Ising Model Optimization Problems on a FPGA Accelerated Restricted Boltzmann Machine

Abstract: Optimization problems, particularly NP-Hard Combinatorial Optimization problems, are some of the hardest computing problems with no known polynomial time algorithm existing. Recently there has been interest in using dedicated hardware to accelerate the solution to these problems, with physical annealers and quantum adiabatic computers being some of the state of the art. In this work we demonstrate usage of the Restricted Boltzmann Machine (RBM) as a stochastic neural network capable of solving these problems efficiently. We show that by mapping the RBM onto a reconfigurable Field Programmable Gate Array (FPGA), we can effectively hardware accelerate the RBM's stochastic sampling algorithm. We benchmark the RBM against the DWave 2000Q Quantum Adiabatic Computer and the Optical Coherent Ising Machine on two such optimization problems: the MAX-CUT problem and finding the ground state of a Sherrington-Kirkpatrick (SK) spin glass. On these problems, the hardware accelerated RBM shows best in class performance compared to these other accelerators, with an empirical scaling performance of $\mathcal{O}(e^{-N})$ for probability of reaching the ground state compared to a similar empirical $\mathcal{O}(e^{-N})$ for the CIM (with the RBM showing a constant factor of improvement over the CIM) and empirical $\mathcal{O}(e^{-N^2})$ for the DWave Annealer. The results show up to $10^7$x and $10^5$x time to solution improvement compared to the DWave 2000Q on the MAX-CUT and SK problems respectively, along with a $150$x and $1000$x performance increase compared to the Coherent Ising Machine annealer on those problems. By using commodity hardware running at room temperature for acceleration, the RBM also has greater potential for immediate and scalable use.