This document explores the connections between singular value decomposition (SVD) and various tensor operations in the context of quantum state calculations and tensor networks. The relationships between SVD and quantities such as trace, trace distance, Frobenius norm, and fidelity are examined, highlighting their similarities and differences. The intuition behind these mathematical connections is explained, providing insights into the underlying principles and applications of SVD and tensor operations in quantum physics and computational mathematics
This document explores the connections between singular value decomposition (SVD) and various tensor operations in the context of quantum state calculations and tensor networks. The relationships between SVD and quantities such as trace, trace distance, Frobenius norm, and fidelity are examined, highlighting their similarities and differences. The intuition behind these mathematical connections is explained, providing insights into the underlying principles and applications of SVD and tensor operations in quantum physics and computational mathematics
Abstract Tensors are increasingly ubiquitous in various areas of applied mathematics and computing, and tensor decompositions are of practical significance and benefit many applications in data completion, image processing, computer vision, collaborative filtering, etc. Recently, Kilmer and Martin propose a new tensor factorization strategy, tensor singular value decomposition (t-svd), which extends the matrix singular value decomposition to tensors. However, computing t-svd for high dimensional tensors costs much computation and thus cannot efficiently handle large scale datasets. Motivated by advantage of quantum computation, in this paper, we present a quantum algorithm of t-svd for third-order tensors and then extend it to order-p tensors. We prove that our quantum t-svd algorithm for a third-order N dimensional tensor runs in time N polylog ( N ) if we do not recover classical information from the quantum output state. Moreover, we apply our quantum t-svd algorithm to context-aware multidimensional recommendation systems, where we just need to extract partial classical information from the quantum output state, thus achieving low time complexity.
Tensor networks have emerged as a powerful theoretical and computational framework for understanding and simulating quantum many-body systems. Their efficiency in representing quantum states, particularly those satisfying area laws of entanglement, is intimately linked to the structure of their constituent tensors. A fundamental aspect of these networks is their spectral properties, which manifest in various forms, such as the singular values of bond tensors or the eigenvalues of associated transfer matrices. Concurrently, entanglement measures provide quantitative insights into the quantum correlations present within these systems. This paper explores the intricate relationship between the spectra arising from tensor network representations and the different entanglement measures used to characterize quantum states. We delve into how the decay rates of singular values directly impact entanglement entropy, how entanglement spectra derived from Schmidt decompositions can reveal topological order, and how the spectral gaps of transfer matrices are connected to correlation lengths. By analyzing various tensor network architectures, including Matrix Product States (MPS) and Projected Entangled Pair States (PEPS), we elucidate the mechanisms through which their spectral features encode profound information about quantum entanglement. The study highlights how a deeper understanding of these spectral properties can not only guide the construction of more efficient tensor network algorithms but also provide fundamental insights into the nature of quantum phases and phase transitions.
Developed as a workhorse for classical simulations of quantum algorithms and quantum many-body systems, Tensor Network methods have entered the scientific mainstream in quantum physics. Among various types of tensor networks, Tensor Trains (commonly know as Matrix Product States in the quantum computing community) have already found applications in machine learning. These methods often rely on a powerful linear algebra tool called the Singular Value Decomposition (SVD). Several conditional GAN architectures for image denoising incorporate SVD as a single-cut decomposition step applied to generator feature maps. In this work we introduce TT-Net, which replaces the per-channel SVD denoising block with a two-cut tensor-train decomposition capable of accessing cross-channel information directly, a capability absent from contemporary alternatives. In a controlled comparison differing only in this decomposition mechanism, TT-Net outperforms SVD-Net on PSNR and SSIM across all three noise types tested (Gaussian, motion blur, and salt-and-pepper), supporting the hypothesis that cross-channel access improves denoising quality. Training-dynamics analysis further shows that TT-Net's adversarial loss term consistently saturates to a stagnant state across all three noise types, more so than SVD-Net's, while reconstruction quality continues to improve regardless, raising an open question about the adversarial component's contribution that this work identifies but does not resolve. Furthermore, for Gaussian noise our method outperforms both the EigenGAN and the state of the art Pix2pix method which does not assume any linear algebra decompositions and does not retain any linear algebra information. Our manuscript shows how quantum inspired tools can be used as practical real world feature filters for deep learning applications.
Needless to emphasize, the home of these quantum states and observables are the linear vector spaces or alternatively the Hilbert spaces. This chapter has crucial recipes relating to operations like inverse of matrices, tridiagonalization, QR decomposition, Gram-Schmidt orthogonalization, a wide spectrum of functions and powers of matrices, trace norm, Hilbert-Schmidt norm, singular value decomposition and so on. The recipies in this chapter can be not only used in quantum information theory but it can also be used as standalone recipies for any person working in linear algebra as well.
Tensor networks were developed in the context of many-body physics as compressed representations of multiparticle quantum states. These representations mitigate the exponential complexity of many-body systems by capturing only the most relevant dependencies. Due to the formal similarity between quantum entanglement and statistical correlations, tensor networks have recently been integrated in machine learning, operating both as alternative learning architectures and as decompositions of components of neural networks. The expectation is that the theoretical understanding of tensor networks developed within quantum many-body physics leads to novel methods that offer advantages in terms of computational efficiency, explainability, or privacy. Here we review the use of tensor networks in the context of machine learning, providing a critical assessment of the state of the art, the potential advantages, and the challenges that must be overcome.
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 A∈RN×N×N achieves the complexity of O(Npolylog(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 of users to the personalized recommendation. We provide recommendations varying with contexts by measuring the output quantum state corresponding to an approximation of this user's preferences. This algorithm runs in expected time O(Npolylog(N)poly(k)), if every frontal slice of the preference tensor has a good rank-k approximation. At last, we provide a quantum algorithm for tensor completion based on a different truncation method which is tested to have a good performance in dynamic video completion.