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Journal of Physics Communications

Quantum tensor singular value decomposition*

Xiaoqiang Wang, Lejia Gu, Heung-wing Lee, Guofeng Zhang

Published Jun 22, 2021• 4 citations
Publisher

Abstract

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.

DOI
10.1088/2399-6528/ac0d5f

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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.
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-ppp tensors. It can be proved that the quantum version of the t-svd for a third-order tensor A∈RN×N×N\mathcal{A} \in \mathbb{R}^{N\times N \times N}A∈RN×N×N achieves the complexity of O(Npolylog(N))\mathcal{O}(N{\rm polylog}(N))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)),\mathcal{O}(N{\rm polylog}(N){\rm poly}(k)),O(Npolylog(N)poly(k)), if every frontal slice of the preference tensor has a good rank-kkk 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.
This research paper explores the application of singular value decomposition (SVD) in quantum image processing (QIP), specifically focusing on the computation of eigenvalues using variational quantum algorithms. SVD is a powerful mathematical tool in image processing, used for tasks such as image compression, noise reduction, and feature extraction. In this study, we propose quantum variational singular value decomposition (QVSVD) based on the variational quantum deflation (VQD) algorithm to determine the eigenvalues that contributes in calculating the singular values of the image matrix and eventually doing the SVD. This facilitates the extraction of eigenvalues with exponential speedup on real hardware compared to classical methods. We detail the implementation of this quantum algorithm within the framework of QIP, highlighting its advantages in terms of computational efficiency and accuracy on a Fault-Tolerant Quantum Computing (FTQC). Furthermore, we present a comparative analysis of the quantum and the classical method of SVD, demonstrating the accuracy of the image data. The simulation results validate the theoretical advantages of using quantum algorithms for SVD in image processing. This work not only underscores the potential of quantum computing in enhancing image processing techniques but also sets the stage for future research in the field, exploring more complex image processing tasks and other quantum algorithms. Our findings suggest that quantum image processing can offer unprecedented capabilities, paving the way for advancements in various applications such as medical imaging, remote sensing, and multimedia processing.
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 Researchers have put a lot of effort into reducing the gap between current quantum processing units (QPU) capabilities and their potential supremacy.One approach is to keep supplementary computations in the CPU, and use QPU only for the core of the problem. In this work, we address the complexity of quantum algorithm of arbitrary quantum state initialization, an important building block of quantum data analysis and machine learning.QPUs do not outperform classical machines with existing precise initialization algorithms.Hence, many studies propose an approximate but robust quantum state initialization.Cutting a quantum state into a product of (almost) independent partitions with the help of CPU reduces the number of two-qubit gates, and correspondingly minimizes the loss of state fidelity in the quantum part of the algorithm.To find the least entangled qubits, current methods compute the singular value decomposition (SVD) for each qubit separately with CPU.In this paper, we optimize CPU usage and memory resource bottlenecks.We consider Tucker tensor decomposition as an alternative to the CPU-based SVD in a single low-entangled qubit detection task without the loss of solution quality.Both proposed methods outperform the SVD in time and memory for systems of at least ten qubits. We achieve an order faster implementation and two orders less memory usage for a system of 15 qubits.
In the context of big data, high-order tensor decompositions have to face a new challenge in terms of storage and computational costs. The tensor train (TT) decomposition provides a very useful graph-based model reduction, whose storage cost grows linearly with the tensor order DDD. The computation of the TT-core tensors and TT-ranks can be done in a stable sequential (i.e., noniterative) way thanks to the popular TT-SVD algorithm. In this paper, we exploit the ideas developed for the hierarchical/tree Tucker decomposition in the context of the TT decomposition. Specifically, a new efficient estimation scheme, called TT-HSVD (Tensor-Train Hierarchical SVD), is proposed as a solution to compute the TT decomposition of a high-order tensor. The new algorithm simultaneously delivers the TT-core tensors and their TT-ranks in a hierarchical way. It is a stable (i.e., noniterative) and computationally more efficient algorithm than TT-SVD, which is very important when dealing with large-scale data. The TT-HSVD algorithm uses a new reshaping strategy and a tailored partial SVD, which allows us to deal with smaller matrices compared to those of the TT-SVD. In addition, TT-HSVD is well suited for a parallel processing architecture. An algebraic analysis of the two algorithms is carried out, showing that TT-SVD and TT-HSVD compute the same TT-ranks and TT-core tensors up to specific bases. Simulation results for different tensor orders and dimensions corroborate the effectiveness of the proposed algorithm.
Singular value decomposition (SVD) is fundamentally important and broadly useful in both quantum and classical computing. There are several quantum algorithms known for SVD. Variational quantum algorithms with parametric quantum circuits (PQC) are the most promising approach to find SVD with near-term quantum computers. This paper reports a new method for quantum SVD (QSVD) that finds the singular vectors ordered by the magnitude of singular values by adding extra CNOT gates and a new local cost function design. This ordering is important because the singular vectors with the larger singular values is more representative for the given information. The ordering property of this approach is investigated with the standard Iris dataset by numerical simulations of 4-qubit states. Methods for choosing appropriate choices for the cost function hyperparameter and cost function terms as well as an application example of a quantum encoder are also discussed.