Hasegawa and Ohzeki compare Ising and QUBO encodings under fixed model, sampler, and step size, finding QUBO has lower Fisher spectral entropy and slower SGD convergence while natural gradient descent makes the encodings converge alike
Under a controlled protocol that fixes the Boltzmann machine model, simulated-annealing sampler, and learning-rate design, the study compares Ising ({−1,+1}) and QUBO ({0,1}) variable encodings, exploits the identity that the Fisher information matrix equals the covariance of sufficient statistics to visualize empirical moments, and finds that QUBO induces larger cross terms between first- and second-order statistics, creating more small-eigenvalue directions and lowering spectral entropy, which explains slower convergence under stochastic gradient descent, whereas natural gradient descent, which rescales updates by the Fisher information matrix metric, achieves similar convergence across encodings due to reparameterization invariance.