A Unified Theoretical Framework for Fractal Computational Complexity and Quantum Phase Transitions

This paper establishes a new paradigm of computational complexity based onfractal dimensions, proposing a rigorous mathematical framework for dimension?covariant complexity, classified as NPDH. By constructing fractal input spaces andverification protocols, we demonstrate that for Hausdorff dimensions DH > Dc,NP-complete problems can be solved in polynomial time, revealing the critical roleof dimension as an order parameter for computational difficulty. Using supercon?ducting quantum chips, we fabricated Sierpi´nski fractal qubit arrays and, for thefirst time, observed the critical dimensional phase transition for the 3-SAT problem(Dc = 3.15±0.02) and quantum acceleration effects (T ∝ n0.92). This work providesboth theoretical and experimental validation for the application of fractal geome?try in quantum computing and proposes an interdisciplinary research paradigm of”quantum phase transitions and dimensional reduction.”