  We present a variational quantum circuit that produces the Singular Value
Decomposition of a bipartite pure state. The proposed circuit, that we name
Quantum Singular Value Decomposer or QSVD, is made of two unitaries
respectively acting on each part of the system. The key idea of the algorithm
is to train this circuit so that the final state displays exact output
coincidence from both subsystems for every measurement in the computational
basis. Such circuit preserves entanglement between the parties and acts as a
diagonalizer that delivers the eigenvalues of the Schmidt decomposition. Our
algorithm only requires measurements in one single setting, in striking
contrast to the $3^n$ settings required by state tomography. Furthermore, the
adjoints of the unitaries making the circuit are used to create the
eigenvectors of the decomposition up to a global phase. Some further
applications of QSVD are readily obtained. The proposed QSVD circuit allows to
construct a SWAP between the two parties of the system without the need of any
quantum gate communicating them. We also show that a circuit made with QSVD and
CNOTs acts as an encoder of information of the original state onto one of its
parties. This idea can be reversed and used to create random states with a
precise entanglement structure.
