Physics
http://hdl.handle.net/1903/2269
2020-02-26T23:04:55ZQUANTUM ALGORITHMS FOR DIFFERENTIAL EQUATIONS
http://hdl.handle.net/1903/25502
QUANTUM ALGORITHMS FOR DIFFERENTIAL EQUATIONS
Ostrander, Aaron Jacob
This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs).
Product formulas are used to simulate Hamiltonians which can be expressed as a sum of terms which can each be simulated individually. By simulating each of these terms in sequence, the net effect approximately simulates the total Hamiltonian.
We find that the error of product formulas can be improved by randomizing over the order in which the Hamiltonian terms are simulated. We prove that this approach is asymptotically better than ordinary product formulas and present numerical comparisons for small numbers of qubits.
The ODE algorithm applies to the initial value problem for time-independent first order linear ODEs. We approximate the propagator of the ODE by a truncated Taylor series, and we encode the initial value problem in a large linear system. We solve this linear system with a quantum linear system algorithm (QLSA) whose output we perform a post-selective measurement on. The resulting state encodes the solution to the initial value problem. We prove that our algorithm is asymptotically optimal with respect to several system parameters.
The PDE algorithms apply the finite difference method (FDM) to Poisson's equation, the wave equation, and the Klein-Gordon equation. We use high order FDM approximations of the Laplacian operator to develop linear systems for Poisson's equation in cubic volumes under periodic, Neumann, and Dirichlet boundary conditions. Using QLSAs, we output states encoding solutions to Poisson's equation. We prove that our algorithm is exponentially faster with respect to the spatial dimension than analogous classical algorithms.
We also consider how high order Laplacian approximations can be used for simulating the wave and Klein-Gordon equations. We consider under what conditions it suffices to use Hamiltonian simulation for time evolution, and we propose an algorithm for these cases that uses QLSAs for state preparation and post-processing.
2019-01-01T00:00:00ZMachine Learning Approaches for Data-Driven Analysis and Forecasting of High-Dimensional Chaotic Dynamical Systems
http://hdl.handle.net/1903/25469
Machine Learning Approaches for Data-Driven Analysis and Forecasting of High-Dimensional Chaotic Dynamical Systems
Pathak, Jaideep
We consider problems in the forecasting of large, complex, spatiotemporal chaotic systems and the possibility that machine learning might be a useful tool for significant improvement of such forecasts. Focusing on weather forecasting as perhaps the most important example of such systems, we note that physics-based weather models have substantial error due to various factors including imperfect modeling of subgrid-scale dynamics and incomplete knowledge of physical processes. In this thesis, we ask if machine learning can potentially correct for such knowledge deficits.
First, we demonstrate the effectiveness of using machine learning for model- free prediction of spatiotemporally chaotic systems of arbitrarily large spatial extent and attractor dimension purely from observations of the systemâ€™s past evolution. We present a parallel scheme with an example implementation based on the reservoir computing paradigm and demonstrate the scalability of our scheme using the Kuramoto-Sivashinsky equation as an example of a spatiotemporally chaotic system. We then demonstrate the use of machine learning for inferring fundamental properties of dynamical systems, namely the Lyapunov exponents, purely from observed data. We obtain results of unprecedented fidelity with our novel technique, making it possible to find the Lyapunov exponents of large systems where previously known techniques have failed.
Next, we propose a general method that combines a physics-informed knowledge-based model and a machine learning technique to build a hybrid forecasting scheme. We further extend our hybrid forecasting approach to the difficult case where only partial measurements of the state of the dynamical system are available. For this purpose, we propose a novel technique that combines machine learning with a data assimilation method called an Ensemble Transform Kalman Filter (ETKF).
2019-01-01T00:00:00ZElectronic and Magnetic Properties of MnP-Type Binary Compounds
http://hdl.handle.net/1903/25436
Electronic and Magnetic Properties of MnP-Type Binary Compounds
Campbell, Daniel James
The interactions between electrons, and the resulting impact on physical properties, are at the heart of present-day materials science. This thesis looks at this idea through the lens of several compounds from a single family: the MnP-type transition metal pnictides. FeAs and FeP show long range magnetic order with some similarities to the high temperature, unconventional iron-based superconductors. CoAs lies on the border of magnetism, with strong fluctuations but no stable ordered state. CoP, in contrast, shows no strong magnetic
fluctuations but serves as a useful baseline in determining the origin (from composition, structure, or magnetic order) of behavior in the other materials.
For this work, single crystals were grown with two different techniques: solvent flux and chemical vapor transport. In the case of FeAs the flux method resulted in the highest quality crystals yet produced. Extensive work was then performed on these samples at the University of Maryland and the National High Magnetic Field Laboratory. Quantum oscillations observed in high magnetic fields, in combination with density functional theory calculations, give insight into the Fermi surfaces of these materials. Large magnetoresistance in the phosphides, but not the arsenides, demonstrates differences in the choice of pnictogen atom that cannot be simply a product of electron count. Angle-dependent linear magnetoresistance in FeP is a sign of a possible Dirac dispersion and topological physics, as has been hinted at in other MnP-type materials. Ultimately, it is possible to examine results for all four compounds and draw conclusions on the role of each of the two elements in the formula, which can be extended to other members of this family.
2019-01-01T00:00:00ZNonequilibrium Dynamics in Open Quantum Systems
http://hdl.handle.net/1903/25432
Nonequilibrium Dynamics in Open Quantum Systems
Young, Jeremy
Due to the variety of tools available to control atomic, molecular, and optical (AMO) systems, they provide a versatile platform for studying many-body physics, quantum simulation, and quantum computation. Although extensive efforts are employed to reduce coupling between the system and the environment, the effects of the environment can never fully be avoided, so it is important to develop a comprehensive understanding of open quantum systems. The system-environment coupling often leads to loss via dissipation, which can be countered by a coherent drive. Open quantum systems subject to dissipation and drive are known as driven-dissipative systems, and they provide an excellent platform for studying many-body nonequilibrium physics.
The first part of this dissertation will focus on driven-dissipative phase transitions. Despite the nonequilibrium nature of these systems, the corresponding phase transitions tend to exhibit emergent equilibrium behavior. However, we will show that in the vicinity of a multicritical point where multiple phase transitions intersect, genuinely nonequilibrium criticality can emerge, even though the individual phase transitions on their own exhibit equilibrium criticality. These nonequilibrium multicritical points can exhibit a variety of exotic phenomena not possible for their equilibrium counterparts, including the emergence of complex critical exponents, which lead to discrete scale invariance and spiraling phase boundaries. Furthermore, the Liouvillian gap can take on complex values, and the fluctuation-dissipation theorem is violated, corresponding to an effective temperature which gets "hotter" and "hotter" at longer and longer wavelengths.
The second part of this dissertation will focus on Rydberg atoms. In particular, we study how the spontaneous generation of contaminant Rydberg states drastically modifies the behavior of a driven-dissipative Rydberg system due to the resultant dipole-dipole interactions. These interactions lead to a complicated competition of both blockade and anti-blockade effects, leading to strongly enhanced Rydberg populations for far-detuned drive and reduced Rydberg populations for resonant drive.
2019-01-01T00:00:00Z