Simulated bifurcation for higher-order cost functions

Ising machines have been developed for solving combinatorial optimization problems mapped to the Ising problem. High-performance Ising machines have been realized with digital processors implementing heuristic algorithms such as simulated bifurcation (SB). In this work, we extend cost functions of SB from second-order ones (Ising energies) to higher-order ones. This higher-order SB allows us to map given problems to cost functions with smaller numbers of spin variables than the second-order SB. We evaluate the performance of the higher-order SB by solving a problem with third-order cost functions, and demonstrate that the higher-order SB can perform better not only than a second-order counterpart, but also than simulated annealing with higher-order cost functions.