Chapter 22 Quantum Dynamical Methods

Luke Edwards

Spin dynamics simulations are used to gain insight into important magnetic resonance experiments in the fields of chemistry, biochemistry, and physics. Presented in this thesis are investigations into how to accelerate these simulations by making them more efficient. Chapter 1 gives a brief introduction to the methods of spin dynamics simulation used in the rest of the thesis. The `exponential scaling problem' that formally limits the size of spin system that can be simulated is described. Chapter 2 provides a summary of methods that have been developed to overcome the exponential scaling problem in liquid state magnetic resonance. The possibility of utilizing the multiple processors prevalent in modern computers to accelerate spin dynamics simulations provides the impetus for the investigation found in Chapter 3. A number of different methods of parallelization leading to acceleration of spin dynamics simulations are derived and discussed. It is often the case that the parameters defining a spin system are time-dependent. This complicates the simulation of the spin dynamics of the system. Chapter 4 presents a method of simplifying such simulations by mapping the spin dynamics into a larger state space. This method is applied to simulations incorporating mechanical spinning of the sample with powder averaging. In Chapter 5, implementations of several magnetic resonance experiments are detailed. In so doing, use of techniques developed in Chapters 2 and 3 are exemplified. Further, specific details of these experiments are utilized to increase the efficiency of their simulation.

Yuri A. Rylov

Kazuma Nagao

Joseph Kleinhenz

In this thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.

Riccardo Manenti, Mário Motta

Abstract This chapter presents some quantum algorithms to simulate the dynamics of a quantum system, one of the most important applications of a quantum computer. We introduce the conditions for efficient simulation of Hamiltonian dynamics. We then present some Trotterization methods, in which the time evolution operator is approximated as a product of terms, each of which can be implemented by a quantum circuit at polynomial cost. We show high-order decompositions and illustrate the use of a Trotterization method to simulate sparse Hamiltonians. We discuss the Taylor series method and the qubitization method. The latter achieves optimal cost in simulation time and target accuracy. We conclude the chapter with a proof that simulating Hamiltonian dynamics is BQP-complete

Other Contributor-Ludwig Streit