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.

提供了一种基于量子变分电路构建奇异值分解的算法，可以在计算基的情况下仅通过一次测量获得两个子系统的精确输出一致性，并保持系统的纠缠状态。

量子奇异值分解器

Quantum Singular Value Decomposer

We study the mapping which occurs when a single qubit in an arbitrary state
interacts with another qubit in a given, fixed state resulting in some unitary
transformation on the two qubit system which, in effect, makes two copies of
the first qubit. The general problem of the quality of the resulting copies is
discussed using a special representation, a generalization of the usual Schmidt
decomposition, of an arbitrary two-dimensional subspace of a tensor product of
two 2-dimensional Hilbert spaces. We exhibit quantum circuits which can
reproduce the results of any two qubit copying machine of this type. A simple
stochastic generalization (using a ``classical'' random signal) of the copying
machine is also considered. These copying machines provide simple embodiments
of previously proposed optimal eavesdropping schemes for the BB84 and B92
quantum cryptography protocols.

本文研究了在任意状态下的单个量子比特与给定状态下的另一量子比特相互作用时发生的映射，旨在通过一个特殊表示法，即一种任意二维向量子空间的广义 Schmidt 分解，讨论复制质量的一般问题。同时展示了可以复制该类型任何两比特复制机的量子电路，并考虑了这种复制机的简单随机泛化。这些复制机为 BB84 和 B92 量子密码协议的先前提出的最优窃听方案提供了简单的实现。