Quantum Higher Order Singular Value Decomposition

Phase Factors in Singular Value Decomposition and Schmidt Decomposition

In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AA† and A†A are unique up to complex phase factors. Thus, the two unitary matrices in SVD are unique up to diagonal matrices of phase factors, the phase-factor matrices. Also, the product of these two phase-factor matrices, or the product of phase factors of the corresponding singular vectors with the same singular value, is unique. In the Schmidt decomposition, a phase-factor matrix is a phase rotation operator acting on a subsystem alone. We summarize here three simple steps to consistent…

基于量子退火的奇异值分解

Singular-value decomposition using quantum annealing

在本研究中，我们演示了如何执行，使用量子退火，奇异值分解和主成分分析。量子退火提供了一种找到系统基态的方法，而奇异值分解需要最大本征态。其核心思想是对最终哈密顿量进行符号变换，并通过量子退火得到最大本征态。此外，绝热时间尺度是通过对最大本征值的近似得到的。

Variational Quantum Singular Value Decomposition

Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although these algorithms are promising, the required quantum subroutines and resources are too costly on near-term quantum devices. In this work, we propose a variational quantum algorithm for singular value decomposition (VQSVD). By exploiting the variational principles for singular values and the Ky Fan Theorem, we design a novel loss function such that two quantu…

Quantum Singular Value Decomposition of Spin Correlation Matrix in One-Dimensional Heisenberg Model

We present singular value decomposition of spin correlation matrix defined from the ground state of one-dimensional antiferromagnetic quantum Heisenberg model. The decomposition creates a data set that coincides with various domain excitations from classical antiferromagnetic state. We determine the scaling relation for the singular values as a function of the domain size. The singular values are closely related to the square of weights of the bases in the ground-state wavefunction. The nature of the singular value decomposition is to precisely estimate the appropriate bases and corresponding …

量子信息理论中的奇异值分解和矩阵重排

Singular value decomposition and matrix reorderings in quantum information theory

我们回顾施密特和克劳斯分解的形式奇异值分解使用的整形，向量化和重组操作。利用引入的符号，借助Jamiolkowski同构分析了量子态与操作之间的对应关系。提出的矩阵重排使我们能够得到简单的公式组成的量子信道和部分操作中使用的量子信息理论。为了提供所讨论的操作的示例，我们利用Mathematica计算系统的软件包来实现与量子信息理论相关的计算中使用的基本功能。

利用奇异值分解对能谱组合系综进行展开（unfolding）

Unfolding a composed ensemble of energy spectra using singular value decomposition

针对存在样本间涨落、局域态密度非单调的能谱组合系综中经典展开方法失效的问题，提出利用奇异值分解实现能谱展开，可适用于参数不受控或态密度强涨落的实验场景。

基于高阶奇异值分解的粗粒度重整化

Coarse-graining renormalization by higher-order singular value decomposition

提出了一种基于高阶奇异值分解的粗粒度张量重整化群方法。这种方法提供了一个准确的，但计算成本低的技术，在二维或三维的经典和量子晶格模型的研究。我们已经证明了这种方法使用伊辛模型的正方形和立方晶格。通过保持多达16个键基态，我们得到了迄今为止最精确的数值重整化群的结果为三维伊辛模型。我们还将该方法应用于二维量子横伊辛模型的基态和有限温度性质的研究，得到了与文献一致的结果.

Quantum tensor singular value decomposition with applications to recommendation systems

In this paper, we present a quantum singular value decomposition algorithm for third-order tensors inspired by the classical algorithm of tensor singular value decomposition (t-svd) and then extend it to order-$p$ tensors. It can be proved that the quantum version of the t-svd for a third-order tensor $\mathcal{A} \in \mathbb{R}^{N\times N \times N}$ achieves the complexity of $\mathcal{O}(N{\rm polylog}(N))$, an exponential speedup compared with its classical counterpart. As an application, we propose a quantum algorithm for recommendation systems which incorporates the contextual situation o…