Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we consider the application of quantum computers to scientific computing and combinatorial optimization. We study five problems. The first three deal with quantum algorithms for computational problems in science and engineering, including quantum simulation of physical systems. In particular, we study quantum algorithms for numerical computation, for the approximation of ground and excited state energies of the Schr\"odinger equation, and for Hamiltonian simulation with applications to physics and chemistry. The remaining two deal with quantum algorithms for approximate optimization. We study the performance of the quantum approximate optimization algorithm (QAOA), and show a generalization of QAOA, the $\textit{quantum}$ $\textit{alternating}$ $\textit{operator}$ $\textit{ansatz}$, particularly suitable for constrained optimization problems and low-resource implementations on near-term quantum devices.

We study quantum computing algorithms for solving certain constrained resource allocation problems we coin as Mission Covering Optimization (MCO). We compare formulations of constrained optimization problems using Quantum Annealing techniques and the Quantum Alternating Operator Ansatz (Hadfield et al. arXiv:1709.03489v2, a generalized algorithm of the Quantum Approximate Optimization Algorithm, Farhi et al. arXiv:1411.4028v1) on D-Wave and IBM machines respectively using the following metrics: cost, timing, constraints held, and qubits used. We provide results from two different MCO scenarios and analyze results.

As we begin to reach the limits of classical computing, quantum computing has emerged as a technology that has captured the imagination of the scientific world. While for many years, the ability to execute quantum algorithms was only a theoretical possibility, recent advances in hardware mean that quantum computing devices now exist that can carry out quantum computation on a limited scale. Thus, it is now a real possibility, and of central importance at this time, to assess the potential impact of quantum computers on real problems of interest. One of the earliest and most compelling applications for quantum computers is Feynman's idea of simulating quantum systems with many degrees of freedom. Such systems are found across chemistry, physics, and materials science. The particular way in which quantum computing extends classical computing means that one cannot expect arbitrary simulations to be sped up by a quantum computer, thus one must carefully identify areas where quantum advantage may be achieved. In this review, we briefly describe central problems in chemistry and materials science, in areas of electronic structure, quantum statistical mechanics, and quantum dynamics that are of potential interest for solution on a quantum computer. We then take a detailed snapshot of current progress in quantum algorithms for ground-state, dynamics, and thermal-state simulation and analyze their strengths and weaknesses for future developments.

The increasing complexity of real-world optimization problems highlights the importance of this research since classical algorithms are unable to provide efficient answers in these cases. Innovative methods for fast and scalable resolution of nonlinear optimization problems are required because these problems are prevalent in many fields. The potential for quantum computing to speed up optimization processes and overcome classical limitations is great, owing to its superposition principles and intrinsic parallelism. The integration of quantum algorithms (I-QA) into real-world applications, however, will not always be smooth sailing. There are significant challenges associated with preserving quantum coherence, correcting errors, and working within hardware limits. To enable the simultaneous exploration of solution spaces through quantum parallelism, this research proposes the Hybrid Quantum Gradient -Classical Approach (HQG-CA), which makes use of parameterized quantum circuits to represent probable solutions. Additionally, improves convergence rates through applying quantum gradient information to direct optimization in the quantum state space. Optimization of portfolios in finance, adjustment of model parameters in machine learning, and optimization of routes in logistics are a few examples of the many industries that find use for HQG-CA. These applications are explored in this abstract, which highlights the revolutionary potential of HQG-CA to solve optimization problems in the real world. The effectiveness of HQG-CA is assessed through a thorough simulation experiment. Performance measures such algorithmic speedup, solution accuracy, and scalability are discussed, which is based on extensive testing and comparison with classical alternatives. The present research provides a comprehensive evaluation of HQG-CA's potential for tackling nonlinear optimization problems.

AbstractDigital quantum computers provide a computational framework for solving the Schrödinger equation for a variety of many‐particle systems. Quantum computing algorithms for the quantum simulation of these systems have recently witnessed remarkable growth, notwithstanding the limitations of existing quantum hardware, especially as a tool for electronic structure computations in molecules. In this review, we provide a self‐contained introduction to emerging algorithms for the simulation of Hamiltonian dynamics and eigenstates, with emphasis on their applications to the electronic structure in molecular systems. Theoretical foundations and implementation details of the method are discussed, and their strengths, limitations, and recent advances are presented.This article is categorized under: Quantum Computing > Algorithms Electronic Structure Theory > Ab Initio Electronic Structure Methods Quantum Computing > Theory Development

This research investigates the potential of quantum computing in production planning and addresses the limitations of conventional computing approaches. Traditional methods have been partially effective, but they struggle to solve complex optimization problems, accurately predict demand, and manage supply chains efficiently. The unique computational capabilities of quantum computing offer promising solutions to surmount these obstacles and revolutionize production planning processes. This study seeks to bridge the gap between quantum computing and production planning by analyzing the benefits, limitations, and challenges of its applicability in this field. It proposes customized algorithms and methodologies for leveraging quantum computation to enhance production planning efficiency, cost reduction, and decision-making processes. The research demonstrates the potential of quantum algorithms to minimize total production costs while appeasing demand and resource constraints through a numerical example and mathematical formulation. The results emphasize the advantages of quantum computing in terms of cost reduction, enhanced efficiency, and scalability. Comparisons with conventional methods illuminate the benefits and drawbacks of quantum computing in production planning. This research contributes to the development of novel strategies to improve production planning efficiency, lower costs, and enhance decision-making processes, allowing organizations to leverage quantum computing for optimized production operations

Observations in quantum mechanics are subject to complex restrictions arising from the principle of energy conservation. Determining such restrictions, however, has been so far an elusive task, and only partial results are known. In this Letter, we discuss how constraints on the energy spectrum of a measurement device translate into limitations on the measurements which we can effect on a target system with a nonconstant energy operator. We provide efficient algorithms to characterize such limitations and, in case the target is a two-level quantum system, we quantify them exactly. Our Letter, thus, identifies the boundaries between what is possible or impossible to measure, i.e., between what we can see or not, when energy conservation is at stake.

With the development of science and technology, it is difficult for traditional computers to solve cutting-edge problems due to the lack of computing power, and the importance of quantum computers is increasing day by day. This article starts with the simple principle of quantum computing, introduces the most advanced quantum computing instruments and quantum computing algorithms, and points out the application prospects in medicine, chemistry and other fields. This paper explains the basic principles of quantum computing algorithms, their efficiency over traditional algorithms, and focuses on the Shor algorithm and its variations. In terms of applications, the quantum computer Zu Chongzhi and its contribution to the sampling problem of quantum random circuits are introduced. This paper makes a certain analysis of the limitations of quantum computing, and gives the future development goals in a targeted manner. Besides, we summarize popular quantum computing algorithms and applications, and make contributions to the promotion and development of quantum computing. Overall, these results shed light on guiding further exploration of how to improve the computational efficiency of quantum computers.

Quantum computers are capable of ultra fast computation in the fields where classical computers fail. Even though quantum computers are nowhere near commercialization, many researchers have developed quantum algorithms in fields such as modern encryption and molecular simulation, which, in theory, are exponentially faster than their classical counterparts. In this case, this paper will discuss the advantages of quantum computers over classical computers in those fields by examining and analyzing the various quantum algorithms. To be specific, the develop routine as well as detail examples will be exhibited to illustrate the differences and preferences. In addition, this study will also fully aware of the challenges that quantum computing researchers are facing. On this basis, possible limitations of quantum computers are also presented. The aim is to promote interest in quantum computing by introducing their supremacy in modern encryption and biological science. These results shed light on guiding further exploration of quantum computing algorithms.

Since the mid-1990s theoretical quadratic exponential and polynomial Quantum Computing (QC) speedup algorithms have been discussed. Recently the advent of relativistic information processing (RIP) introducing a relativistic qubit (r-qubit) with additional degrees of freedom beyond the current Hilbert space Bloch 2-sphere qubit formalism extended theory has appeared. In this work a penultimate form of QC speedup – Instantaneous Quantum Computing Algorithms (IQCA) is proposed. Discussion exists on passing beyond the quantum limits of locality and unitarity heretofore restricting the evolution of quantum systems to the standard Copenhagen Interpretation. In that respect as introduced in prior work an ontological-phase topological QC avails itself of extended modeling. As well-known by EPR experiments instantaneous connectivity exists inherently in the nonlocal arena. As our starting point we utilize Bohm’s super-implicate order where inside a wave packet a super-quantum potential introduces nonlocal connectivity. Additionally EPR experiments entangle simultaneously emitted photon pairs by parametric down-conversion. Operating an IQCA requires a parametric up-conversion cycle an M-Theoretic Unified Field Mechanical (MUFM) set of topological transformations beyond the current Galilean Lorentz-Poincairé transforms of the standard model (SM). Yang-Mills Kaluza-Klein (YM-KK) correspondence is shown to provide a path beyond the semi-quantum limit to realize the local-nonlocal duality required to implement IQCA.