# Simulated bifurcation for higher-order cost functions

arXiv: https://arxiv.org/abs/2211.09296 · HTML: https://mlsanity.com/paper/2211.09296/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2022.

## Built on by (most cited first, 5 of 5)

- [Enhancing In-vehicle Multiple Object Tracking Systems with Embeddable Ising Machines](https://mlsanity.com/paper/2410.14093/) (2024)
- [Augmented Electronic Ising Machine as an Effective SAT Solver](https://mlsanity.com/paper/2305.01623/) (2023)
- [Machine Learning-assisted High-speed Combinatorial Optimization with Ising Machines for Dynamically Changing Problems](https://mlsanity.com/paper/2503.23966/) (2025)
- [Binary Quadratic Quantization: Beyond First-Order Quantization for Real-Valued Matrix Compression](https://mlsanity.com/paper/2510.18650/) (2025)
- [Training Energy-Based Models with Non-MCMC Samplers and Efficient Temperature Estimation](https://mlsanity.com/paper/2512.02323/) (2025)

## Abstract (arXiv)

High-performance Ising machines for solving combinatorial optimization problems have been developed with digital processors implementing heuristic algorithms such as simulated bifurcation (SB). Although Ising machines have been designed for second-order cost functions, there are practical problems expressed naturally by higher-order cost functions. In this work, we extend SB to such higher-order cost functions. By solving a problem having third-order cost functions, we show that the higher-order SB can outperform not only the second-order SB with additional spin variables, but also simulated annealing applied directly to the third-order cost functions. This result suggests that the higher-order SB can be practically useful.