# Constrained Maximum Likelihood Hamiltonian Fitting

Probabilistic-bit invertible logic encodes a Boolean circuit as an energy and samples it to compute in any direction, but conventional encodings only enforce legal ground states and widen the minimum gap, while finite-time samplers spend their runs in excited states. This work casts Hamiltonian design as inverse statistical mechanics, fitting interactions up to fourth order by constrained maximum likelihood under a gap floor so that the Gibbs law approaches uniformity on each clamp case's legal states. On 44- to 77-bit multipliers in factor mode, the learned energies cut updates to 99% verified success to 0.55, 0.44, 0.52 and 0.46 of the hand-written encoding across nine tuned samplers, at the cost of more terms evaluated per update.

Probabilistic-bit invertible logic encodes a Boolean circuit as an energy and samples it to compute in any direction, but conventional encodings only enforce legal ground states and widen the minimum gap, while finite-time samplers spend their runs in excited states. This work casts Hamiltonian design as inverse statistical mechanics, fitting interactions up to fourth order by constrained maximum likelihood under a gap floor so that the Gibbs law approaches uniformity on each clamp case's legal states. On 44- to 77-bit multipliers in factor mode, the learned energies cut updates to 99% verified success to 0.55, 0.44, 0.52 and 0.46 of the hand-written encoding across nine tuned samplers, at the cost of more terms evaluated per update.