Matrix encoding method in variational quantum singular value decomposition

Abstract: We propose the variational quantum singular value decomposition based on encoding the elements of the considered matrix into the state of a quantum system of appropriate dimension. This method doesn't use the expansion of this matrix in terms of the unitary matrices. Conditional measurement is involved to avoid small success probability in ancilla measurement. The objective function for maximization algorithm can be obtained probabilistically via measurement of the state of a one-qubit subsystem. The circuit requires $O(\log N)$ qubits for realization of this algorithm whose depths is $O(\log N)$ as well.