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hhl algorithm without ancilla postselection for Toeplitz and Hankel systems

PHYSICS LETTERS A 2025年 531卷

作者： Li, Yong-Le Peng, Bo Wang, Yu-Han Zhang, Zhi-Chao Wang, Yu-Kun Univ Sci & Technol Beijing Sch Math & Phys Beijing 100083 Peoples R China China Univ Petr Beijing Key Lab Petr Data Min Beijing 102249 Peoples R China

Recently, Babukhin [D. V. Babukhin, Phys. Rev. A, 107. 042408(2023)] present the Harrow-Hassidim-Lloyd (hhl) algorithm without ancilla postselection which can work without postselection of the ancilla qubit. However, the remaining questions are to find more useful problems, which satisfy the postselection free condition. In this paper, we first prove that the symmetric Toeplitz matrices and their corresponding observable satisfy the postselection free condition. Then we show that the persymmetric Hankel matrix and the selected observable also satisfy this condition. Furthermore, we provide a specific example of a Toeplitz matrix (which is also a Hankel matrix) and a given observable. Finally, we compare the expectation values of the observable under the two different outcomes of the auxiliary qubit measurements with classical results and find excellent consistency. These results can significantly extend the application scope of postselection free hhl algorithm.

关键词： hhl algorithm Toeplitz matrix Hankel matrix Quantum computing

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A survey on hhl algorithm: From theory to application in quantum machine learning

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PHYSICS LETTERS A 2020年第24期384卷 126595-126595页

作者： Duan, Bojia Yuan, Jiabin Yu, Chao-Hua Huang, Jianbang Hsieh, Chang-Yu Nanjing Univ Aeronaut & Astronaut Coll Comp Sci & Technol Nanjing 211106 Peoples R China Jiangxi Univ Finance & Econ Sch Informat Technol Nanchang 330032 Jiangxi Peoples R China Tencent Quantum Lab Shenzhen 518000 Peoples R China

The Harrow-Hassidim-Lloyd (hhl) algorithm is a method to solve the quantum linear system of equations that may be found at the core of various scientific applications and quantum machine learning models including the linear regression, support vector machines and recommender systems etc. After reviewing the necessary background on elementary quantum algorithms, we provide detailed account of how hhl is exploited in different quantum machine learning (QML) models, and how it provides the desired quantum speedup in all these models. At the end, we briefly discuss some of the remaining challenges ahead for hhl-based QML models and related methods. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Quantum computation hhl algorithm Quantum machine learning Quantum circuit

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Research on quantum dialogue protocol based on the hhl algorithm

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QUANTUM INFORMATION PROCESSING 2023年第9期22卷 1-23页

作者： Yang, Xiaolong Li, Dongfen Zhou, Jie Tan, Yuqiao Zheng, Yundan Liu, Xiaofang Chengdu Univ Technol Coll Comp Sci & Cyber Secur Oxford Brookes Coll Chengdu 610059 Peoples R China

This paper presents an improved quantum dialogue protocol based on the hhl algorithm. In this protocol, Alice and Bob want to transmit their classical information to each other. Combining the hhl algorithm and quantum linear equation ensures that eavesdroppers cannot steal the data of communicators. In addition, communicators will connect come hhl quantum circuit constructed in this scheme with the swap test circuit to calculate the solution rather than directly measuring the result, which can improve the solution speed. Security analysis shows that this scheme is more secure than other general quantum dialogue schemes, and it can realize information protection in extreme cases, with good security and higher transmission efficiency.

关键词： Quantum communication Quantum dialogue hhl algorithm Swap test circuit Fidelity calculation

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The hhl algorithm: Implementation and research directions

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INTERNATIONAL JOURNAL OF QUANTUM INFORMATION 2024年第6期22卷

作者： Sambhaje, Varsha Chaurasia, Anju DY Patil Int Univ Sch Interdisciplinary Studies & Res Pune 411044 Maharashtra India

Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed "NP-completeness" as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow-Hassidim-Lloyd (hhl) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the hhl algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the hhl algorithm's potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the hhl algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the hhl algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the hhl algorithm. Identifying these roadblocks will pave the way for future research and development efforts.

关键词： System of equations quantum state preparation quantum phase estimation measurement condition number hhl algorithm runtime complexity

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Quantum algorithm for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems

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Journal of Systems Science & Complexity 2022年第1期35卷 373-412页

作者： CHEN Yu-Ao GAO Xiao-Shan Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190China University of Chinese Academy of Sciences Beijing 100049China

This paper presents a quantum algorithm to decide whether a Boolean equation system F has a solution and to compute one if F does have solutions with any given success *** runtime complexity of the algorithm is polynomial in the size of F and the condition number of certain Macaulay matrix associated with *** a consequence,the authors give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are polynomial in the size of *** authors apply the proposed quantum algorithm to the cryptanalysis of several important cryptosystems:The stream cipher Trivum,the block cipher AES,the hash function SHA-3/Keccak,the multivariate public key cryptosystems,and show that they are secure under quantum algebraic attack only if the corresponding condition numbers are *** leads to a new criterion for designing such cryptosystems which are safe against the attack of quantum computers:The corresponding condition number.

关键词： Block cipher AES Boolean equation solving condition number hash function SHA3/Keccak hhl algorithm MPKC polynomial system solving quantum algorithm stream cipher Trivum

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Quantum fast Poisson solver: the algorithm and complete and modular circuit design

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QUANTUM INFORMATION PROCESSING 2020年第6期19卷 1-25页

作者： Wang, Shengbin Wang, Zhimin Li, Wendong Fan, Lixin Wei, Zhiqiang Gu, Yongjian Ocean Univ China Coll Informat Sci & Engn Dept Phys Qingdao 266100 Peoples R China Ocean Univ China Coll Informat Sci & Engn Dept Comp Sci & Technol Qingdao 266100 Peoples R China

The Poisson equation has applications across many areas of physics and engineering, such as the dynamic process simulation of ocean current. Here we present a quantum algorithm for solving Poisson equation, as well as a complete and modular circuit design. The algorithm takes the hhl algorithm as the framework (where hhl is for solving linear equations). A more efficient way of implementing the controlled rotation, one of the crucial steps in hhl, is developed based on the arc cotangent function. The key point is that the inverse trigonometric function can be evaluated in a very simple recursive way by a binary expansion method. Quantum algorithms for solving square root and reciprocal functions are proposed based on the classical non-restoring method. These advances not only reduce the algorithm's complexity, but more importantly make the circuit more complete and practical. We demonstrate our circuits on a quantum virtual computing system installed on the Sunway TaihuLight supercomputer. This is an important step toward practical applications of the present circuits as a fast Poisson solver in the near-term hybrid classical/quantum devices.

关键词： Quantum circuit Poisson equation hhl algorithm Trigonometric function Function-value binary expansion

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QUANTUM algorithm FOR MATRIX FUNCTIONS BY CAUCHY'S INTEGRAL FORMULA

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QUANTUM INFORMATION & COMPUTATION 2020年第1-2期20卷 14-36页

作者： Takahira, Souichi Ohashi, Asuka Sogabe, Tomohiro Usuda, Tsuyoshi Sasaki Aichi Prefectural Univ Grad Sch Informat Sci & Technol 1522-3 Ibaragabasama Nagakute Aichi 4801198 Japan Ritsumeikan Univ Coll Sci & Engn 1-1-1 Noji Higashi Kusatsu Shiga 5258577 Japan Nagoya Univ Grad Sch Engn Chikusa Ku Furo Cho Nagoya Aichi 4648603 Japan

For matrix A, vector b and function f, the computation of vector f (A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state vertical bar f > corresponding to vector f (A)b. There is a quantum algorithm to compute state vertical bar f > using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e(iAt). However, the algorithm based on eigenvalue estimation needs poly(1/epsilon) runtime, where epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, e(iAt) is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is poly(log(1/epsilon) and the algorithm outputs state vertical bar f > even if A is not Hermitian.

关键词： Quantum algorithm Matrix functions hhl algorithm Cauchy's integral formula

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Advanced Quantum Poisson Solver in the NISQ era 3

Advanced Quantum Poisson Solver in the NISQ era

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3rd IEEE International Conference on Quantum Computing and Engineering (QCE)

作者： Robson, Walter Saha, Kamal K. Howington, Connor Suh, In-Saeng Nabrzyski, Jaroslaw Univ Notre Dame Dept Comp Sci & Engn Notre Dame IN 46556 USA Univ Notre Dame Ctr Res Comp Notre Dame IN 46556 USA Oak Ridge Natl Lab Natl Ctr Computat Sci Oak Ridge TN 37380 USA

ISBN: (数字)9781665491136

ISBN: (纸本)9781665491136

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far, either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to the linear systems through the finite difference method, we adopt the Harrow-Hassidim-Lloyd (hhl) algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the hhl algorithm. We show that our algorithm not only increases the accuracy of the solutions, but also composes more practical and scalable circuits by dynamically controlling problem size in the NISQ devices. We present both simulated and experimental results, and discuss the sources of errors. Finally, we conclude that overall results on the quantum hardware are dominated by the error in the CNOT gates.

关键词： Poisson Equation Quantum algorithm Quantum Circuit hhl algorithm

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Quantum Linear Prediction For System Identification And Spectral Estimation Applications 57

Quantum Linear Prediction For System Identification And Spec...

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57th Asilomar Conference on Signals, Systems and Computers

作者： Sharma, Aradhita Uehara, Glen Spanias, Andreas Arizona State Univ SenSIP Ctr Sch ECEE Tempe AZ 85281 USA

ISBN: (纸本)9798350325744

This paper presents the design and implementation of quantum algorithms and circuits for linear prediction. The intended applications are system identification, spectral estimation, and speech processing. A frequency-domain method that uses the quantum Fourier transform is developed to estimate the autocorrelation sequence of the signal and a quantum circuit is designed to estimate the linear prediction parameters. The quantum linear prediction performance is evaluated in system identification, autoregressive (AR) spectral estimation, and speech analysis applications. More specifically, we evaluate the performance of the quantum linear predictor in terms of quantum noise effects, qubit precision, and overall computational requirements. Comparisons of quantum versus classical linear prediction are presented.

关键词： Quantum linear prediction quantum Fourier transform hhl algorithm qubits

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A New Hybrid algorithm for Multivariate Polynomial System Solving

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SN Computer Science 2024年第4期5卷 341页

作者： Roy, Debasish IIT Kharagpur India

The method of solving a polynomial system of equations plays a crucial role in various domains of cryptography. For instance, multivariate cryptography using public keys relies on the computational difficulty of solving a multivariate polynomial problem over a finite field. Aram Harrow, Avinatan Hassidam, and Seth Loyd proposed a quantum algorithm (hhl) to solve the Equation Ax = b, where A is a Hermitian matrix. Compared to the fastest classical approach, which has an O(Nκ) run time, the hhl algorithm has a poly(LogN,κ) runtime that offers exponential speed-up. To solve a set of multivariate polynomial equations, we turned to the hhl method for assistance. In particular, our proposed algorithm will be a building block for future quantum XL algorithms. Here, we have proposed a new hybrid quantum XL algorithm and used it to solve a standard problem to demonstrate its efficacy. © The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd 2024.

关键词： hhl algorithm Quantum algorithms Quantum computation XL algorithm

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