First hardware demonstration of a continuous-state probabilistic computer with programmable energy-defined sampling

Abstract: Probabilistic computing has proved useful for combinatorial optimization, but conventional implementations rely on binary stochastic units and sigmoid-based updates. Extending these systems to continuous variables requires binary encoding. Here we introduce a continuous probabilistic computer (CPC) whose continuous probabilistic bits sample bounded states from conditional distributions defined by local energies. Separating random-source generation from conditional sampling provides a route to using physical random sources with different native distributions. A field-programmable gate array (FPGA) prototype updates finite-precision states using digital randomness. The same framework supports product constraints, memory recovery and solution-space exploration. For the tested 7-bit factorization case, fixed-temperature FPGA CPC reduces mean core solving time approximately 33,000-fold relative to the evaluated quadratic unconstrained binary optimization software implementation. The FPGA recovers stored MNIST patterns without pretrained networks, achieving final success rates of 100%. Software CPC reaches 90% reference coverage of the feasible area in approximately 18% of the time required by restarted Adam.