Nebula: Network Enhanced Boltzmann Machine With Universal Local Search Architecture

We propose Nebula, a computational method for combinatorial optimization and stochastic sampling. The proposed method is designed for a generic local search engine utilized for digital annealing in an optimization system for multidimensional binary variables, and it can handle a variety of cost functions of multidimensional binary variables with multi-body interactions in a unified manner. In addition to optimization, the method can be used to perform sampling to reproduce the Boltzmann distribution. To achieve this, we extend the network of conventional Boltzmann machines or Ising machines to include dependent variables mediating various interactions. The extended network enables fast operation by predicting the total energy change due to state transitions in all one-flip neighborhoods. The energy function, which is limited to the quadratic form in conventional Ising machines, is extended here to represent inequality constraints and higher-order products. The ability to handle higher-order spin products enables the implementation of arbitrary energy functions with $k$ -body interactions based on the Walsh transform. We conducted numerical experiments to demonstrate the concept of our proposed method and show that it can solve problems with inequality constraints and higher order terms, which are difficult to solve with conventional Ising machines. The numerical experiments also show that our method can exploit the expansion of the energy landscape by the Walsh function and reproduce the Boltzmann distribution by sampling.