In optimization, one of the well-known classical algorithms is power iterations. Simply stated, the algorithm recovers the dominant eigenvector of some diagonalizable matrix. Since numerous optimization problems can be formulated as an eigenvalue/eigenvector search, this algorithm features wide applicability. Operationally, power iterations consist of performing repeated matrix-to-vector multiplications (or ) followed by a renormalization step in order to converge to the dominant eigenvalue/eigenvector. However, classical tensor network realizations can, in the worst case, exhibit an exponential growth of the tensor ranks with the number of iterations. In this study, we propose a quantum realization that offers a possible route around this bottleneck. Our method involves casting low-rank representations — Matrix Product Operators (MPO) for matrices and Matrix Product States (MPS) for vectors — into quantum circuits. Specifically, we recover a unitary approximation by variationally minimizing the Frobenius distance between a target MPO and an MPO ansatz wherein the tensor cores are constrained to unitaries. Such unitary MPO can be implemented as a quantum circuit with the addition of ancillary qubits. Thereafter, with appropriate initialization and post-selection on the ancillary space, we realize a single iteration of the classical algorithm. With our proposed methodology, power iterations can be realized entirely on a quantum computer via repeated, static circuit blocks. Moreover, by exploiting Riemannian optimization and cross-approximation techniques, our methodology becomes instance agnostic and thus allows one to address black-box optimization of functions admitting low-rank tensor network representations within the framework of quantum computing. We emphasize that the present work establishes the construction and analyzes the conditions under which it operates reliably; it does not demonstrate an end-to-end computational advantage over classical tensor network optimization, which we discuss explicitly.