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 $U = D_1 V_1 D_2 V_2 \cdots V_{M-1}D_M$, 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.