Mathematical Validation of the Parametric Trotter Expression and the Unitary Parametric Trotter Operator

This is the derivation of the Parametric Trotter Expression (PTE), a parameterized operator ansatz for approximating time evolution in quantum circuits by combining intensity (weights) and timing design variables. I demonstrate how PTE arises naturally from Schr¨odinger time evolution through Hamiltonian decomposition and trotterization, explain why the PTE in its raw sum- of-exponentials form is generally nonunitary, and provide a rigorous and numerically stable construction of the U-PTE (unitary parametric trotter expression) via polar (SVD) projection with appropriate regularization. I show how the PTE’s parameters (ρ, τ) influence output states and describe how the U-PTE operator can be embedded in optimization frameworks using methods such as the Method of Moving Asymptotes. All derivations are worked out explicitly with equal emphasis on theoretical foundations and mathematical rigor. This validation study establishes the mathematical foundations for the Parametric Trot- ter Expression as a key component of topology-optimized quantum circuit frameworks. The PTE enables weighted superposition of circuit primitives which can potentially provide noise re- silience and error suppression protocols as well. This paper, however, focuses exclusively on the mathematical validation of the PTE ansatz, with full integration into optimized architectures presented in the companion work