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Toffoli gate count optimized space-efficient quantum circuit for binary field multiplication

quantum information processing 2024年第10期23卷 1-32页

作者： Kim, Sunyeop Kim, Insung Kim, Seonggyeom Hong, Seokhie Korea Univ Inst Cyber Secur & Privacy ICSP Seoul 02855 South Korea Samsung Elect Syst LSI Div Hwaseong 18448 South Korea

Shor's algorithm solves the elliptic curve discrete logarithm problem (ECDLP) in polynomial time. To optimize Shor's algorithm for binary elliptic curves, reducing the cost of binary field multiplication is essential because it is the most cost-critical arithmetic operation. In this paper, we propose Toffoli gate count-optimized, space-efficient (i.e., no ancilla qubits are used) quantum circuits for binary field ((F2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {F}_{2<^>{n}})$$\end{document}) multiplication. To achieve this, we leverage the Karatsuba-like formulae and demonstrate that its application can be implemented without the need for ancillary qubits. We optimize these circuits in terms of CNOT gate count and depth. Building upon the Karatsuba-like formulae, we develop a space-efficient CRT-based multiplication technique utilizing two types of out-of-place multiplication algorithms to reduce the CNOT gate count. Our quantum circuits exhibit an extremely low Toffoli gate count of O(n2log2 & lowast;n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n2<^>{\log {2}<^>{*}n})$$\end{document}, where log2 & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log _{2}<^>{*}$$\end{document} represents the iterative logarithmic function that grows very slowly. When compared to recent Karatsuba-based space-efficient quantum circuit, our approach requires only (10-25 %) of the Toffoli gate count and Toffoli depth for c

关键词： quantum computing Qubit Algorithm quantum circuits

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Logarithmic-Regret quantum Learning algorithms for Zero-Sum Games 37

Logarithmic-Regret Quantum Learning Algorithms for Zero-Sum ...

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37th Conference on Neural information processing Systems (NeurIPS)

作者： Gao, Minbo Ji, Zhengfeng Li, Tongyang Wang, Qisheng Chinese Acad Sci State Key Lab Comp Sci Inst Software Beijing Peoples R China Univ Chinese Acad Sci Beijing Peoples R China Tsinghua Univ Dept Comp Sci & Technol Beijing Peoples R China Peking Univ Ctr Frontiers Comp Studies Beijing Peoples R China Peking Univ Sch Comp Sci Beijing Peoples R China Nagoya Univ Grad Sch Math Nagoya Japan

ISBN: (纸本)9781713899921

We propose the first online quantum algorithm for solving zero-sum games with (O) over tilde (1) regret under the game setting.(1) Moreover, our quantum algorithm computes an epsilon-approximate Nash equilibrium of an m x n matrix zero-sum game in quantum time (O) over tilde(root m + n/epsilon(2.5)). Our algorithm uses standard quantum inputs and generates classical outputs with succinct descriptions, facilitating end-to-end applications. Technically, our online quantum algorithm "quantizes" classical algorithms based on the optimistic multiplicative weight update method. At the heart of our algorithm is a fast quantum multi-sampling procedure for the Gibbs sampling problem, which may be of independent interest.

关键词： Game theory

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Variational quantum algorithms for machine learning: Theory and Applications

Variational quantum algorithms for machine learning: Theory ...

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作者： Stefano Mangini University of Pavia

学位级别：博士

The topic of the present Ph. D. thesis is quantum Computation and information processing, with a specific focus on the relatively new fields of Variational quantum algorithms and quantum Machine Learning. These disciplines not only offer a unique perspective on studying quantum information processing tasks, but also have the potential to provide a useful computational quantum advantage even with the currently available first-generation small and noisy quantum computing devices. Variational quantum algorithms involve a hybrid quantum-classical computational loop, where a quantum computer is used only for some specific ideally quantum-native subroutines, and a classical computer runs an optimisation procedure on variational parameters to minimise a cost function whose minimum corresponds to the solution to the problem to be solved. This framework shares the same foundational idea as state-of-the-art Deep Learning models, where complex parametric models are tuned via optimisation methods to solve various tasks. The intersection of quantum computing and machine learning has led to the development of quantum machine learning, an interdisciplinary area that explores the benefits of combining quantum computation and artificial intelligence. This thesis provides a comprehensive analysis of the state of the art of the field, including numerous original contributions, from the study of quantum models for artificial neurons to the characterisation of entanglement created in quantum architectures for neural networks, up to discussing the effect of measurement noise on a more quantum information perspective. The first chapters are devoted to a careful review of the basics of quantum computing and a thorough discussion of variational quantum algorithms. Then the discussion is moved to quantum machine learning, where an introduction to the elements of machine learning and statistical learning theory is followed by a review of the most common quantum counterparts of machine learnin

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Approximate real-time evolution operator for potential with one ancillary qubit and application to first-quantized Hamiltonian simulation

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quantum information processing 2025年第3期24卷 1-51页

作者： Huang, Xinchi Kosugi, Taichi Nishi, Hirofumi Matsushita, Yu-ichiro Univ Tokyo Dept Phys 7-3-1 HongoBunkyo Ku Tokyo 1138654 Japan Quemix Inc 2-11-2 Nihombashi Chuo ku Tokyo 1030027 Japan Natl Inst Quantum Sci & Technol QST Quantum Mat & Applicat Res Ctr 2-12-1 OokayamaMeguro ku Tokyo 1528550 Japan Tokyo Inst Technol Inst Innovat Res Lab Mat & Struct Yokohama 2268503 Japan

In many quantum algorithms, including Hamiltonian simulation, efficient quantum circuit implementation of diagonal unitary matrices is an important issue. For small unitary diagonal matrices, a method based on Walsh operators is known and allows an exact implementation. Whereas, as the matrix size increases, the required resources increase linearly regarding the matrix size, so an efficient approximate implementation is indispensable. In this study, we specify the approximation using piecewise polynomials when the diagonal unitary matrix is generated by a known underlying function. It accelerates the implementation by an exponential factor compared to the exact one. In more detail, we modify a previous method, which we call PPP (phase gate for piecewise-defined polynomial), and propose a novel one called LIU (linearly interpolated unitary diagonal matrix). By introducing a coarse-graining parameter, calculated from the underlying function and the desired error bound, we evaluate the explicit gate counts for different methods as functions of some norms of the given function, the grid parameter, and the allowable error. It helps us to figure out the efficient quantum circuits in practical settings of different grid parameters and error bounds, in addition to an asymptotic speedup when the grid parameter goes to infinity. As an application, we apply our method to the first-quantized Hamiltonian simulation and estimate the quantum resources (gate count and ancillary qubits). It reveals that the error coming from the approximation of the potential function is not negligible compared to the error from the Trotter-Suzuki formula.

关键词： quantum circuit Diagonal unitary matrix First-quantized Hamiltonian simulation Spline interpolation

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Towards Atomic MIMO Receivers

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 2025年第3期43卷 659-673页

作者： Cui, Mingyao Zeng, Qunsong Huang, Kaibin Univ Hong Kong Dept Elect & Elect Engn Hong Kong Peoples R China

The advancement of Rydberg atoms in quantum information technology is driving a paradigm shift from classical radio-frequency (RF) receivers to Rydberg atomic receivers. Capitalizing on the extreme sensitivity of Rydberg atoms to external electromagnetic fields, Rydberg atomic receivers are capable of realizing more precise radio-wave measurements than RF receivers to support high-performance wireless communication and sensing. Although the atomic receiver is developing rapidly in quantum-physics domain, its integration with wireless communications is at a nascent stage. In particular, systematic methods to enhance communication performance through this integration are yet to be discovered. Motivated by this observation, we propose in this paper to incorporate Rydberg atomic receivers into multiple-input-multiple-output (MIMO) communication, a prominent 5G technology, as the first attempt on implementing atomic MIMO receivers. To begin with, we provide a comprehensive introduction on the principles of Rydberg atomic receivers and build on them to design the atomic MIMO receivers. Our findings reveal that signal detection of atomic MIMO receivers corresponds to a non-linear biased phase retrieval (PR) problem, as opposed to the linear Gaussian model adopted in classical MIMO systems. Then, to recover signals from this non-linear model, we modify the Gerchberg-Saxton (GS) algorithm, a typical PR solver, into a biased GS algorithm to solve the biased PR problem. Moreover, we propose a novel Expectation-Maximization GS (EM-GS) algorithm to cope with the unique Rician distribution of the biased PR model. Our EM-GS algorithm introduces a high-pass filter constructed by the ratio of Bessel functions into the iteration procedure of GS, thereby improving the detection accuracy without sacrificing the computational efficiency. Finally, the effectiveness of the devised algorithms and the feasibility of atomic MIMO receivers are demonstrated by theoretical analysis and numerica

关键词： Receivers Atomic measurements Rydberg atoms Radio frequency Sensors Electromagnetic fields quantum sensing Energy states Symbols Signal processing algorithms Rydberg atomic receivers multiple-input-multiple-output (MIMO) quantum sensing signal detection

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Analysis of a Programmable quantum Annealer as a Random Number Generator

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IEEE TRANSACTIONS ON information FORENSICS AND SECURITY 2024年 19卷 3636-3643页

作者： Pelofske, Elijah Los Alamos Natl Lab CCS 3 Informat Sci Los Alamos NM 87545 USA

quantum devices offer a highly useful function - that is generating random numbers in a non-deterministic way since the measurement of a quantum state is not deterministic. This means that quantum devices can be constructed that generate qubits in a uniform superposition and then measure the state of those qubits. If the preparation of the qubits in a uniform superposition is unbiased, then quantum computers can be used to create high entropy, secure random numbers. Typically, preparing and measuring such quantum systems requires more time compared to classical pseudo random number generators (PRNGs) which are inherently deterministic algorithms. Therefore, the typical use of quantum random number generators (QRNGs) is to provide high entropy secure seeds for PRNGs. quantum annealing (QA) is a type of analog quantum computation that is a relaxed form of adiabatic quantum computation and uses quantum fluctuations in order to search for ground state solutions of a programmable Ising model. Here we present extensive experimental random number results from a D-Wave 2000Q quantum annealer, totaling over 20 billion bits of QA measurements, which is significantly larger than previous D-Wave QA random number generator studies. Current quantum annealers are susceptible to noise from environmental sources and calibration errors, and are not in general unbiased samplers. Therefore, it is of interest to quantify whether noisy quantum annealers can effectively function as an unbiased QRNG. The amount of data that was collected from the quantum annealer allows a comprehensive analysis of the random bits to be performed using the NIST SP 800-22 Rev 1a testsuite, as well as min-entropy estimates from NIST SP 800-90B. The randomness tests show that the generated random bits from the D-Wave 2000Q are biased, and not unpredictable random bit sequences. With no server-side sampling post-processing, the 1 microsecond annealing time measurements had a min-entropy of 0.824.

关键词： Annealing quantum annealing Generators Qubit Hardware Computational modeling Servers quantum computing random number generator Ising model min-entropy chimera graph transverse-field Ising model D-Wave 2000Q

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Resource analysis of quantum computing with noisy qubits for Shor's factoring algorithms

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quantum information processing 2022年第2期21卷 1-19页

作者： Ha, Jinyoung Lee, Jonghyun Heo, Jun Korea Univ Seoul South Korea

We decompose two implementations of Shor's algorithm for prime factorization into universal gate units at the logical level and predict the number of physical qubits and execution time when surface codes are used. Logical qubit encoding using a rotated surface code and logical qubits with all-to-all connectivity are assumed. We express the number of physical qubits and execution time in terms of the bit length of the number to be factorized and error rate of the physical quantum gate. We confirm the relationship between the number of qubits and the execution time by analyzing two algorithms using various bit lengths and physical gate error rates .

关键词： quantum algorithm quantum resource analysis quantum computation quantum gates quantum error correction

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Data rotation and its influence on quantum encoding

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quantum information processing 2023年第1期22卷 1-15页

作者： Sierra-Sosa, Daniel Pal, Soham Telahun, Michael Hood Coll Dept Comp Sci & Informat Technol Frederick MD 21701 USA Iowa State Univ Dept Phys & Astron Ames IA USA Univ Louisville Dept Comp Sci & Engn Louisville KY USA

Parametric quantum machine learning (QML) has been vastly studied over the last several years. These algorithms rely on hybrid implementations, where quantum methods define the models, and the parameters are update on classical devices. The encoding of classical data into quantum states within the Hilbert space is fundamental to training these hybrid models;this can be achieved in a number of ways. In this work, we focus on two of these methods, amplitude encoding and encoding via a second-order Pauli feature map. We compared their performances across two near-term QML models, quantum support vector classifier and variational quantum classifier. We found that amplitude encoding is significantly resilient to classical transformations of data. This work additionally introduces the concept of a rotation, applied to classical data as a preprocessing step. In our results, we observe that other encoding methods can significantly benefit from certain Cartesian rotations of the data. We expand this rotation to a larger n - D dataset and show the method's performance.

关键词： quantum computing Machine learning Encoding Preprocessing Distribution Rotation

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A high-capacity and robust steganography algorithm for quantum images

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CHINESE JOURNAL OF PHYSICS 2023年 85卷 89-103页

作者： Khan, Mubashar Rasheed, Amer Lahore Univ Management Sci Sch Sci & Engn Dept Math Opposite Sect U Opposite Sect U DHA Cantt 54792 Pakistan

The improved flexible representation of a quantum image (IFRQI) employs an effective method to encode grayscale image information in a normalized quantum state vector, which allows for accurate retrieval of image information through projective measurements. In this paper, we propose a high-capacity and robust steganography algorithm based on the IFRQI model. We embed the grayscale information of a 2(n) x2(n) sized secret image in a 2(n+1) x2(n+1) sized cover image using controlled rotations. The secret image is divided into four image planes, each with a bit depth of 2. For each image plane, an array of angle values encoding the 2-bit color information is prepared. The encoded information is then embedded in the IFRQI state of the cover image using controlled rotations determined by the key K, which is only known to the operator. The secret image extraction algorithm is the inverse process of the embedding algorithm, which requires the inverse key K '. Analyses of the embedding capacity, time complexity, and visual effects reveal that the proposed steganography algorithm is comparable with some state-of-the-art algorithms.

关键词： quantum image processing quantum image steganography Embedding capacity quantum image scrambling Erbium-doped fiber laser chaotic system

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Three quantum neural network architectures for multi-class image classification 3

Three quantum neural network architectures for multi-class i...

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3rd International Conference on algorithms, Network, and Communication Technology, ICANCT 2024

作者： Qu, Guangkai Yu, Xianglong Wang, Zhimin Gu, Yongjian Faculty of Information Science and Engineering Ocean University of China Qingdao266100 China

ISBN: (数字)9781510688902

ISBN: (纸本)9781510688896

quantum neural networks (QNNs) integrate the advantages of quantum information with the formidable capabilities of classical neural networks, exhibiting notable potential in image classification tasks. However, the performance of current QNNs in processing multi-class image classification tasks still leaves room for improvement. In this paper, we propose three QNN architectures to enhance the accuracy of multi-class image classification by utilizing techniques of overparameterization, dense pooling, and local feature extraction. These networks are capable of mapping conventional data to the amplitudes of quantum states via amplitude encoding, eschewing the need for traditional dimensionality reduction techniques. Following processing with parameterized quantum circuits (PQCs), we conduct quantum measurements to extract feature information. We validate the proposed architectures using the MNIST dataset for eight-class image classification problems. The experimental results unequivocally demonstrate that the three techniques we employed significantly enhance the accuracy of QNNs in multi-class image classification tasks. © 2025 SPIE.

关键词： quantum electronics

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