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arXiv:2203.10780

Tripartite entanglement and matrix inversion quantum algorithm

Mi-Ra Hwang, MuSeong Kim, Eylee Jung, Chang-Yong Woo, DaeKil Park

Published Nov 23, 2022• 2 citations
arXivHTMLPDF

Abstract

Abstract The role of entanglement is discussed in the Harrow-Hassidim-Lloyd (HHL) algorithm. We compute all tripartite entanglement at every steps of the HHL algorithm. The tripartite entanglement is generated at the first quantum phase estimation (QPE) step. However, it turns out that amount of the generated entanglement is not maximal except very rare cases. In the second rotation step some tripartite entanglement is used. Thus, the net tripartite entanglement is diminished. At the final inverse-QPE step the matrix inversion task is completed at the price of complete annihilation of the entanglement. An implication of this result is briefly discussed.

DOI
10.21203/rs.3.rs-2287755/v1

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Abstract The role of entanglement is discussed in the Harrow-Hassidim-Lloyd (HHL) algorithm. We compute all tripartite entanglement at every steps of the HHL algorithm. The tripartite entanglement is generated at the first quantum phase estimation (QPE) step. However, it turns out that amount of the generated entanglement is not maximal except very rare cases. In the second rotation step some tripartite entanglement is used. Thus, the net tripartite entanglement is diminished. At the final inverse-QPE step the matrix inversion task is completed at the price of complete annihilation of the entanglement. An implication of this result is briefly discussed.
After learning basic quantum computing concepts, it is desirable to reinforce the learning using an important and relatively complex algorithm through which students can observe and appreciate how qubits evolve and interact with each other. Harrow-Hassidim-Lloyd (HHL) quantum algorithm, which can solve linear system problems with exponential speed-up over the classical method and is the basis of many important quantum computing algorithms, is used to serve this purpose. The HHL algorithm is explained analytically followed by a 4-qubit numerical example in bra-ket notation. Matlab code corresponding to the numerical example is available for students to gain a deeper understanding of the HHL algorithm from a pure matrix point of view. A quantum circuit programmed using qiskit is also provided for real hardware execution in IBM quantum computers. After going through the material, students are expected to have a better appreciation of the concepts such as basis transformation, bra-ket and matrix representations, superposition, entanglement, controlled operations, measurement, quantum Fourier transformation, quantum phase estimation, and quantum programming. To help readers review these basic concepts, brief explanations augmented by the HHL numerical examples in the main text are provided in the Appendix.
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Quantum computing has generated significant expectations within the scientific community. However, the development of noise-resilient quantum computers capable of addressing a broad class of practical problems is expected to require at least another decade. During this ongoing evolution, rigorous step-by-step tutorials are essential not only to clarify fundamental concepts but also to disseminate them beyond the core quantum computing community to the wider engineering audience interested in applications. In many engineering fields, the efficient solution of linear systems of equations is a central computational task. For this reason, although it remains an open question whether the Harrow–Hassidim–Lloyd (HHL) algorithm will ultimately play a decisive role in future fault-tolerant application-scale quantum (FASQ) computers, clear and rigorous tutorials on this paradigmatic algorithm remain of fundamental importance. While well-structured introductions to elementary implementations of HHL already exist, certain simplifying assumptions—appropriate for pedagogical clarity—may limit their extension to more general or application-oriented settings. In this work, the HHL algorithm is presented through a detailed analytical treatment, followed by a complete example including a generic module for eigenvalue inversion. A corresponding Jupyter Notebook implementation in Qiskit is provided to support practical understanding. After working through the material, readers are expected to gain a clearer appreciation of how quantum computing concepts operate in practice and why quantum approaches are of interest for combinatorial optimization and machine learning problems.
Abstract This study explores the application of the Harrow–Hassidim–Lloyd (HHL) algorithm for solving linear systems within the context of information security and quantum computing. Linear systems are fundamental in scientific, engineering, financial, and economic applications, impacting tasks such as differential equation solving, regression analysis, and Fourier transforms. In the quantum domain, effective implementation of the HHL algorithm relies on proper utilization of Pauli matrices and the choice of computational bases, including the computational basis and Hadamard basis. This work highlights how basis selection influences algorithmic efficiency and solution accuracy, offering insights for optimizing quantum computation workflows. By examining the HHL algorithm, we demonstrate its potential for accelerating linear system solutions and enhancing information processing capabilities, which may contribute to more secure and efficient computational methodologies in diverse application areas.
The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.