Quantum singular value decomposer

We present a variational quantum circuit that produces the singular value decomposition of a bipartite pure state. The proposed circuit, which 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 us 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.

Figure 1

Parametrized unitary transformations implementing the quantum singular value decomposer (QSVD). Training is based on demanding exact output coincidence for both parties and for every measurement.

Figure 2

Left: Von Neumann entropy computed from the variational form of QSVD vs. exact entropy, for random states (including a product state and AME state) of six qubits and natural bipartition. As the number of layers increases, we observe convergence towards the exact entropy. Right: Mean relative error in the estimation of the entropy vs. number of layers (error bars represent the standard deviation). The error decreases exponentially with the depth of the circuit, as suggested by the Solovay-Kitaev theorem.

Figure 3

Application of the QSVD followed by the adjoint $U^{†}$ and $V^{†}$ gates acting on opposite subsystems, mediated by classical communication ($CC$) of the optimal parameters, allows us to perform a long-distance SWAP operation without the need of any quantum communication between subsystems.

Figure 4

Further use of CNOT gates makes QSVD an encoder of the original quantum state $|\psi {\rangle }_{AB}$ onto one of its parts $|\varphi {\rangle }_B$.

Figure 5

Architecture of the variational circuit employed in the QSVD. Several layers of gates are applied in a consecutive manner in order to improve accuracy, and the circuit always ends with a final set of single-qubit rotations prior to measurement. The notation stands for $R\left({\theta }_{\alpha ,\beta ,\gamma }\right)\equiv Rz\left({\theta }_{\alpha }\right)Rx\left({\theta }_{\beta }\right)Rz\left({\theta }_{\gamma }\right)$. If the number of qubits is odd in a given subsystem, then an extra CZ between the first and last qubit of the subsystem is added after each complete rotation.