Programming the Vacuum: A Unified Hamiltonian Engineering Framework for Optimization and Spectral Synthesis

The exponential growth of Hilbert space dimensions presents a fundamental barrier to the classical simulation of quantum systems, necessitating a paradigm shift from algorithmic simulation to physical instantiation. This study validates the “Inverted Church-Turing-Deutsch” (CTD) framework, which posits that physical Hamiltonian systems can serve as efficient computational substrates for problems intractable to Turing machines. We operationalize this framework through two modalities: “Class A” discrete optimization using Ising models and “Class B” spectral engineering targeting the Riemann zeros. Through rigorous computational validation on synthetic data ($N=3-12$), we demonstrate a decisive scaling divergence: Hamiltonian relaxation exhibits polynomial scaling (T ∝ N2.02), whereas classical brute-force search follows an exponential trajectory (T ∝ 20.55N), supported by a Bayes Factor > 10²⁵. Furthermore, we successfully engineer a 1D physical potential $V(x)$ that reproduces the first five Riemann zeros with a Mean Absolute Percentage Error of 0.033%, effectively functioning as a “Physical Oracle” for number theory. While acknowledging the limitations of synthetic noise models and fabrication challenges, these findings establish a robust theoretical and algorithmic foundation for “Programming the Vacuum,” suggesting that the universe is not merely a simulator but a universal computational engine.