Quantum Field Theory Universality Criterion for Layered Programmable Decompositions

The decomposition of arbitrary unitary transformations into sequences of simpler, physically realizable operations is a foundational problem in quantum information science, quantum control, and linear optics. We establish a 1D Quantum Field Theory model for justifying the universality of a broad class of such factorizations. We consider parametrizations of the form U = D₁ V₁ D₂ V₂ cdots V_M-1D_M, where \{D_j\} are programmable diagonal unitary matrices and \{V_j\} are fixed mixing matrices. By leveraging concepts like the anomalies of our effective model, we establish universality criteria given the set of mixer matrices. This approach yields a rigorous proof grounded in physics for the conditions required for the parametrization to cover the entire group of special unitary matrices. This framework provides a unified method to verify the universality of various proposed architectures and clarifies the nature of the "generic" mixers required for such constructions. We also provide a deterministic algorithm for verifying this genericity condition and a geometry-aware optimization method for finding the parameters of a decomposition.