A Comparative Analysis of Ising Formulations for Neuromorphic Maximum-Likelihood Channel Decoding
Neuromorphic computing has so far been driven predominantly by machine-learning workloads, yet its underlying properties also make it particularly well suited to combinatorial optimization problems expressed in Ising or QUBO form. While neuromorphic Ising solvers have been demonstrated, how a given problem should be formulated to best suit neuromorphic dynamics has received far less attention. Maximum-likelihood (ML) channel decoding can be expressed as an Ising/QUBO problem, and two distinct formulations already exist in the quantum-annealing literature: a squared-penalty formulation that uses few spins but produces dense intra-check couplings, and a chain-product formulation that improves locality at the cost of additional auxiliary spins. Both place the ML codeword at the ground state under sufficient constraint enforcement, but they have not been compared under the constraints that neuromorphic hardware imposes. This work provides the first systematic side-by-side comparison of QUBO/Ising formulations of ML decoding for linear codes. We show that the two formulations impose fundamentally different tradeoffs in neuron count, synaptic density, locality, and convergence behavior. The preferred formulation is inseparable from the choice of solver, and the two must be considered jointly. Finally, we show that ground-state correctness alone is an insufficient design criterion, and that signal processing tasks should ideally be co-formulated with their neuromorphic hardware models if neuromorphic computing is to extend into the receiver pipeline.