and the dependence of ground-state probability on weight resolution and energy gap.
- Circuit implementation: The circuit uses differential LC oscillators with coupling applied through transistors M3 and M4, while digital potentiometers implement the inter-oscillator gain ratios.The experimental setup programs six unique weights through an Arduino-controlled I2C interface and monitors analog oscillator voltages.
- Circuit implementation: Super-harmonic injection locking forces oscillator phases to 0° or 180°, producing binary states for the Ising solution.The signal also replaces the rounding feature used in the model and is described as required for extracting sensible answers in larger systems.
- Solution dynamics: 100 µs solution time corresponds to 5 cycles at 50 kHz, while reducing the oscillator frequency to 5 kHz increases time-to-solution by approximately one order of magnitude.The observed scaling supports increasing oscillator frequency as a pathway to reduce solution time.
- Accuracy characterization: 98% and 84% ground-state probabilities are measured for randomized problems with 1-bit and 5-bit weights, respectively, across more than 2000 experiments.The solutions follow a Boltzmann distribution whose peak is at the ground state, and accuracy decreases with increasing weight bit resolution.
- Accuracy characterization: 13 oscillator periods are required on average to reach the ground state with 99% certainty at 5-bit weight resolution.The estimate uses the measured single-run ground-state probability and the number of oscillator periods per run.
- Accuracy characterization: For randomly generated 5-bit problems, lower ΔE between the ground and next-highest energy states is associated with lower single-trial ground-state probability.The measured behavior is reasonably matched by stochastic differential-equation simulations; higher bit resolutions reduce minimum ΔE and accuracy.
4. Discussion
Scaling simulations show that coupled oscillators can retain ground-state solutions across increasing problem sizes, with performance trends becoming favorable against prior Ising-machine and GPU-based approaches. The analysis also indicates that near-ground-state solutions may remain useful for optimization.
- Scaling analysis: ~100% ground-state probability at V=8 falls to roughly 3% at V=300 for simulated Mobius ladder graphs.The scaling study used an annealing schedule of τ = 5 cycles and 500 independent runs per problem set.
- Scaling analysis: A 1 MHz oscillator trend line highlights potential speedup over previous Ising-machine implementations on the identical problem set.The comparison uses data from a previous Ising-machine implementation and relies on demonstrated direct scaling with oscillator frequency.
- Solution quality: When the exact ground state is missed, the system often settles near the absolute minimum, which may suffice for many optimization problems.This observation is reported for cases where the ground-state solution is not reached.
- Randomized cubic graphs: At V ~ 150, coupled-oscillator solution-time scaling crosses over with GPU-accelerated mean-field performance and trends more favorably at larger sizes.The comparison used randomized cubic graphs, 100 problem instances per node size, and a 1 MHz oscillator assumption.
5. Conclusion
The paper demonstrates a parallel, all-to-all coupled oscillator system that correctly maps to the weighted Ising model and solves randomized MAX-CUT instances. It also identifies larger-scale integration and dense interconnects as important next steps.
- 5. Conclusion: The system correctly maps a parallel, all-to-all connected coupled oscillator network to the weighted Ising model.The work includes simulation, design, experiment, and characterization of the system.
- 5. Conclusion: 98% success probability is achieved on random MAX-CUT problems with binary weights.The result is reported for the proof-of-concept system.
- 5. Conclusion: Scaling simulations show favorable behavior compared with existing optoelectronic implementations and GPU-based algorithms for similar problem sets.The comparison is based on predicted behavior at larger node sizes.
- 5. Conclusion: Densely connecting very large oscillator networks above 1e3 nodes remains a major interconnect challenge.The paper identifies time-, frequency-, and code-division multiplexing as possible pathways for allocating connection resources.
7. Author Contributions Statement
The author contributions statement assigns responsibility for manuscript writing, circuit construction and testing, scaling analysis, advisory work, and manuscript review.
- 7. Author Contributions Statement: J.C. and S.B. wrote the manuscript, while J.C. conceived, built, and tested the breadboard circuit system.S.G. assisted with building the breadboard circuit.
- 7. Author Contributions Statement: S.B. performed the scaling calculations and plots, and W.H. advised the program.All authors reviewed the manuscript.
8. Competing Financial Interests
The authors declare a U.S. patent application associated with the work.
- The authors declare U.S. Patent Application No. 62/826,080.
9. Additional Information
The work was approved for public release with unlimited distribution and supported by the United States Air Force under a specified contract.
- The material is approved for public release, and distribution is unlimited.
- The work was supported by the United States Air Force under Air Force Contract No. FA8702-15-D-0001.
- The authors state that the expressed opinions, findings, conclusions, and recommendations do not necessarily reflect United States Air Force views.