Comparative Monte Carlo Efficiency by Monte Carlo Analysis

We propose a modified power method for computing the subdominant eigenvalue $\lambda_2$ of a matrix or continuous operator. Here we focus on defining simple Monte Carlo methods for its application. The methods presented use random walkers of mixed signs to represent the subdominant eigenfuction. Accordingly, the methods must cancel these signs properly in order to sample this eigenfunction faithfully. We present a simple procedure to solve this sign problem and then test our Monte Carlo methods by computing the $\lambda_2$ of various Markov chain transition matrices. We first computed ${\lambda_2}$ for several one and two dimensional Ising models, which have a discrete phase space, and compared the relative efficiencies of the Metropolis and heat-bath algorithms as a function of temperature and applied magnetic field. Next, we computed $\lambda_2$ for a model of an interacting gas trapped by a harmonic potential, which has a mutidimensional continuous phase space, and studied the efficiency of the Metropolis algorithm as a function of temperature and the maximum allowable step size $\Delta$. Based on the $\lambda_2$ criterion, we found for the Ising models that small lattices appear to give an adequate picture of comparative efficiency and that the heat-bath algorithm is more efficient than the Metropolis algorithm only at low temperatures where both algorithms are inefficient. For the harmonic trap problem, we found that the traditional rule-of-thumb of adjusting $\Delta$ so the Metropolis acceptance rate is around 50% range is often sub-optimal. In general, as a function of temperature or $\Delta$, $\lambda_2$ for this model displayed trends defining optimal efficiency that the acceptance ratio does not. The cases studied also suggested that Monte Carlo simulations for a continuum model are likely more efficient than those for a discretized version of the model.Comment: 23 pages, 8 figure

Introduction to Quantum Information Processing

As a result of the capabilities of quantum information, the science of quantum information processing is now a prospering, interdisciplinary field focused on better understanding the possibilities and limitations of the underlying theory, on developing new applications of quantum information and on physically realizing controllable quantum devices. The purpose of this primer is to provide an elementary introduction to quantum information processing, and then to briefly explain how we hope to exploit the advantages of quantum information. These two sections can be read independently. For reference, we have included a glossary of the main terms of quantum information.Comment: 48 pages, to appear in LA Science. Hyperlinked PDF at http://www.c3.lanl.gov/~knill/qip/prhtml/prpdf.pdf, HTML at http://www.c3.lanl.gov/~knill/qip/prhtm

An Infinite Swapping Approach to the Rare-Event Sampling Problem

We describe a new approach to the rare-event Monte Carlo sampling problem. This technique utilizes a symmetrization strategy to create probability distributions that are more highly connected and thus more easily sampled than their original, potentially sparse counterparts. After discussing the formal outline of the approach and devising techniques for its practical implementation, we illustrate the utility of the technique with a series of numerical applications to Lennard-Jones clusters of varying complexity and rare-event character.Comment: 24 pages, 16 figure

Magnetic impurity affected by spin-orbit coupling: Behavior near a topological phase transition

We investigate the effect of spin-orbit coupling on the behavior of a magnetic impurity at the edge of a zigzag graphene ribbon by means of quantum Monte Carlo simulations. A peculiar interplay of Kane-Mele type spin-orbit and impurity-host coupling is found to greatly affect properties for the local moment. The local characters of the impurity are mainly dominated by the local density of states at the edge, such as double occupancy, magnetic moment, and spin susceptibilities. The special helical nature of the topological insulator on the boundary is found to affect nonlocal quantities, such as the two-particle and spin-spin correlation functions linking electrons on the impurity with those in the conduction band; in particular, due to the spin-orbit coupling, the symmetry of the spin rotation in the Kondo cloud around the impurity is partly broken.Peer reviewe