Last May 7th the members of Quantic Carlos Bravo Prieto, Diego García Martín and José Ignacio Latorre published an arXiv preprint named Quantum Singular Value Decomposer.
In this article we propose a hybrid classical-quantum algorithm that produces the Singular Value Decomposition of a bipartite pure state. The proposed algorithm (Quantum Singular Value Decomposer or QSVD) forces a diagonal form of the state by demanding exact output coincidence.
By training the unitaries to force that subtle diagonalization (the more layers the more precision), we recover the singular values from the probabilities of obtaining the computational-basis vectors after measurement. It follows that entropies can be estimated
A peculiar spinoff of the QSVD circuit is the possibility of performing a SWAP operation between parties A and B without using any gate that connects both subsystems. We just have to apply the adjoints of the unitaries in both parties by classical communication of the parameters.
We consider a further spinoff. We can use the QVSD and CNOTs as a quantum 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.