Mathematical aspects of the decomposition of diagonal U(N) operators
M. M. Fedin, A. A. Morozov
TL;DR
This work develops a recursive, diagrammatic method to decompose diagonal unitary matrices in $U(2^n)$ into products of smaller diagonal and SU-block components, formalized through $D_n$, $U_{tail}$, and the tail-control matrices $L(k)$. A linear, bijective map $\mathcal{L}$ links the original diagonal phases to the decomposition parameters via a matrix $r_n$, with the key result $r_n^{-1} = \tfrac{1}{2^{n-1}} r_n^T$ ensuring invertibility when $A_n$ is chosen as a binary-tree sequence; this underpins the recurrent decomposition $D_n = (D_{n-1}\otimes I) \cdot U_{tail}$. The paper analyzes the structure and invertibility of the parameter map, derives explicit inverse relations, and demonstrates diagrammatic symmetries that yield minimal or extended-tail constructions for diagonal unitaries. It also discusses practical implications for quantum circuit synthesis and potential generalizations to broader groups, supported by appendix results on embedding $U(N)$ into $U(2^n)$ and a combinatorial lemma. Overall, the approach provides a unified, algebraic framework for efficient diagonal operator decompositions with clear symmetry insights and diagrammatic intuition, relevant for optimization in quantum computing and related areas.
Abstract
We prove the decomposition of arbitrary diagonal operators into tensor and matrix products of smaller matrices, focusing on the analytic structure of the resulting formulas and their inherent symmetries. Diagrammatic representations are introduced, providing clear visualizations of the structure of these decompositions. We also discuss symmetries of the suggested decomposition. Methods and representations developed in this paper can be applied in different areas, including optimization of quantum computing algorithms, complex biological analysis, crystallography, optimization of AI models, and others.