# Quantum Singular Value Decomposer

## TL;DR

The paper introduces QSVD, a variational quantum circuit that efficiently performs singular value decomposition of bipartite states, enabling entanglement analysis and state manipulation with fewer measurements than traditional tomography.

## Contribution

It presents the first variational quantum circuit for SVD of bipartite states that requires only one measurement setting, improving efficiency over existing methods.

## Key findings

- QSVD accurately decomposes bipartite states into eigenvalues and eigenvectors.
- The circuit enables entanglement-preserving transformations like SWAP without additional gates.
- QSVD-based circuits can encode and generate states with specific entanglement properties.

## Abstract

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.