Unitary transformations depending on a small parameter

We formulate a unitary perturbation theory for quantum mechanics inspired by the LieDeprit formulation of canonical transformations. The original Hamiltonian is converted into a solvable one by a transformation obtained through a Magnus expansion. This ensures unitarity at every order in a small parameter. A comparison with the standard perturbation theory is provided. We work out the scheme up to order ten with some simple examples

A Brief Review on Mathematical Tools Applicable to Quantum Computing for Modelling and Optimization Problems in Engineering

Since its emergence, quantum computing has enabled a wide spectrum of new possibilities and advantages, including its efficiency in accelerating computational processes exponentially. This has directed much research towards completely novel ways of solving a wide variety of engineering problems, especially through describing quantum versions of many mathematical tools such as Fourier and Laplace transforms, differential equations, systems of linear equations, and optimization techniques, among others. Exploration and development in this direction will revolutionize the world of engineering. In this manuscript, we review the state of the art of these emerging techniques from the perspective of quantum computer development and performance optimization, with a focus on the most common mathematical tools that support engineering applications. This review focuses on the application of these mathematical tools to quantum computer development and performance improvement/optimization. It also identifies the challenges and limitations related to the exploitation of quantum computing and outlines the main opportunities for future contributions. This review aims at offering a valuable reference for researchers in fields of engineering that are likely to turn to quantum computing for solutions. Doi: 10.28991/ESJ-2023-07-01-020 Full Text: PD

Non-adiabatic dynamics of electrons and atoms under non-equilibrium conditions

An approach to non-adiabatic dynamics of atoms in molecular and condensed matter systems under general non-equilibrium conditions is proposed. In this method interaction between nuclei and electrons is considered explicitly up to the second order in atomic displacements defined with respect to the mean atomic trajectory. This method enables one to consider movement of atoms beyond their simple vibrations. Both electrons and nuclei are treated fully quantum-mechanically using a combination of path integrals applied to nuclei and non-equilibrium Green's functions (NEGF) to elections. Our method is partition-less: initially, the entire system is coupled and assumed to be at thermal equilibrium. Then, the exact application of the Hubbard-Stratanovich transformation in mixed real and imaginary times enables us to obtain, without doing any additional approximations, an exact expression for the reduced density matrix for nuclei and hence an effective quantum Liouville equation for them, both containing Gaussian noises. It is shown that the time evolution of the expectation values for atomic positions is described by an infinite hierarchy of stochastic differential equations for atomic positions and momenta and their various fluctuations. The actual dynamics is obtained by sampling all stochastic trajectories. It is expected that applications of the method may include photo-induced chemical reactions (e.g. dissociation), electromigration, atomic manipulation in scanning tunneling microscopy, to name just a few.Comment: 30 pages, 1 fugur

Quantum Algorithm Implementations for Beginners

As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims to explain the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM's quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations

Explicit Determination of Robin Parameters in Optimized Schwarz Waveform Relaxation Methods for Schrödinger Equations Based on Pseudodifferential Operators

International audienceThe Optimized Schwarz Waveform Relaxation algorithm, a domain decomposition method based on Robin transmission condition, is becoming a popular computational method for solving evolution partial differential equations in parallel. Along with well-posedness, it offers a good balance between convergence rate, computational complexity and simplicity of the implementation. The fundamental question is the selection of the Robin parameter to optimize the convergence of the algorithm. In this paper, we propose an approach to explicitly estimate the Robin parameter which is based on the approximation of the transmission operators at the subdomain interfaces, for the linear/nonlinear SchrödingerSchr¨Schrödinger equation. Some illustrating numerical experiments are proposed for the one-and two-dimensional problems

From low dimensions to full configuration space: Generalising models for nonadiabatic molecular dynamics

This thesis aims to bridge the development of nonadiabatic dynamics methods and their application for studies of real molecular systems. First, this work explores fundamental concepts of photochemistry by investigating two different pictures, arising from the Born-Oppenheimer and the exact factorisation representation. Based on a simplistic model, a photochemical experiment from the excitation up to the formation of photoproducts is simulated. This study then compares the Born-Oppenheimer and exact factorisation representations of the processes. Subsequently, the influence of the Born-Oppenheimer picture for approximate nonadiabatic dynamics is investigated on two-dimensional model systems around conical intersections. The effects of neglected couplings and geometric phase are evaluated for ab initio multiple spawning (AIMS), a method for nonadiabatic molecular dynamics based on classically moving Gaussians. Afterwards, this work introduces a standardised test set of molecules to connect between tests of newly developed nonadiabatic dynamics methods on one-dimensional model systems and their intended application to full-dimensional molecules. Inspired by the widely used one-dimensional Tully models, three molecules are selected to form the molecular Tully models, which undergo similar photophysical processes, but in a high-dimensional space. In addition, the recently proposed stochastic-selection AIMS framework is also tested on two molecules undergoing ring-opening reactions to explore the strengths and limitations of the method. Finally, a direct comparison between experimental and computational results is presented. The photochemistry of 2(5H)-thiophenone is probed during and after the initial ring opening using time-resolved photoelectron spectroscopy. Static and dynamic calculations unravel the photoprocesses and identify a variety of photoproducts. Using the computational results, the experimental signal can be translated to insights into the ongoing photochemistry. Overall, this thesis aims to bring models in nonadiabatic dynamics in a real-world context. This work contributes to facilitating the transfer of new nonadiabatic dynamics methods towards the study of molecules in their full dimensionality

Quantum Optimization From a Computer Science Perspective

Optimization problems are ubiquitous in but not limited to the sciences, engineering, and applied mathematics. Examples range from the fastest way USPS can route packages through a delivery network to the best way an autonomous vehicle can navigate through a given traffic environment. Classical optimization algorithms dominate the way we solve these problems. However, with the rapid advance of quantum computers, we are looking at novel, quantum-inspired ways of solving old problems to achieve some speedup over classical algorithms. Specifically, we are looking at the Quantum Approximate Optimization Algorithm (QAOA). We show that QAOA provides a tunable, optimization algorithm whose quantum circuit grows linearly with the number of constraints for MAXSAT, an NP-complete problem

Differential galois theory for some spectral problems

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 14-12-201

Simulating Quantum Computations on Classical Machines: A Survey

We present a comprehensive study of quantum simulation methods and quantum simulators for classical computers. We first study an exhaustive set of 150+ simulators and quantum libraries. Then, we short-list the simulators that are actively maintained and enable simulation of quantum algorithms for more than 10 qubits. As a result, we realize that most efficient and actively maintained simulators have been developed after 2010. We also provide a taxonomy of the most important simulation methods, namely Schrodinger-based, Feynman path integrals, Heisenberg-based, and hybrid methods. We observe that most simulators fall in the category of Schrodinger-based approaches. However, there are a few efficient simulators belonging to other categories. We also make note that quantum frameworks form their own class of software tools that provide more flexibility for algorithm designers with a choice of simulators/simulation method. Another contribution of this study includes the use and classification of optimization methods used in a variety of simulators. We observe that some state-of-the-art simulators utilize a combination of software and hardware optimization techniques to scale up the simulation of quantum circuits. We summarize this study by providing a roadmap for future research that can further enhance the use of quantum simulators in education and research.Comment: 20 pages, 8 figures, under revie