Exploration of Quantum Algorithms for Singular Value Decomposition Problems

Abstract

This thesis explores quantum algorithms for Singular Value Decomposition (SVD) problems, and focuses on both theoretical developments and practical implementations. First, a detailed description of a known purely quantum algorithm for SVD is provided, along with its application to Latent Semantic Analysis (LSA). Following this, we take a known variational quantum approach to SVD and identify a drawback in its objective function. A solution is presented in the form of a modification to this objective function and an application of this modified algorithm for LSA is also proposed. Comparative simulations between the original and modified algorithms are conducted, alongside experimental validation of the LSA algorithm on quantum hardware. We then study the Quantum Singular Value Transform (QSVT) and understand its relation to Quantum Signal Processing with the aid of an example. Various block encodings are explored, including a novel proposal. Simulations employing QSVT for solving linear systems and Topological Data Analysis are carried out for various block encodings and the results are presented. Finally, we extend our study to tensors, proposing both a purely quantum algorithm and a hybrid variational quantum algorithm to find the t-SVD of a third-order tensor. Simulations are conducted to validate their efficacy.