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Optimal deterministic quantum algorithm for the promised element distinctness problem ☆

THEORETICAL COMPUTER SCIENCE 2024年 999卷

作者： Li, Guanzhong Li, Lvzhou Sun Yat Sen Univ Inst Quantum Comp & Software Sch Comp Sci & Engn Guangzhou 510006 Peoples R China Quantum Sci Ctr Guangdong Hong Kong Macao Greater Shenzhen 518045 Peoples R China

The element distinctness problem is to determine whether a string of N elements, i.e. x = ( x 1 , ... , x N ) where x;is an element of {1 , ... , M } and M >= N , contains two elements of the same value (a.k.a colliding pair), for which Ambainis proposed an optimal quantum algorithm. The idea behind Ambainis' algorithm is to first reduce the problem to the promised version in which x is promised to contain at most one colliding pair, and then design an algorithm A requiring O ( N 2 / 3 ) queries based on quantum walk search for the promise problem. However, A is probabilistic and may fail to give the right answer. We thus, in this work, design a deterministic quantum algorithm for the promise problem which never errs and requires O ( N 2 / 3 ) queries. This algorithm is proved optimal. Technically, we modify the quantum walk search operator on quasi -Johnson graph to have arbitrary phases, and then use Jordan's lemma as the analyzing tool to reduce the quantum walk search operator to the generalized Grover's operator. This allows us to utilize the recently proposed fixed -axis -rotation (FXR) method for deterministic quantum search, and hence achieve 100% success.

关键词： quantum computing quantum algorithm Element distinctness problem

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Efficient and fail-safe quantum algorithm for the transport equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2024年 502卷

作者： Schalkers, Merel A. Moller, Matthias Delft Univ Technol Dept Appl Math Delft Netherlands

In this paper we present a scalable algorithm for fault-tolerant quantum computers for solving the transport equation in two and three spatial dimensions for variable grid sizes and discrete velocities, where the object walls are aligned with the Cartesian grid, the relative difference of velocities in each dimension is bounded by 1 and the total simulated time is dependent on the discrete velocities chosen. We provide detailed descriptions and complexity analyses of all steps of our quantum transport method (QTM) and present numerical results for 2D flows generated in Qiskit as a proof of concept. Our QTM is based on a novel streaming approach which leads to a reduction in the amount of CNOT gates required in comparison to state-of-the-art quantum streaming methods. As a second highlight of this paper we present a novel object encoding method, that reduces the complexity of the amount of CNOT gates required to encode walls, which now becomes independent of the size of the wall. Finally we present a novel quantum encoding of the particles' discrete velocities that enables a linear speed-up in the costs of reflecting the velocity of a particle, which now becomes independent of the amount of velocities encoded. Our main contribution consists of a detailed description of a fail-safe implementation of a quantum algorithm for the reflection step of the transport equation that can be readily implemented on a physical quantum computer. This fail-safe implementation allows for a variety of initial conditions and particle velocities and leads to physically correct particle flow behavior around the walls, edges and corners of obstacles. Combining these results we present a novel and fail-safe quantum algorithm for the transport equation that can be used for a multitude of flow configurations and leads to physically correct behavior. ( ) We finally show that our approach only requires O ��2 ��+ �� 2CNOT gates, which is �� max

关键词： quantum computing quantum computational fluid dynamics Transport equation quantum algorithm

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A quantum algorithm Framework for Discrete Probability Distributions With Applications to Renyi Entropy Estimation

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IEEE TRANSACTIONS ON INFORMATION THEORY 2024年第5期70卷 3399-3426页

作者： Wang, Xinzhao Zhang, Shengyu Li, Tongyang Peking Univ Ctr Frontiers Comp Studies Beijing 100871 Peoples R China Peking Univ Sch Comp Sci Beijing 100871 Peoples R China Tencent Quantum Lab Shenzhen 518057 Peoples R China

Estimating statistical properties is fundamental in statistics and computer science. In this paper, we propose a unified quantum algorithm framework for estimating properties of discrete probability distributions, with estimating R & eacute;nyi entropies as specific examples. In particular, given a quantum oracle that prepares an n-dimensional quantum state & sum;(n )(i=1)root pi|(i)> , for alpha > 1 and 0 < alpha < 1 , our algorithm framework estimates alpha -R & eacute;nyi entropy H-alpha(p) to within additive error & varepsilon;with probability at least 2/3 using O(n(1-1/2 alpha)/ & varepsilon;+ root n-/& varepsilon;(1+1/2 alpha)) and O(n(1/2 alpha)/& varepsilon;(1+1/2 alpha)) queries, respectively. This improves the best known dependence in & varepsilon;as well as the joint dependence between n and 1/& varepsilon;. Technically, our quantum algorithms combine quantum singular value transformation, quantum annealing, and variable-time amplitude estimation. We believe that our algorithm framework is of general interest and has wide applications.-

关键词： Entropy quantum algorithm Complexity theory Probability distribution Estimation Additives Amplitude estimation Entropy estimation R & eacute nyi entropy quantum algorithms quantum query complexity

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Learning to Learn Variational quantum algorithm

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023年第11期34卷 8430-8440页

作者： Huang, Rui Tan, Xiaoqing Xu, Qingshan Jinan Univ Coll Informat Sci & Technol Guangzhou 510632 Peoples R China Jinan Univ Coll Cyber Secur Guangzhou 510632 Peoples R China Adv Cryptog & Syst Secur Key Lab Sichuan Prov Chengdu 610025 Peoples R China

Variational quantum algorithms (VQAs) use classical computers as the quantum outer loop optimizer and update the circuit parameters to obtain an approximate ground state. In this article, we present a meta-learning variational quantum algorithm (meta-VQA) by recurrent unit, which uses a technique called "meta-learner." Motivated by the hybrid quantum-classical algorithms, we train classical recurrent units to assist quantum computing, learning to find approximate optima in the parameter landscape. Here, aiming to reduce the sampling number more efficiently, we use the quantum stochastic gradient descent method and introduce the adaptive learning rate. Finally, we deploy on the TensorFlow quantum processor within approximate quantum optimization for the Ising model and variational quantum eigensolver for molecular hydrogen (H-2), lithium hydride (LiH), and helium hydride cation (HeH+). Our algorithm can be expanded to larger system sizes and problem instances, which have higher performance on near-term processors.

关键词： Meta-learning quantum algorithm quantum computing quantum information quantum machine learning (QML)

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A quantum algorithm for obtaining the lowest eigenstate of a Hamiltonian assisted with an ancillary qubit system

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quantum INFORMATION PROCESSING 2015年第1期14卷 103-118页

作者： Bang, Jeongho Lee, Seung-Woo Lee, Chang-Woo Jeong, Hyunseok Seoul Natl Univ Ctr Macroscop Quantum Control Dept Phys & Astron Seoul 151747 South Korea Texas A&M Univ Qatar Dept Phys Doha Qatar Univ Queensland Sch Math & Phys Ctr Quantum Computat & Commun Technol Brisbane Qld 4072 Australia Hanyang Univ Dept Phys Seoul 133791 South Korea

We propose a quantum algorithm to obtain the lowest eigenstate of any Hamiltonian simulated by a quantum computer. The proposed algorithm begins with an arbitrary initial state of the simulated system. A finite series of transforms is iteratively applied to the initial state assisted with an ancillary qubit. The fraction of the lowest eigenstate in the initial state is then amplified up to similar or equal to 1. We prove that our algorithm can faithfully work for any arbitrary Hamiltonian in the theoretical analysis. Numerical analyses are also carried out. We firstly provide a numerical proof-of-principle demonstration with a simple Hamiltonian in order to compare our scheme with the so-called "Demon-like algorithmic cooling (DLAC)", recently proposed in Xu (Nat Photonics 8:113, 2014). The result shows a good agreement with our theoretical analysis, exhibiting the comparable behavior to the best 'cooling' with the DLAC method. We then consider a random Hamiltonian model for further analysis of our algorithm. By numerical simulations, we show that the total number n(c) of iterations is proportional to similar or equal to O(D-1 epsilon(-0.19)) , where D is the difference between the two lowest eigenvalues and is an error defined as the probability that the finally obtained system state is in an unexpected (i.e., not the lowest) eigenstate.

关键词： quantum algorithm algorithmic cooling Ground state

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Three-Dimensional Reconstruction of Fuzzy Medical Images Using quantum algorithm

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IEEE ACCESS 2020年 8卷 218279-218288页

作者： Li, Xiaofeng Sui, Jun Wang, Yanwei Heilongjiang Int Univ Dept Informat Engn Harbin 150025 Peoples R China Heilongjiang Int Univ Off Acad Res Harbin 150025 Peoples R China Purdue Univ Dept Mech Engn W Lafayette IN 47906 USA

In order to deal with the problems of poor anti-interference, low reconstruction clarity and large errors in the traditional medical image reconstruction methods, we proposed a fuzzy medical images three-dimensional (3D) reconstruction method using quantum algorithm. First of all, a feature matching model of fuzzy medical image was built. Secondly, this study decomposed the edge contour features by using the Gaussian mixture feature matching method, extracted the edge contour vectors of fuzzy medical image, and enhanced the information of fuzzy medical images by adopting the region edge sharpening. Thirdly, this study reorganized the 3D texture structure of images and reconstructed the sparse scattered points according to its texture and detail regions. Finally, we combined with the gray histogram of fuzzy medical images to achieve the adaptive pixel reconstruction of fuzzy medical images, and completed the 3D reconstruction of fuzzy medical images by employing the quantum algorithm. The results show that the proposed method is characterized by high matching degree of image features and balanced distribution of point clouds, and the self-similarity coefficient of the reconstructed texture can reach 0.994;in addition, the SINR value of the reconstruction result can be maintained around 100dB, and it has lower error rate than the traditional method, thereby improving the detection and recognition capability of medical images, and the algorithm has certain practical application.

关键词： Biomedical imaging Three-dimensional displays Image reconstruction Feature extraction Image edge detection Solid modeling Topology quantum algorithm fuzzy medical images three-dimensional reconstruction characteristics of the reconstruction

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An Improved quantum algorithm for Ridge Regression

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IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 2021年第3期33卷 858-866页

作者： Yu, Chao-Hua Gao, Fei Wen, Qiao-Yan Beijing Univ Posts & Telecommun State Key Lab Networking & Switching Technol Beijing 100876 Peoples R China State Key Lab Cryptol POB 5159 Beijing 100878 Peoples R China Jiangxi Univ Finance & Econ Sch Informat Technol Nanchang 330032 Jiangxi Peoples R China Univ Western Australia Sch Phys Nedlands WA 6009 Australia Peng Cheng Lab Ctr Quantum Comp Shenzhen 518055 Peoples R China

Ridge regression (RR) is an important machine learning technique which introduces a regularization hyperparameter a to ordinary multiple linear regression for analyzing data suffering from multicollinearity. In this paper, we present a quantum algorithm for RR, where the technique of parallel Hamiltonian simulation to simulate a number of Hermitian matrices in parallel is proposed and used to develop a quantum version of K-fold cross-validation approach, which can efficiently estimate the predictive performance of RR. Our algorithm consists of two phases: (1) using quantum K-fold cross-validation to efficiently determine a good a with which RR can achieve good predictive performance, and then (2) generating a quantum state encoding the optimal fitting parameters of RR with such a, which can be further utilized to predict new data. Since indefinite dense Hamiltonian simulation has been adopted as a key subroutine, our algorithm can efficiently handle non-sparse data matrices. It is shown that our algorithm can achieve exponential speedup over the classical counterpart for (low-rank) data matrices with low condition numbers. But when the condition numbers of data matrices are large to be amenable to full or approximately full ranks of data matrices, only polynomial speedup can be achieved.

关键词： quantum algorithm ridge regression regularization hyperparameter parallel Hamiltonian simulation quantum K-fold cross validation

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An Exact quantum algorithm for the 2-Junta Problem

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INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS 2021年第1期60卷 80-91页

作者： Chen, Chien-Yuan Natl Univ Kaohsiung Dept Comp Sci & Informat Engn 700 Kaohsiung Univ Rd Kaohsiung Taiwan

This paper proposes a novel quantum learning algorithm based on Bernstein and Vazirani's quantum circuit to find the dependent variables of the 2-junta problem. Typically, for a given Boolean function f : {0, 1}(n) -> {0, 1} that depends on only 2 out of n variables, the dependent variables are obtained by evaluating the function 4n times in the worst-case. However, the proposed quantum algorithm only requires O(log(2)n) function operations in the worst-case. Moreover, the algorithm requires an average of 5.3 function operations at the most when n >= 8.

关键词： quantum algorithm quantum learning algorithm The Bernstein-Vazirani algorithm

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A strategy for quantum algorithm design assisted by machine learning

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NEW JOURNAL OF PHYSICS 2014年第7期16卷 1-14页

作者： Bang, Jeongho Ryu, Junghee Yoo, Seokwon Pawlowski, Marcin Lee, Jinhyoung Seoul Natl Univ Ctr Macroscop Quantum Control Seoul 151747 South Korea Seoul Natl Univ Dept Phys & Astron Seoul 151747 South Korea Hanyang Univ Dept Phys Seoul 133791 South Korea Univ Gdansk Inst Theoret Phys & Astrophys PL-80952 Gdansk Poland Univ Bristol Dept Math Bristol BS8 Avon England

We propose a method for quantum algorithm design assisted by machine learning. The method uses a quantum-classical hybrid simulator, where a 'quantum student' is being taught by a 'classical teacher'. In other words, in our method, the learning system is supposed to evolve into a quantum algorithm for a given problem, assisted by a classical main-feedback system. Our method is applicable for designing quantum oracle-based algorithms. We chose, as a case study, an oracle decision problem, called a Deutsch-Jozsa problem. We showed by using Monte Carlo simulations that our simulator can faithfully learn a quantum algorithm for solving the problem for a given oracle. Remarkably, the learning time is proportional to the square root of the total number of parameters, rather than showing the exponential dependence found in the classical machine learning-based method.

关键词： quantum learning quantum automatic control quantum algorithm

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A quantum algorithm for System Specifications Verification

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IEEE INTERNET OF THINGS JOURNAL 2024年第14期11卷 24775-24794页

作者： Zidan, Mohammed Eisa, Ahmed M. Qasymeh, Montasir Shoman, Mahmoud A. Ismail South Valley Univ Fac Comp & Artificial Intelligence Dept Artificial Intelligence Hurghada 1950003 Egypt Egyptian E Learning Univ Fac Comp & Informat Technol Giza 12611 Egypt Abu Dhabi Univ Elect & Comp Engn Dept Abu Dhabi U Arab Emirates Cairo Univ Fac Comp & Informat Cairo 12613 Egypt

Verifying system specifications is a crucial requirement in science and technology, as it improves accuracy and efficiency in systems engineering. However, traditional computers are time-consuming and cannot efficiently check the consistency of system specifications for classical and quantum systems due to exponential complexity costs. Currently, there is no quantum approach available for verifying consistency in quantum systems. This article introduces a novel quantum algorithm that utilizes the virtual degree of entanglement to efficiently check the consistency of unknown system specifications represented by m oracles. The proposed algorithm has a time complexity of O(m epsilon(-2)), which is independent of the input size n and depends only on a predetermined error margin epsilon, providing exponential speedup compared to the classical approach with a time complexity of O(m2(n)). Moreover, the proposed algorithm can handle weighted superposition input states, which are impossible for classical computers to handle. Thus, the algorithm has the potential to check the consistency of both quantum and classical system specifications. The proposed algorithm is realized experimentally using IBM's Aer simulator with an input size of n = 12. The experimental results closely aligned with the theoretical predictions, demonstrating the algorithm's efficiency in checking the consistency of system specifications. Specifically, the proposed algorithm achieved a quantum supremacy ratio of 300% to 3100% compared to the classical algorithm, highlighting its significant advantage in terms of time complexity. Additionally, the simulation results confirmed the efficiency of the proposed algorithm in checking the consistency of the quantum system's specifications.

关键词： quantum computing quantum entanglement Computers Qubit quantum algorithm quantum system Prediction algorithms Concurrence entanglement quantum algorithms quantum computing sensors systems engineering

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