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In this paper, we present a new and algebraically simple algorithm that prepares any pure three-qubit state using only local gates and at most three controlled-Z gates. In 2008, Znidaric et al. already showed that three is the optimal number of controlled-Z gates required for the preparation of three-qubit states if in addition to those gates only local gates are used. If we restrict the local gates to Ry(휃) gates, our algorithm provides a way to prepare any pure three-qubit state |ϕ〉 with real amplitudes using at most four controlled-Z gates when its hyperdeterminant is negative and three or fewer controlled-Z gates otherwise. We conjecture the existence of three-qubit states with real amplitudes that cannot be prepared using only Ry(휃) gates and less than four controlled-Z gates. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Private set union (PSU) allows several parties to obtain the union of their private sets without disclosing each party's private information. Existing PSU protocols often have polynomial complexity for the complete set size or complicated process. In this paper, a quantum multi-party PSU protocol based on least common multiple (LCM) and Shor's algorithm is proposed, which enables the union of multiple sets to be computed all at once. In order to increase the one-time success probability of the protocol, we first improved Shor's period-finding algorithm, which is used in LCM computation and integer factoring. Each party's private set is encoded into an integer obtained by multiplying several prime numbers, thus the PSU problem is transformed into an LCM problem. The LCM of these integers is computed by using the improved Shor's period-finding algorithm, and then factored to derived the union set. We prove the correctness of the proposed protocol, and its unconditional security against semi-honest attacks. Complexity analysis shows that our protocol has logarithmic complexity for the complete set size. [ABSTRACT FROM AUTHOR]

In this paper, a new algorithm for solving the Navier–Stokes equations (NSE) on a quantum device is presented. For the fluid flow equations, the stream function-vorticity formulation is adopted, while the lattice Boltzmann method (LBM) is utilized for solving the corresponding system of equations numerically for one time step. Following the nature of the lattice Boltzmann method, the proposed quantum algorithm consists of five major sections: initialization, collision, propagation, boundary condition implementation and calculation of macroscopic quantities. The collision and boundary condition step is quantumly implemented by applying the standard-form encoding approach, while the quantum walk procedure is applied for the propagation step. The algorithm is implemented by using IBM's quantum computing software development framework Qiskit, while for the verification purposes, two-dimensional (2D) cavity flow is simulated and compared with classical code. [ABSTRACT FROM AUTHOR]