Let the Qudit Do the Jacobi: A Structured Quantum Algorithm for Spectral Decomposition

Jacobi diagonalization is a long-established numerical algorithm for the spectral decomposition of Hermitian and, more generally, normal matrices. In this work, we develop a qudit-native quantum realization of the Jacobi diagonalization algorithm for unknown unitary operators. The proposed framework avoids explicit reconstruction of the operator and controlled-unitary operations. Each elementary two-parameter Givens rotation is implemented through two sequential single-parameter quantum variational optimizations that are directly accessible experimentally. An interferometric protocol is further introduced for extracting the eigenvalues of the diagonalized unitary, up to an overall global phase. Numerical simulations on ensembles of Haar-random unitary matrices demonstrate that the proposed algorithm preserves the characteristic convergence behavior of the classical Jacobi method and exhibits the expected quadratic scaling in the number of elementary operations with the dimension $d$, comparable to its classical counterpart. The results establish the proposed algorithm as a structured quantum numerical linear algebra algorithm naturally suited to qudit architectures and provide a bridge between classical iterative matrix algorithms and their quantum realizations.