Abugattas Formalizes Heisenberg-Bayesian Equivalence via Matern Eigenfunctions on arXiv

The work maps the quadratic likelihood norm in Hilbert space onto the Heisenberg picture by treating the direct model as a time-dependent operator. Karhunen-Loeve expansion of the Matern covariance supplies the eigenbasis in which the likelihood becomes a quadratic form over quantum states. This yields explicit expressions for the evolution operator's unitarity, the Hamiltonian's self-adjointness, observable time-invariance, and posterior equivalence, all verified numerically for limiting length scales.

The correspondence reframes classical inverse problems as closed quantum dynamics, suggesting quantum circuits could evaluate high-dimensional posteriors without explicit sampling. It also supplies a new interpretation of the Matern kernel as a quantum observable whose spectrum encodes regularization strength. Existing quantum-Bayesian literature has lacked this operator-level bridge, so the four theorems close a formal gap rather than merely restating known analogies.

The principal limitation is restriction to Gaussian noise and stationary kernels; non-stationary or heavy-tailed observation models remain unaddressed. Strengthening evidence requires an experimental test on quantum hardware measuring estimation variance reduction against classical MCMC baselines within controlled Matérn length regimes.

Next steps include extending the formalism to Lindblad evolution for open systems and testing whether the same operator correspondence accelerates variational inference on near-term devices.